openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Repetition /CleanPhysicalReplayLaw.lean
| import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalReplay | |
| import OAI.Computability.PerfectCompleteness.Sampling.FilledUniformSampler | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CleanPhysicalReplayLaw | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| noncomputable section | |
| variable {branch : Nat → Nat} {root h t : Nat} | |
| (rows repeats : Nat → Nat) (p : Path branch root (h + 1)) | |
| (outside : Slots branch root → Fin t → MixedSupport.Slot) | |
| (placeholder inside : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| theorem physicalTape_eq_filledSplitEquiv | |
| (external : CleanPhysicalReplay.Exterior rows repeats p outside placeholder) | |
| (children : CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows) : | |
| CleanPhysicalReplay.physicalTape rows repeats p outside placeholder inside external children = | |
| FilledUniformSampler.filledSplitEquiv rows repeats p outside placeholder inside | |
| (external, children) := by | |
| apply congrArg (WholeCutGrouping.splitTape rows repeats p | |
| (CutSlotAssembly.fill p outside inside)).symm | |
| apply Prod.ext | |
| · rfl | |
| · funext i | |
| exact eq_of_heq ((CleanPhysicalReplay.fillChild_heq rows repeats p outside inside i | |
| (children i)).trans | |
| (FilledUniformSampler.childrenEquiv_heq rows repeats p outside inside children i).symm) | |
| theorem physicalTape_law : | |
| ((FiniteDistribution.uniform | |
| (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product | |
| (CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows)).pushforward | |
| (fun x => CleanPhysicalReplay.physicalTape rows repeats p outside placeholder inside x.1 x.2) = | |
| WholeCutSampler.tapeLaw rows repeats p (CutSlotAssembly.fill p outside inside) := by | |
| have hmap : (fun x : CleanPhysicalReplay.Exterior rows repeats p outside placeholder × | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows => | |
| CleanPhysicalReplay.physicalTape rows repeats p outside placeholder inside | |
| x.1 x.2) = FilledUniformSampler.filledSplitEquiv rows repeats p outside placeholder inside := by | |
| funext x | |
| exact physicalTape_eq_filledSplitEquiv rows repeats p outside placeholder inside x.1 x.2 | |
| rw [hmap] | |
| exact FilledUniformSampler.filledSplitEquiv_tapeLaw rows repeats p outside placeholder inside | |
| theorem evaluate_physicalTape_law : | |
| ((FiniteDistribution.uniform | |
| (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product | |
| (CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) inside rows)).pushforward | |
| (fun x => WholeCutSampler.evaluate rows repeats p (CutSlotAssembly.fill p outside inside) | |
| (CleanPhysicalReplay.physicalTape rows repeats p outside placeholder inside x.1 x.2)) = | |
| WholeArraySampler.law rows repeats p (CutSlotAssembly.fill p outside inside) := by | |
| have hmap : (fun x : CleanPhysicalReplay.Exterior rows repeats p outside placeholder × | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) inside rows => | |
| WholeCutSampler.evaluate rows repeats p | |
| (CutSlotAssembly.fill p outside inside) | |
| (CleanPhysicalReplay.physicalTape rows repeats p outside placeholder inside x.1 x.2)) = | |
| FilledUniformSampler.evaluate rows repeats p outside placeholder inside := by | |
| funext x | |
| exact congrArg (WholeCutSampler.evaluate rows repeats p (CutSlotAssembly.fill p outside inside)) | |
| (physicalTape_eq_filledSplitEquiv rows repeats p outside placeholder inside x.1 x.2) | |
| rw [hmap] | |
| exact FilledUniformSampler.evaluate_law rows repeats p outside placeholder inside | |
| end | |
| end PerfectCompleteness.CleanPhysicalReplayLaw | |
| end | |
| end OAI | |