openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /NumberedUniformCut.lean
| import OAI.Computability.PerfectCompleteness.Foundations.OriginalWholeCutBridge | |
| import OAI.Computability.PerfectCompleteness.Reduction.CompletionSoundness | |
| import OAI.Computability.PerfectCompleteness.Sampling.OriginalUniformCut | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.NumberedUniformCut | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open OriginalWholeCutTape OriginalWholeCut | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| noncomputable section | |
| attribute [local instance 2000] OriginalWholeCutLaw.valuesBelowFintype | |
| private theorem sigma_product_pushforward {E A Γ : Type*} | |
| [Fintype E] [Fintype A] [Fintype Γ] | |
| (μ : FiniteDistribution E) (ν : FiniteDistribution A) (f : E × A → Γ) : | |
| (CompletionSoundness.sigmaLaw μ (fun _ => ν)).pushforward | |
| (fun z => f (z.1, z.2)) = | |
| (μ.product ν).pushforward f := by | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro z | |
| simp only [FiniteDistribution.pushforward, CompletionSoundness.sigmaLaw, | |
| FiniteDistribution.product, Fintype.sum_sigma, Fintype.sum_prod_type] | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| def numberRecordEquiv (rows repeats : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| Record rows repeats p slots ≃ OriginalWholeCutBridge.NumberedRecord rows repeats p slots := | |
| Equiv.prodCongr (Equiv.refl (Exterior rows repeats p slots)) | |
| ((Equiv.prodCongr | |
| (Equiv.arrowCongr (callNumbering rows repeats p) (Equiv.refl (H (cutSlots p slots)))) | |
| (Equiv.refl (BelowArrays rows p slots))).trans | |
| (OriginalWholeCutBridge.assembledEquiv | |
| (OriginalCutCalls.count rows repeats n (m + 1)) rows (cutSlots p slots))) | |
| @[simp] theorem numberRecordEquiv_apply (rows repeats : Nat → Nat) | |
| (p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| (record : Record rows repeats p slots) : | |
| numberRecordEquiv rows repeats p slots record = | |
| OriginalWholeCutBridge.numberRecord rows repeats p slots record := rfl | |
| theorem reference_numberRecord_law (rows repeats : Nat → Nat) | |
| (p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| (OriginalUniformCut.referenceLaw rows repeats p slots).pushforward | |
| (OriginalWholeCutBridge.numberRecord rows repeats p slots) = | |
| (exteriorLaw rows repeats p slots).product | |
| (FiniteDistribution.uniform (CutChildGrouping.Assembled | |
| (C := Fin (OriginalCutCalls.count rows repeats n (m + 1))) (cutSlots p slots) rows)) := by | |
| rw [OriginalUniformCut.referenceLaw_uniform] | |
| change (FiniteDistribution.uniform (Record rows repeats p slots)).pushforward | |
| (numberRecordEquiv rows repeats p slots) = _ | |
| rw [← FiniteDistribution.transport_eq_pushforward, UniformConditioning.uniform_transport] | |
| exact WholeCutSampler.uniform_product.symm | |
| theorem reconstruct_product_uniform (rows repeats : Nat → Nat) | |
| (p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| ((exteriorLaw rows repeats p slots).product | |
| (FiniteDistribution.uniform (CutChildGrouping.Assembled | |
| (C := Fin (OriginalCutCalls.count rows repeats n (m + 1))) (cutSlots p slots) rows))).pushforward | |
| (OriginalWholeCutBridge.reconstruct rows repeats p slots) = | |
| WholeArraySampler.law rows repeats p slots := by | |
| rw [← reference_numberRecord_law rows repeats p slots, FiniteDistribution.pushforward_comp] | |
| have hmap : (fun record : Record rows repeats p slots => | |
| OriginalWholeCutBridge.reconstruct rows repeats p slots | |
| (OriginalWholeCutBridge.numberRecord rows repeats p slots record)) = | |
| reconstructRecord rows repeats p slots := | |
| funext (OriginalWholeCutBridge.reconstruct_numberRecord rows repeats p slots) | |
| rw [hmap] | |
| exact OriginalUniformCut.reconstruct_referenceLaw rows repeats p slots | |
| theorem reconstruct_sigma_uniform (rows repeats : Nat → Nat) | |
| (p : Path branch n (m + 1)) (slots : Slots branch n → Fin t → MixedSupport.Slot) : | |
| (CompletionSoundness.sigmaLaw (exteriorLaw rows repeats p slots) (fun _ => | |
| FiniteDistribution.uniform (CutChildGrouping.Assembled | |
| (C := Fin (OriginalCutCalls.count rows repeats n (m + 1))) (cutSlots p slots) rows))).pushforward | |
| (fun z => OriginalWholeCutBridge.reconstruct rows repeats p slots (z.1, z.2)) = | |
| WholeArraySampler.law rows repeats p slots := by | |
| rw [sigma_product_pushforward] | |
| exact reconstruct_product_uniform rows repeats p slots | |
| end | |
| end PerfectCompleteness.NumberedUniformCut | |
| end | |
| end OAI | |