openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Transport /ChoiceContextEnumeration.lean
| import OAI.Combinatorics.ContingencyTables.Transport.ContextWeightedEnumeration | |
| import OAI.Combinatorics.ContingencyTables.Sampling.PhysicalPrefixEnergies | |
| import OAI.Combinatorics.ContingencyTables.Transport.FiniteChoiceFibres | |
| namespace OAI | |
| /-! # Literal finite choice fibres equal the transport child sums -/ | |
| namespace ContingencyTables.SmallContextEnumeration | |
| open scoped BigOperators Classical | |
| open SmallContextCoordinates SmallContextFibres PhysicalCompletionFibres | |
| open LargePairEnumeration BalancedEnumeration BalancedFamily | |
| open FirstPaperContextTransport FirstPaperPhysicalMarginal ExcessContextProfiles SmallGraphProfiles | |
| open FiniteExposureVariance | |
| open IntegerWeightedTransport | |
| universe u | |
| variable {I J : Type u} [Fintype I] [Fintype J] [LinearOrder I] [LinearOrder J] | |
| def rawChoice (small : I → J → Bool) (U : ℕ) (x : Cells small → Fin (U+1)) : | |
| SmallProfile small := fun p => if p.2 then U-(x p.1).val else (x p.1).val | |
| def earlierChoices (small : I → J → Bool) (s : I×J) (U : ℕ) | |
| (x : Cells small → Fin (U+1)) : Earlier small s → Fin (U+1) := fun a => x a.val | |
| omit [Fintype I] [Fintype J] in | |
| lemma earlierChoices_eq_iff (small : I → J → Bool) (s : I×J) (U : ℕ) | |
| (x : Cells small → Fin (U+1)) (τ : Earlier small s → Fin (U+1)) : | |
| earlierChoices small s U x=τ ↔ ∀ a : Earlier small s,x a.val=τ a := | |
| funext_iff | |
| def ChoiceContext (small : I → J → Bool) (s : I×J) (hs : small s.1 s.2=true) (U : ℕ) | |
| (τ : Earlier small s → Fin (U+1)) (k : Fin (U+1)) := | |
| {x : Cells small → Fin (U+1) // earlierChoices small s U x=τ ∧ x ⟨s,hs⟩=k} | |
| noncomputable instance choiceContextFintype (small : I → J → Bool) (s : I×J) | |
| (hs : small s.1 s.2=true) (U : ℕ) (τ : Earlier small s → Fin (U+1)) (k : Fin (U+1)) : | |
| Fintype (ChoiceContext small s hs U τ k) := by unfold ChoiceContext; infer_instance | |
| noncomputable def choiceContextEquiv (small : I → J → Bool) (s : I×J) | |
| (hs : small s.1 s.2=true) (U : ℕ) (τ : Earlier small s → Fin (U+1)) (k : Fin (U+1)) : | |
| (Later small s → Fin (U+1)) ≃ ChoiceContext small s hs U τ k where | |
| toFun y := ⟨splitChoices small s hs U τ k y,funext (splitChoices_prefix small s hs U τ k y), | |
| splitChoices_special small s hs U τ k y⟩ | |
| invFun x := fun a => x.val a.val | |
| left_inv y := by funext a; exact splitChoices_later small s hs U τ k y a | |
| right_inv x := by | |
| apply Subtype.ext | |
| funext a | |
| obtain ⟨v,rfl⟩ := (cellSplitEquiv small s hs).surjective a | |
| rcases v with p|(_|t) | |
| · exact (splitChoices_prefix small s hs U τ k _ p).trans (congrFun x.property.1 p).symm | |
| · exact (splitChoices_special small s hs U τ k _).trans x.property.2.symm | |
| · exact splitChoices_later small s hs U τ k _ t | |
| lemma choice_fibre_sum (small : I → J → Bool) (s : I×J) (hs : small s.1 s.2=true) | |
| (U : ℕ) (τ : Earlier small s → Fin (U+1)) (k : Fin (U+1)) | |
| (F : (Cells small → Fin (U+1)) → ℝ) : | |
| (∑ x,fibreWeight (fibreWeight F (earlierChoices small s U) τ) (fun x => x ⟨s,hs⟩) k x)= | |
| ∑ x : ChoiceContext small s hs U τ k,F x.val := by | |
| exact double_fibre_sum F (earlierChoices small s U) (fun x => x ⟨s,hs⟩) τ k | |
| omit [Fintype I] [Fintype J] in | |
| lemma prefixChoices_extension (small : I → J → Bool) (s : I×J) (U : ℕ) | |
| (τ : Earlier small s → Fin (U+1)) : | |
| prefixChoices small s (prefixExtension small s U τ) U (prefixExtension_bounded small s U τ)=τ := by | |
| funext a | |
| apply Fin.ext | |
| exact prefixExtension_at small s U τ a | |
| /-- The child mass and every child moment are the actual finite fibre | |
| sums of balanced physical choices, retaining all large completions. -/ | |
| theorem eliminate_context_eq_choice_fibre (small : I → J → Bool) (B : I → J → ℕ) | |
| (s : I×J) (hs : small s.1 s.2=true) (U : ℕ) | |
| (hB : ∀ i j,small i j=true → B i j=U) | |
| (τ : Earlier small s → Fin (U+1)) (k : Fin (U+1)) | |
| {n : ℕ} (e : Fin n ≃ Later small s) (F : SmallProfile small → ℝ) : | |
| eliminate (widths (fun _ : Fin n => U)) | |
| (fun p => F (assemble small s hs (prefixExtension small s U τ) U e p)) (childProfile U k.val)= | |
| ∑ x,fibreWeight (fibreWeight (fun x => F (rawChoice small U x)) | |
| (earlierChoices small s U) τ) (fun x => x ⟨s,hs⟩) k x := by | |
| rw [eliminate_context_eq_weighted_sum small B s hs _ U hB | |
| (prefixExtension_bounded small s U τ),choice_fibre_sum] | |
| have he := Fintype.sum_equiv | |
| (balancedContextEquiv small B s hs (prefixExtension small s U τ) U hB | |
| (prefixExtension_bounded small s U τ) k) | |
| (fun y => F (rawChoice small U (splitChoices small s hs U τ k y))) | |
| (fun z => F (naturalProfile z.val)) (fun y => by | |
| change F (rawChoice small U (splitChoices small s hs U τ k y))= | |
| F (rawChoice small U (splitChoices small s hs U | |
| (prefixChoices small s (prefixExtension small s U τ) U (prefixExtension_bounded small s U τ)) k y)) | |
| rw [prefixChoices_extension]) | |
| have hc := Fintype.sum_equiv (choiceContextEquiv small s hs U τ k) | |
| (fun y => F (rawChoice small U (splitChoices small s hs U τ k y))) | |
| (fun x => F (rawChoice small U x.val)) (fun _ => rfl) | |
| exact he.symm.trans hc | |
| end ContingencyTables.SmallContextEnumeration | |
| end OAI | |