openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /CutCallOriginalLaw.lean
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutPairLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutCallOriginalLaw | |
| open RecursiveSpaces TreeSourceSpaces HierarchicalArrays WholeCutGrouping | |
| open UniqueGamesTheorem.Foundations.Games | |
| open scoped Classical | |
| noncomputable section | |
| section Original | |
| variable {branch : Nat → Nat} {n t : Nat} {C : Type*} [Fintype C] | |
| (slots : Slots branch (n + 1) → Fin t → MixedSupport.Slot) | |
| (rows : Nat → Nat) | |
| def originalLinear : CutCallExtension.Extended (C := C) slots rows →ₗ[ZMod 2] | |
| CutChildGrouping.Raw (C := C) slots rows where | |
| toFun := CutCallExtension.original slots rows | |
| map_add' _ _ := rfl | |
| map_smul' _ _ := rfl | |
| omit [Fintype C] in | |
| theorem originalLinear_surjective : Function.Surjective | |
| (originalLinear (C := C) slots rows) := by | |
| intro old | |
| refine ⟨fun i => (fun c => match c with | |
| | none => 0 | |
| | some c => (old i).1 c, (old i).2), ?_⟩ | |
| rfl | |
| theorem original_law : | |
| (CutChildGrouping.rawLaw (C := Option C) slots rows).pushforward | |
| (CutCallExtension.original slots rows) = | |
| CutChildGrouping.rawLaw (C := C) slots rows := by | |
| unfold CutChildGrouping.rawLaw | |
| rw [UniformLinearImage.law_uniform, UniformLinearImage.law_uniform] | |
| exact UniformLinearImage.uniform_pushforward_linearMap | |
| (originalLinear slots rows) (originalLinear_surjective slots rows) | |
| theorem original_probability (event : CutChildGrouping.Raw (C := C) slots rows → Bool) : | |
| (CutChildGrouping.rawLaw (C := Option C) slots rows).probability | |
| (fun raw => event (CutCallExtension.original slots rows raw)) = | |
| (CutChildGrouping.rawLaw (C := C) slots rows).probability event := by | |
| rw [← FiniteDistribution.probability_pushforward, original_law] | |
| end Original | |
| section First | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| (rows repeats : Nat → Nat) (p : DescendantSpaces.Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) | |
| theorem first_law (exterior : Exterior rows repeats p slots) : | |
| (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p)) | |
| (cutSlots p slots) rows).pushforward | |
| (LowerCutPair.first rows repeats p slots exterior) = | |
| (CutChildGrouping.rawLaw (C := LowerCutPair.Calls rows repeats p) | |
| (cutSlots p slots) rows).pushforward | |
| (LowerCutRows.reconstruct rows repeats p slots exterior) := by | |
| change (CutChildGrouping.rawLaw | |
| (C := Option (LowerCutPair.Calls rows repeats p)) (cutSlots p slots) rows).pushforward | |
| (fun raw => LowerCutRows.reconstruct rows repeats p slots exterior | |
| (CutCallExtension.original (cutSlots p slots) rows raw)) = _ | |
| rw [← FiniteDistribution.pushforward_comp, original_law] | |
| theorem first_probability (exterior : Exterior rows repeats p slots) | |
| (event : Arrays slots rows → Bool) : | |
| (CutChildGrouping.rawLaw (C := Option (LowerCutPair.Calls rows repeats p)) | |
| (cutSlots p slots) rows).probability | |
| (fun raw => event (LowerCutPair.first rows repeats p slots exterior raw)) = | |
| (CutChildGrouping.rawLaw (C := LowerCutPair.Calls rows repeats p) | |
| (cutSlots p slots) rows).probability | |
| (fun raw => event (LowerCutRows.reconstruct rows repeats p slots exterior raw)) := by | |
| exact original_probability (cutSlots p slots) rows | |
| (fun raw => event (LowerCutRows.reconstruct rows repeats p slots exterior raw)) | |
| end First | |
| end | |
| end PerfectCompleteness.CutCallOriginalLaw | |
| end | |
| end OAI | |