openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /FixedStoppedPhysicalLawLemmas.lean
| import OAI.Computability.PerfectCompleteness.Decoding.FixedStoppedDecoderContext | |
| import OAI.Computability.PerfectCompleteness.Foundations.SourcePhysicalSourceSwapLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.StoppedPhysicalPrefixSwap | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedStoppedPhysicalLaw | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| attribute [local instance] linearMapFintype | |
| attribute [local instance] StoppedProjectedPhysicalAdvice.adviceFintype | |
| StoppedProjectedPhysicalAdvice.physicalAdviceFintype | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ) | |
| (i j : Fin parameters.plan.depth) (hij : i < j) (input : List Bool) | |
| abbrev clauses := PCPSource.clauseFamily (BinaryLanguage.totalRename input) | |
| def designated : Fin (FixedParameters.branch parameters i.val) → | |
| Slots (FixedParameters.branch parameters) i.val := | |
| fun _ => StoppedCleanGeometry.firstLeaf (FixedParameters.branch_pos parameters) i.val | |
| def flag : Fin (FixedParameters.branch parameters i.val) → FiniteDistribution Bool := | |
| fun _ => ProjectedCleanRate.projectionFlag (parameters.cubeSize i.val) | |
| (FixedDecoderCleanRate.cubePositive parameters i.val) | |
| abbrev Prefixes := FixedStoppedDecoderContext.Prefixes parameters i j | |
| abbrev Context := FixedStoppedDecoderContext.Context parameters i j hij input | |
| def path (pref : Prefixes parameters i j) := | |
| (FixedStoppedDecoderContext.upperPath parameters i j pref).append | |
| (FixedStoppedDecoderContext.lowerPath parameters i j hij pref) | |
| abbrev ChildSample (pref : Prefixes parameters i j) := | |
| SourceChildKernel.Sample | |
| (C := WholeCutCalls.Index (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (path parameters i j hij pref)) | |
| (t := FixedRows.sourceLength parameters.plan hδ) | |
| (FixedRows.rows parameters.plan) (clauses input) (designated parameters i) | |
| abbrev Sample := (context : Context parameters i j hij input) × | |
| ChildSample parameters i j hij input context.1 | |
| abbrev PhysicalSample := StoppedProjectedPhysicalAdvice.Sample | |
| (t := FixedRows.sourceLength parameters.plan hδ) | |
| (clauses input) (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) parameters.plan.order | |
| def childLaw (pref : Prefixes parameters i j) : | |
| FiniteDistribution (ChildSample parameters i j hij input pref) := | |
| SourceChildKernel.originalLaw | |
| (C := WholeCutCalls.Index (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (path parameters i j hij pref)) | |
| (t := FixedRows.sourceLength parameters.plan hδ) | |
| (FixedRows.rows parameters.plan) (clauses input) (designated parameters i) (flag parameters i) | |
| def law : FiniteDistribution (Sample parameters i j hij input) := | |
| CompletionSoundness.sigmaLaw (FixedStoppedDecoderContext.law parameters i j hij input) | |
| (fun context => childLaw parameters i j hij input context.1) | |
| def read (sample : Sample parameters i j hij input) : | |
| PhysicalSample parameters i j hij input := | |
| let context := sample.1 | |
| let inside := SourceQuestionRawSwap.insideQuestions (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) sample.2 | |
| let questions := SourceQuestionCutSplit.join (path parameters i j hij context.1) | |
| inside context.2.1 | |
| let positions := SourceQuestionRawSwap.positions (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) sample.2 | |
| let exterior := SourcePhysicalSourceSwap.joinedExterior (clauses input) | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (path parameters i j hij context.1) inside context.2.1 context.2.2.2.2 | |
| let raw := SourceQuestionRawSwap.raw (FixedRows.rows parameters.plan) | |
| (clauses input) (designated parameters i) sample.2 | |
| ⟨(questions, context.1.1), | |
| (⟨(context.1.2, (positions, context.2.2.2.1)), (exterior, raw)⟩, context.2.2.1)⟩ | |
| abbrev physicalLaw := StoppedProjectedPhysicalAdvice.law | |
| (t := FixedRows.sourceLength parameters.plan hδ) | |
| (clauses input) (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => parameters.plan.rows_pos (parameters.plan.depth - (k + 1))) | |
| (flag parameters i) parameters.plan.order | |
| theorem expectation_read (value : PhysicalSample parameters i j hij input → ℝ) : | |
| (physicalLaw parameters i j hij input).expectation value = | |
| (law parameters i j hij input).expectation | |
| (fun sample => value (read parameters i j hij input sample)) := by | |
| refine (StoppedPhysicalPrefixSwap.expectation_prefixes | |
| (t := FixedRows.sourceLength parameters.plan hδ) | |
| (clauses input) (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => parameters.plan.rows_pos (parameters.plan.depth - (k + 1))) | |
| (flag parameters i) parameters.plan.order value).trans ?_ | |
| simp only [law, FixedStoppedDecoderContext.law, CandidateCoupling.expectation_sigmaLaw, | |
| FixedStoppedDecoderContext.fiberLaw] | |
| apply FiniteDistribution.expectation_congr | |
| intro pref | |
| simp only [FiniteDistribution.expectation_product] | |
| let questionLaw := PreliminarySampler.questionsLaw | |
| (branch := FixedParameters.branch parameters) (n := parameters.plan.depth) | |
| (t := FixedRows.sourceLength parameters.plan hδ) | |
| (m := (PCPSource.normalizedFormula (BinaryLanguage.totalRename input)).clauses.length) | |
| let adviceLaw := FiniteDistribution.uniform (FixedStoppedDecoderContext.Advice parameters i j pref) | |
| let directionsLaw := FixedStoppedDecoderContext.directionLaw parameters | |
| let outsideLaw := FiniteDistribution.uniform | |
| (FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref) | |
| let statistic := fun | |
| (questions : PreliminarySampler.Questions (FixedParameters.branch parameters) | |
| parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ) | |
| (PCPSource.normalizedFormula (BinaryLanguage.totalRename input)).clauses.length) | |
| (A : FixedStoppedDecoderContext.Advice parameters i j pref) | |
| (directions : FixedStoppedDecoderContext.Directions parameters) => | |
| (SourceProjectedTag.positionLaw (branch := FixedParameters.branch parameters) | |
| (n := i.val) (t := FixedRows.sourceLength parameters.plan hδ)).expectation | |
| (fun positions => (StoppedProjectedExperiment.physicalFiberLaw | |
| (clauses input) (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij (designated parameters i) (flag parameters i) | |
| (questions, pref.1) (pref.2, (positions, directions))).expectation | |
| (fun raw => value ⟨(questions, pref.1), (⟨(pref.2, (positions, directions)), raw⟩, A)⟩)) | |
| let rebuilt := fun (external : FixedStoppedDecoderContext.OutsideQuestions parameters i j hij input pref) | |
| (A : FixedStoppedDecoderContext.Advice parameters i j pref) | |
| (directions : FixedStoppedDecoderContext.Directions parameters) => | |
| (FiniteDistribution.uniform | |
| (FixedStoppedDecoderContext.Exterior parameters i j hij input pref external)).expectation | |
| (fun exterior => (childLaw parameters i j hij input pref).expectation | |
| (fun sample => value (read parameters i j hij input | |
| ⟨⟨pref, external, A, directions, exterior⟩, sample⟩))) | |
| change questionLaw.expectation (fun questions => adviceLaw.expectation (fun A => | |
| directionsLaw.expectation (fun directions => statistic questions A directions))) = | |
| outsideLaw.expectation (fun external => adviceLaw.expectation (fun A => | |
| directionsLaw.expectation (fun directions => rebuilt external A directions))) | |
| calc | |
| _ = adviceLaw.expectation (fun A => directionsLaw.expectation (fun directions => | |
| questionLaw.expectation (fun questions => statistic questions A directions))) := by | |
| rw [FiniteDistribution.expectation_comm] | |
| apply FiniteDistribution.expectation_congr | |
| intro A | |
| exact FiniteDistribution.expectation_comm _ _ _ | |
| _ = adviceLaw.expectation (fun A => directionsLaw.expectation (fun directions => | |
| outsideLaw.expectation (fun external => rebuilt external A directions))) := by | |
| apply FiniteDistribution.expectation_congr | |
| intro A | |
| apply FiniteDistribution.expectation_congr | |
| intro directions | |
| exact SourcePhysicalSourceSwap.expectation_swap (clauses input) | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (path parameters i j hij pref) (designated parameters i) (flag parameters i) | |
| (fun questions positions exterior raw => | |
| value ⟨(questions, pref.1), (⟨(pref.2, (positions, directions)), (exterior, raw)⟩, A)⟩) | |
| _ = _ := by | |
| calc | |
| _ = adviceLaw.expectation (fun A => outsideLaw.expectation (fun external => | |
| directionsLaw.expectation (fun directions => rebuilt external A directions))) := by | |
| apply FiniteDistribution.expectation_congr | |
| intro A | |
| exact FiniteDistribution.expectation_comm _ _ _ | |
| _ = _ := FiniteDistribution.expectation_comm _ _ _ | |
| theorem read_law : | |
| (law parameters i j hij input).pushforward (read parameters i j hij input) = | |
| physicalLaw parameters i j hij input := by | |
| apply SigmaObservation.eq_of_probability_eq | |
| intro event | |
| rw [FiniteDistribution.probability_pushforward, | |
| CandidateCoupling.probability_eq_expectation, CandidateCoupling.probability_eq_expectation] | |
| exact (expectation_read parameters i j hij input (fun sample => if event sample then 1 else 0)).symm | |
| end | |
| end PerfectCompleteness.FixedStoppedPhysicalLaw | |
| end | |
| section | |
| namespace PerfectCompleteness.FixedStoppedNativeLaw | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| attribute [local instance] StoppedProjectedPhysicalAdvice.adviceFintype | |
| StoppedProjectedPhysicalAdvice.physicalAdviceFintype | |
| section PathTransport | |
| variable {branch : Nat → Nat} {root h t v m : Nat} | |
| (rows repeats : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch h) → Slots branch h) | |
| {p q : Path branch root (h + 1)} | |
| def sourceSampleEquiv (hp : p = q) : | |
| SourceChildKernel.Sample (C := WholeCutCalls.Index rows repeats p) (t := t) | |
| rows clauses designated ≃ | |
| SourceChildKernel.Sample (C := WholeCutCalls.Index rows repeats q) (t := t) | |
| rows clauses designated := | |
| Equiv.cast (congrArg (fun path : Path branch root (h + 1) => | |
| SourceChildKernel.Sample (C := WholeCutCalls.Index rows repeats path) (t := t) | |
| rows clauses designated) hp) | |
| theorem sourceSampleEquiv_law [NeZero m] (hp : p = q) | |
| (flag : Fin (branch h) → FiniteDistribution Bool) : | |
| (SourceChildKernel.originalLaw (C := WholeCutCalls.Index rows repeats p) (t := t) | |
| rows clauses designated flag).pushforward | |
| (sourceSampleEquiv rows repeats clauses designated hp) = | |
| SourceChildKernel.originalLaw (C := WholeCutCalls.Index rows repeats q) (t := t) | |
| rows clauses designated flag := by | |
| cases hp | |
| exact FiniteDistribution.pushforward_id _ | |
| end PathTransport | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ) | |
| (i j : Fin parameters.plan.depth) (hij : i < j) (input : List Bool) | |
| local instance contextAdviceFintype | |
| (pref : FixedStoppedDecoderContext.Prefixes parameters i j) : | |
| Fintype (FixedStoppedDecoderContext.Advice parameters i j pref) := | |
| UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype | |
| abbrev Context := FixedStoppedDecoderContext.Context parameters i j hij input | |
| abbrev setup (context : Context parameters i j hij input) := | |
| FixedDecoderCleanRate.setup parameters | |
| (FixedStoppedDecoderContext.geometry parameters i j hij input context) input | |
| abbrev Sample := (context : Context parameters i j hij input) × | |
| CleanDecoderRate.Sample (setup parameters i j hij input context) | |
| theorem path_eq (context : Context parameters i j hij input) : | |
| CleanDecoderRate.path (setup parameters i j hij input context) = | |
| FixedStoppedPhysicalLaw.path parameters i j hij context.1 := | |
| StoppedCleanGeometry.geometry_path parameters input | |
| (FixedStoppedDecoderContext.upperPath parameters i j context.1) | |
| (FixedStoppedDecoderContext.lowerPath parameters i j hij context.1) hij | |
| context.2.1 context.2.2.2.2 | |
| def childRead (context : Context parameters i j hij input) : | |
| CleanDecoderRate.Sample (setup parameters i j hij input context) ≃ | |
| FixedStoppedPhysicalLaw.ChildSample parameters i j hij input context.1 := | |
| sourceSampleEquiv (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (FixedStoppedPhysicalLaw.clauses input) (FixedStoppedPhysicalLaw.designated parameters i) | |
| (path_eq parameters i j hij input context) | |
| theorem childRead_law (context : Context parameters i j hij input) : | |
| (CleanDecoderRate.rawLaw (setup parameters i j hij input context) | |
| (parameters.cubeSize i.val) (FixedDecoderCleanRate.cubePositive parameters i.val)).pushforward | |
| (childRead parameters i j hij input context) = | |
| FixedStoppedPhysicalLaw.childLaw parameters i j hij input context.1 := | |
| sourceSampleEquiv_law (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (FixedStoppedPhysicalLaw.clauses input) (FixedStoppedPhysicalLaw.designated parameters i) | |
| (path_eq parameters i j hij input context) (FixedStoppedPhysicalLaw.flag parameters i) | |
| def law : FiniteDistribution (Sample parameters i j hij input) := | |
| CompletionSoundness.sigmaLaw (FixedStoppedDecoderContext.law parameters i j hij input) | |
| (fun context => CleanDecoderRate.rawLaw (setup parameters i j hij input context) | |
| (parameters.cubeSize i.val) (FixedDecoderCleanRate.cubePositive parameters i.val)) | |
| def sourceRead (sample : Sample parameters i j hij input) : | |
| FixedStoppedPhysicalLaw.Sample parameters i j hij input := | |
| ⟨sample.1, childRead parameters i j hij input sample.1 sample.2⟩ | |
| theorem sourceRead_law : | |
| (law parameters i j hij input).pushforward (sourceRead parameters i j hij input) = | |
| FixedStoppedPhysicalLaw.law parameters i j hij input := by | |
| have hsplit := SigmaObservation.pushforward_fiber | |
| (X := fun context : Context parameters i j hij input => | |
| CleanDecoderRate.Sample (setup parameters i j hij input context)) | |
| (Y := fun context : Context parameters i j hij input => | |
| FixedStoppedPhysicalLaw.ChildSample parameters i j hij input context.1) | |
| (FixedStoppedDecoderContext.law parameters i j hij input) | |
| (fun context => CleanDecoderRate.rawLaw (setup parameters i j hij input context) | |
| (parameters.cubeSize i.val) (FixedDecoderCleanRate.cubePositive parameters i.val)) | |
| (fun context => childRead parameters i j hij input context) | |
| exact hsplit.trans (congrArg | |
| (fun P : (context : Context parameters i j hij input) → | |
| FiniteDistribution (FixedStoppedPhysicalLaw.ChildSample parameters i j hij input | |
| context.1) => | |
| CompletionSoundness.sigmaLaw (FixedStoppedDecoderContext.law parameters i j hij input) P) | |
| (funext (fun context => childRead_law parameters i j hij input context))) | |
| def read (sample : Sample parameters i j hij input) : | |
| FixedStoppedPhysicalLaw.PhysicalSample parameters i j hij input := | |
| FixedStoppedPhysicalLaw.read parameters i j hij input | |
| (sourceRead parameters i j hij input sample) | |
| theorem read_law : | |
| (law parameters i j hij input).pushforward (read parameters i j hij input) = | |
| FixedStoppedPhysicalLaw.physicalLaw parameters i j hij input := by | |
| change (law parameters i j hij input).pushforward | |
| (fun sample => FixedStoppedPhysicalLaw.read parameters i j hij input | |
| (sourceRead parameters i j hij input sample)) = _ | |
| rw [← FiniteDistribution.pushforward_comp, sourceRead_law] | |
| exact FixedStoppedPhysicalLaw.read_law parameters i j hij input | |
| theorem expectation_read | |
| (value : FixedStoppedPhysicalLaw.PhysicalSample parameters i j hij input → ℝ) : | |
| (law parameters i j hij input).expectation | |
| (fun sample => value (read parameters i j hij input sample)) = | |
| (FixedStoppedPhysicalLaw.physicalLaw parameters i j hij input).expectation value := by | |
| rw [← read_law parameters i j hij input, FiniteDistribution.expectation_pushforward] | |
| end | |
| end PerfectCompleteness.FixedStoppedNativeLaw | |
| end | |
| end OAI | |