openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /SourcePhysicalAssemblyLaw.lean
| import OAI.Computability.PerfectCompleteness.Construction.SourceChildKernelLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.DependentPredictionDifferenceLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.SourceChildKernelMarginalLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.SourceChildQuestionLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.SourcePhysicalAssemblyLaw | |
| open UniqueGamesTheorem.Foundations.Games | |
| open RecursiveSpaces TreeSourceSpaces SourceChildKernel | |
| open scoped Classical | |
| noncomputable section | |
| private theorem kernelJoint_pushforward_sigma {S R : Type*} {Y : S → Type*} | |
| [Fintype S] [Fintype R] [∀ s, Fintype (Y s)] | |
| (μ : FiniteDistribution S) (P : S → FiniteDistribution R) | |
| (f : (s : S) → R → Y s) : | |
| (CleanConditioning.kernelJoint μ P).pushforward | |
| (fun x => (⟨x.1, f x.1 x.2⟩ : Σ s, Y s)) = | |
| CompletionSoundness.sigmaLaw μ (fun s => (P s).pushforward (f s)) := by | |
| apply SigmaObservation.eq_of_probability_eq | |
| intro event | |
| rw [FiniteDistribution.probability_pushforward, CleanConditioning.probability_kernelJoint, | |
| CompletionSoundness.sigmaLaw_probability] | |
| simp only [FiniteDistribution.expectation, FiniteDistribution.probability_pushforward] | |
| variable {branch : Nat → Nat} {n t v m : Nat} {C : Type*} [Fintype C] | |
| (rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch n) → Slots branch n) | |
| def assembleLeft (sources : Sources (m := m) (t := t) designated) | |
| (raw : (i : Fin (branch n)) → Raw (C := C) (t := t) rows clauses designated i) : | |
| CutChildGrouping.Assembled (C := C) (parentLeftSlots clauses designated sources) rows := | |
| CutChildGrouping.assemble (parentLeftSlots clauses designated sources) rows | |
| (leftChildren rows clauses designated sources raw) | |
| theorem assembleLeft_eq_sourcePullback | |
| (sources : Sources (m := m) (t := t) designated) | |
| (raw : (i : Fin (branch n)) → Raw (C := C) (t := t) rows clauses designated i) : | |
| assembleLeft rows clauses designated sources raw = | |
| ChildAssemblyProjection.assembledPullback rows | |
| (parentProjection rows clauses designated sources raw) | |
| (CutChildGrouping.assemble (parentRightSlots rows clauses designated sources raw) rows | |
| (rightChildren rows clauses designated sources raw)) := | |
| SourceChildKernel.assemble_leftChildren rows clauses designated sources raw | |
| theorem kernels_assembleLeft (flag : Fin (branch n) → FiniteDistribution Bool) | |
| (sources : Sources (m := m) (t := t) designated) : | |
| (FiniteProduct.law (fun i => | |
| kernel (C := C) (t := t) rows clauses designated flag i (sources i))).pushforward | |
| (assembleLeft rows clauses designated sources) = | |
| (FiniteProduct.law (fun i => | |
| SourceChildNativeLaw.leftLaw (C := C) (t := t) rows clauses designated flag i (sources i))).pushforward | |
| (CutChildGrouping.assemble (C := C) (parentLeftSlots clauses designated sources) rows) := by | |
| unfold assembleLeft | |
| rw [← FiniteDistribution.pushforward_comp | |
| (FiniteProduct.law (fun i => | |
| kernel (C := C) (t := t) rows clauses designated flag i (sources i))) | |
| (leftChildren rows clauses designated sources) | |
| (CutChildGrouping.assemble (C := C) (parentLeftSlots clauses designated sources) rows), | |
| SourceChildNativeLaw.kernels_leftChildren] | |
| rfl | |
| theorem kernels_assembleLeft_false | |
| (sources : Sources (m := m) (t := t) designated) : | |
| (FiniteProduct.law (fun i => | |
| kernel (C := C) (t := t) rows clauses designated | |
| (fun _ => SourceChildNativeLaw.falseFlag) i (sources i))).pushforward | |
| (assembleLeft rows clauses designated sources) = | |
| FiniteDistribution.uniform | |
| (CutChildGrouping.Assembled (C := C) (parentLeftSlots clauses designated sources) rows) := by | |
| unfold assembleLeft | |
| rw [← FiniteDistribution.pushforward_comp | |
| (FiniteProduct.law (fun i => | |
| kernel (C := C) (t := t) rows clauses designated | |
| (fun _ => SourceChildNativeLaw.falseFlag) i (sources i))) | |
| (leftChildren rows clauses designated sources) | |
| (CutChildGrouping.assemble (C := C) (parentLeftSlots clauses designated sources) rows), | |
| SourceChildNativeLaw.kernels_leftChildren_false] | |
| exact CutChildGrouping.assemble_law (C := C) (parentLeftSlots clauses designated sources) rows | |
| abbrev Observation := | |
| Σ sources : Sources (m := m) (t := t) designated, | |
| CutChildGrouping.Assembled (C := C) (parentLeftSlots clauses designated sources) rows | |
| def observe (sample : Sample (C := C) (t := t) rows clauses designated) : | |
| Observation (C := C) (t := t) rows clauses designated := | |
| ⟨(SourceChildKernelJoint.groupEquiv rows clauses designated sample).1, | |
| assembleLeft rows clauses designated | |
| (SourceChildKernelJoint.groupEquiv rows clauses designated sample).1 | |
| (SourceChildKernelJoint.groupEquiv rows clauses designated sample).2⟩ | |
| private theorem original_observe_kernel_law [NeZero m] | |
| (flag : Fin (branch n) → FiniteDistribution Bool) : | |
| (originalLaw (C := C) (t := t) rows clauses designated flag).pushforward | |
| (observe rows clauses designated) = | |
| CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated) | |
| (fun sources => | |
| (FiniteProduct.law (fun i => | |
| kernel (C := C) (t := t) rows clauses designated flag i (sources i))).pushforward | |
| (assembleLeft rows clauses designated sources)) := by | |
| unfold observe | |
| rw [← FiniteDistribution.pushforward_comp | |
| (originalLaw (C := C) (t := t) rows clauses designated flag) | |
| (SourceChildKernelJoint.groupEquiv rows clauses designated) | |
| (fun x => (⟨x.1, assembleLeft rows clauses designated x.1 x.2⟩ : | |
| Observation (C := C) (t := t) rows clauses designated)), | |
| SourceChildKernelJoint.originalLaw_group] | |
| exact kernelJoint_pushforward_sigma (SourceChildQuestionLaw.sourceLaw m t designated) | |
| (fun sources => FiniteProduct.law (fun i => | |
| kernel (C := C) (t := t) rows clauses designated flag i (sources i))) | |
| (assembleLeft rows clauses designated) | |
| theorem original_assembleLeft_law [NeZero m] | |
| (flag : Fin (branch n) → FiniteDistribution Bool) : | |
| (originalLaw (C := C) (t := t) rows clauses designated flag).pushforward | |
| (observe rows clauses designated) = | |
| CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated) | |
| (fun sources => | |
| (FiniteProduct.law (fun i => SourceChildNativeLaw.leftLaw (C := C) (t := t) | |
| rows clauses designated flag i (sources i))).pushforward | |
| (CutChildGrouping.assemble (C := C) (parentLeftSlots clauses designated sources) rows)) := by | |
| rw [original_observe_kernel_law] | |
| exact congrArg (CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated)) | |
| (funext fun sources => kernels_assembleLeft rows clauses designated flag sources) | |
| theorem original_assembleLeft_false_law [NeZero m] : | |
| (originalLaw (C := C) (t := t) rows clauses designated | |
| (fun _ => SourceChildNativeLaw.falseFlag)).pushforward | |
| (observe rows clauses designated) = | |
| CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated) | |
| (fun sources => FiniteDistribution.uniform | |
| (CutChildGrouping.Assembled (C := C) (parentLeftSlots clauses designated sources) rows)) := by | |
| rw [original_observe_kernel_law] | |
| exact congrArg (CompletionSoundness.sigmaLaw (SourceChildQuestionLaw.sourceLaw m t designated)) | |
| (funext fun sources => kernels_assembleLeft_false (C := C) rows clauses designated sources) | |
| end | |
| end PerfectCompleteness.SourcePhysicalAssemblyLaw | |
| end | |
| end OAI | |