openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /SourcePhysicalWholeLaw.lean
| import OAI.Computability.PerfectCompleteness.Repetition.CleanPhysicalReplayLaw | |
| import OAI.Computability.PerfectCompleteness.Sampling.RationalFiniteLawLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.SourceChildKernelMarginalLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.SourcePhysicalWholeLaw | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| noncomputable section | |
| variable {branch : Nat → Nat} {root h t v m : Nat} | |
| (rows repeats : Nat → Nat) (p : Path branch root (h + 1)) | |
| (outside : Slots branch root → Fin t → MixedSupport.Slot) | |
| (placeholder : Slots branch (h + 1) → Fin t → MixedSupport.Slot) | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| abbrev Raw := CleanPhysicalReplay.Exterior rows repeats p outside placeholder × | |
| ((i : Fin (branch h)) → SourceChildKernel.Raw | |
| (C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated i) | |
| def rawLaw (flag : Fin (branch h) → FiniteDistribution Bool) | |
| (sources : SourceChildKernel.Sources (m := m) (t := t) designated) : | |
| FiniteDistribution (Raw rows repeats p outside placeholder clauses designated) := | |
| (FiniteDistribution.uniform (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product | |
| (FiniteProduct.law (fun i => SourceChildKernel.kernel | |
| (C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated flag i (sources i))) | |
| def nativeBlockLaw (flag : Fin (branch h) → FiniteDistribution Bool) | |
| (sources : SourceChildKernel.Sources (m := m) (t := t) designated) : | |
| FiniteDistribution (CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) | |
| (SourceChildKernel.parentLeftSlots clauses designated sources) rows) := | |
| FiniteProduct.law (fun i => SourceChildNativeLaw.leftLaw | |
| (C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated flag i (sources i)) | |
| def observeLeft (sources : SourceChildKernel.Sources (m := m) (t := t) designated) | |
| (sample : Raw rows repeats p outside placeholder clauses designated) : | |
| Arrays (CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources)) rows := | |
| WholeCutSampler.evaluate rows repeats p | |
| (CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources)) | |
| (CleanPhysicalReplay.physicalTape rows repeats p outside placeholder | |
| (SourceChildKernel.parentLeftSlots clauses designated sources) sample.1 | |
| (SourceChildKernel.leftChildren rows clauses designated sources sample.2)) | |
| theorem observeLeft_law (flag : Fin (branch h) → FiniteDistribution Bool) | |
| (sources : SourceChildKernel.Sources (m := m) (t := t) designated) : | |
| (rawLaw rows repeats p outside placeholder clauses designated flag sources).pushforward | |
| (observeLeft rows repeats p outside placeholder clauses designated sources) = | |
| ((FiniteDistribution.uniform (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product | |
| (nativeBlockLaw rows repeats p clauses designated flag sources)).pushforward | |
| (fun x => WholeCutSampler.evaluate rows repeats p | |
| (CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources)) | |
| (CleanPhysicalReplay.physicalTape rows repeats p outside placeholder | |
| (SourceChildKernel.parentLeftSlots clauses designated sources) x.1 x.2)) := by | |
| let decode := fun sample : Raw rows repeats p outside placeholder clauses designated => | |
| (sample.1, SourceChildKernel.leftChildren rows clauses designated sources sample.2) | |
| let evaluate := fun x : CleanPhysicalReplay.Exterior rows repeats p outside placeholder × | |
| CutChildGrouping.Raw (C := WholeCutCalls.Index rows repeats p) | |
| (SourceChildKernel.parentLeftSlots clauses designated sources) rows => | |
| WholeCutSampler.evaluate rows repeats p | |
| (CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources)) | |
| (CleanPhysicalReplay.physicalTape rows repeats p outside placeholder | |
| (SourceChildKernel.parentLeftSlots clauses designated sources) x.1 x.2) | |
| calc | |
| _ = ((rawLaw rows repeats p outside placeholder clauses designated flag sources).pushforward | |
| decode).pushforward evaluate := (FiniteDistribution.pushforward_comp _ _ _).symm | |
| _ = _ := by | |
| have hdecode : (rawLaw rows repeats p outside placeholder clauses designated flag sources).pushforward | |
| decode = | |
| (FiniteDistribution.uniform (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product | |
| (nativeBlockLaw rows repeats p clauses designated flag sources) := by | |
| unfold rawLaw | |
| change ((FiniteDistribution.uniform | |
| (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)).product | |
| (FiniteProduct.law (fun i => SourceChildKernel.kernel | |
| (C := WholeCutCalls.Index rows repeats p) (t := t) rows clauses designated flag i (sources i)))).pushforward | |
| (fun sample => ((id : CleanPhysicalReplay.Exterior rows repeats p outside placeholder → | |
| CleanPhysicalReplay.Exterior rows repeats p outside placeholder) sample.1, | |
| SourceChildKernel.leftChildren rows clauses designated sources sample.2)) = _ | |
| rw [FiniteDistribution.product_pushforward, | |
| UniqueGamesTheorem.Foundations.Games.FiniteDistribution.pushforward_id | |
| (FiniteDistribution.uniform (CleanPhysicalReplay.Exterior rows repeats p outside placeholder)), | |
| SourceChildNativeLaw.kernels_leftChildren] | |
| rfl | |
| rw [hdecode] | |
| theorem nativeBlockLaw_false | |
| (sources : SourceChildKernel.Sources (m := m) (t := t) designated) : | |
| nativeBlockLaw rows repeats p clauses designated (fun _ => SourceChildNativeLaw.falseFlag) sources = | |
| CutChildGrouping.rawLaw (C := WholeCutCalls.Index rows repeats p) | |
| (SourceChildKernel.parentLeftSlots clauses designated sources) rows := by | |
| unfold nativeBlockLaw CutChildGrouping.rawLaw | |
| apply congrArg FiniteProduct.law | |
| funext i | |
| exact SourceChildNativeLaw.leftLaw_false rows clauses designated i (sources i) | |
| theorem observeLeft_false_law | |
| (sources : SourceChildKernel.Sources (m := m) (t := t) designated) : | |
| (rawLaw rows repeats p outside placeholder clauses designated | |
| (fun _ => SourceChildNativeLaw.falseFlag) sources).pushforward | |
| (observeLeft rows repeats p outside placeholder clauses designated sources) = | |
| WholeArraySampler.law rows repeats p | |
| (CutSlotAssembly.fill p outside (SourceChildKernel.parentLeftSlots clauses designated sources)) := by | |
| rw [observeLeft_law, nativeBlockLaw_false] | |
| exact CleanPhysicalReplayLaw.evaluate_physicalTape_law rows repeats p outside placeholder | |
| (SourceChildKernel.parentLeftSlots clauses designated sources) | |
| end | |
| end PerfectCompleteness.SourcePhysicalWholeLaw | |
| end | |
| end OAI | |