openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Construction /SourcePhysicalChildExpectation.lean
| import OAI.Computability.PerfectCompleteness.Construction.SourcePhysicalTaggedChildren | |
| import OAI.Computability.PerfectCompleteness.Construction.SourceQuestionRawSwap | |
| import OAI.Computability.PerfectCompleteness.Sampling.CandidateCoupling | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.SourcePhysicalChildExpectation | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces | |
| open UniqueGamesTheorem.Foundations.Games | |
| 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 observe (q : PreliminarySampler.Questions branch (n + 1) t m) | |
| (choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t)) | |
| (raw : SourceChildKernelJoint.RawTuple (C := C) (t := t) rows clauses designated) : | |
| Σ flags : SourceProjectedTag.Flags (branch := branch) (n := n), | |
| CutChildGrouping.Raw (C := C) | |
| (SourceProjectedTag.mixedInside clauses designated q choices flags) rows := | |
| SourcePhysicalTaggedChildren.observe rows clauses designated | |
| (SourceQuestionKernelJoint.sources designated q choices) raw | |
| @[simp] theorem observe_flags | |
| (q : PreliminarySampler.Questions branch (n + 1) t m) | |
| (choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t)) | |
| (raw : SourceChildKernelJoint.RawTuple (C := C) (t := t) rows clauses designated) : | |
| (observe rows clauses designated q choices raw).1 = | |
| fun child => SourceChildKernel.rawProjected rows clauses designated child (raw child) := rfl | |
| theorem observe_sample_heq | |
| (sample : SourceChildKernel.Sample (C := C) (t := t) rows clauses designated) : | |
| HEq (observe rows clauses designated | |
| (SourceQuestionRawSwap.insideQuestions rows clauses designated sample) | |
| (SourceQuestionRawSwap.positions rows clauses designated sample) | |
| (SourceQuestionRawSwap.raw rows clauses designated sample)) | |
| (SourcePhysicalTaggedChildren.observe rows clauses designated | |
| (fun child => ((sample child).1, (sample child).2.1)) | |
| (fun child => (sample child).2.2)) := by | |
| have congrObserve (sources other : SourceChildKernel.Sources (m := m) (t := t) designated) | |
| (hs : sources = other) | |
| (raw : SourceChildKernelJoint.RawTuple (C := C) (t := t) rows clauses designated) : | |
| HEq (SourcePhysicalTaggedChildren.observe rows clauses designated sources raw) | |
| (SourcePhysicalTaggedChildren.observe rows clauses designated other raw) := by | |
| cases hs | |
| rfl | |
| exact congrObserve _ _ | |
| (SourceQuestionRawSwap.sources_insideQuestions_positions rows clauses designated sample) | |
| (SourceQuestionRawSwap.raw rows clauses designated sample) | |
| theorem kernel_expectation | |
| (flag : Fin (branch n) → FiniteDistribution Bool) | |
| (q : PreliminarySampler.Questions branch (n + 1) t m) | |
| (choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t)) | |
| (F : (flags : SourceProjectedTag.Flags (branch := branch) (n := n)) → | |
| CutChildGrouping.Raw (C := C) | |
| (SourceProjectedTag.mixedInside clauses designated q choices flags) rows → ℝ) : | |
| (SourceQuestionRawSwap.kernelLaw (C := C) rows clauses designated flag q choices).expectation | |
| (fun raw => | |
| let observed := observe rows clauses designated q choices raw | |
| F observed.1 observed.2) = | |
| (SourceProjectedTag.flagLaw flag).expectation (fun flags => | |
| (CutChildGrouping.rawLaw (C := C) | |
| (SourceProjectedTag.mixedInside clauses designated q choices flags) rows).expectation | |
| (F flags)) := by | |
| let statistic : | |
| (Σ flags : SourceProjectedTag.Flags (branch := branch) (n := n), | |
| CutChildGrouping.Raw (C := C) | |
| (SourceProjectedTag.mixedInside clauses designated q choices flags) rows) → ℝ := | |
| fun observed => F observed.1 observed.2 | |
| let physical := fun flags : SourceProjectedTag.Flags (branch := branch) (n := n) => | |
| CutChildGrouping.rawLaw (C := C) | |
| (SourceProjectedTag.mixedInside clauses designated q choices flags) rows | |
| have hlaw : | |
| (SourceQuestionRawSwap.kernelLaw (C := C) rows clauses designated flag q choices).pushforward | |
| (observe rows clauses designated q choices) = | |
| CompletionSoundness.sigmaLaw (SourceProjectedTag.flagLaw flag) physical := | |
| SourcePhysicalTaggedChildren.kernels_tagged (C := C) rows clauses designated flag | |
| (SourceQuestionKernelJoint.sources designated q choices) | |
| calc | |
| _ = ((SourceQuestionRawSwap.kernelLaw (C := C) rows clauses designated flag q choices).pushforward | |
| (observe rows clauses designated q choices)).expectation statistic := | |
| (FiniteDistribution.expectation_pushforward | |
| (SourceQuestionRawSwap.kernelLaw (C := C) rows clauses designated flag q choices) | |
| (observe rows clauses designated q choices) statistic).symm | |
| _ = (CompletionSoundness.sigmaLaw (SourceProjectedTag.flagLaw flag) physical).expectation | |
| statistic := congrArg (fun μ => FiniteDistribution.expectation μ statistic) hlaw | |
| _ = _ := CandidateCoupling.expectation_sigmaLaw | |
| (SourceProjectedTag.flagLaw flag) physical statistic | |
| theorem originalLaw_expectation [NeZero m] | |
| (flag : Fin (branch n) → FiniteDistribution Bool) | |
| (F : (q : PreliminarySampler.Questions branch (n + 1) t m) → | |
| (choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t)) → | |
| (flags : SourceProjectedTag.Flags (branch := branch) (n := n)) → | |
| CutChildGrouping.Raw (C := C) | |
| (SourceProjectedTag.mixedInside clauses designated q choices flags) rows → ℝ) : | |
| (SourceChildKernel.originalLaw (C := C) (t := t) rows clauses designated flag).expectation | |
| (fun sample => | |
| let q := SourceQuestionRawSwap.insideQuestions rows clauses designated sample | |
| let choices := SourceQuestionRawSwap.positions rows clauses designated sample | |
| let observed := observe rows clauses designated q choices | |
| (SourceQuestionRawSwap.raw rows clauses designated sample) | |
| F q choices observed.1 observed.2) = | |
| (PreliminarySampler.questionsLaw | |
| (branch := branch) (n := n + 1) (t := t) (m := m)).expectation | |
| (fun q => (SourceProjectedTag.positionLaw | |
| (branch := branch) (n := n) (t := t)).expectation | |
| (fun choices => (SourceProjectedTag.flagLaw flag).expectation | |
| (fun flags => (CutChildGrouping.rawLaw (C := C) | |
| (SourceProjectedTag.mixedInside clauses designated q choices flags) rows).expectation | |
| (F q choices flags)))) := by | |
| rw [SourceQuestionRawSwap.originalLaw_expectation rows clauses designated flag | |
| (fun q choices raw => | |
| let observed := observe rows clauses designated q choices raw | |
| F q choices observed.1 observed.2)] | |
| apply FiniteDistribution.expectation_congr | |
| intro q | |
| apply FiniteDistribution.expectation_congr | |
| intro choices | |
| exact kernel_expectation rows clauses designated flag q choices (F q choices) | |
| end | |
| end PerfectCompleteness.SourcePhysicalChildExpectation | |
| end | |
| end OAI | |