openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /FixedStoppedUsefulMass.lean
| import OAI.Computability.PerfectCompleteness.Algebra.PrefixMatrixAcceptance | |
| import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalAdviceExperimentLemmas | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedPhysicalStoppedComparison | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedStoppedPair | |
| import OAI.Computability.PerfectCompleteness.Repetition.CutChildCleanLawLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.StoppedProjectedGlobalLawLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedStoppedUsefulMass | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| private theorem probability_cast_array_law | |
| {branch : Nat → Nat} {n h t : Nat} (rows repeats : Nat → Nat) | |
| (p : Path branch n (h + 1)) | |
| {slots slots' : Slots branch n → Fin t → MixedSupport.Slot} | |
| (hs : slots = slots') (event : Arrays slots' rows → Bool) : | |
| (WholeArraySampler.law rows repeats p slots).probability | |
| (fun arrays => event (cast (congrArg (fun s => Arrays s rows) hs) arrays)) = | |
| (WholeArraySampler.law rows repeats p slots').probability event := by | |
| cases hs | |
| rfl | |
| 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) | |
| abbrev Outer := StoppedProjectedExperiment.Outer | |
| (branch := FixedParameters.branch parameters) (n := parameters.plan.depth) | |
| (j := j.val) (t := FixedRows.sourceLength parameters.plan hδ) | |
| (m := (PCPSource.normalizedFormula (BinaryLanguage.totalRename input)).clauses.length) | |
| abbrev Base := StoppedProjectedExperiment.PhysicalBase | |
| (branch := FixedParameters.branch parameters) (n := parameters.plan.depth) | |
| (i := i.val) (j := j.val) (t := FixedRows.sourceLength parameters.plan hδ) | |
| (FixedRows.rows parameters.plan) | |
| def flags : Fin (FixedParameters.branch parameters i.val) → FiniteDistribution Bool := | |
| fun _ => ProjectionPosterior.bernoulli (FixedParameters.projectionProbability parameters i.val : ℝ) | |
| (FixedPhysicalStoppedComparison.beta_nonneg parameters) | |
| (FixedPhysicalStoppedComparison.beta_le_one parameters) | |
| theorem flags_eq_projectionFlag (hcube : 0 < parameters.cubeSize i.val) : | |
| flags parameters i = fun _ => | |
| ProjectedCleanRate.projectionFlag (parameters.cubeSize i.val) hcube := by | |
| have hprob : (FixedParameters.projectionProbability parameters i.val : ℝ) = | |
| ProjectedCleanRate.projectionDensity (parameters.cubeSize i.val) := by | |
| simp [FixedParameters.projectionProbability, ProjectedCleanRate.projectionDensity] | |
| funext child | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro b | |
| cases b <;> simp [flags, ProjectionPosterior.bernoulli, | |
| ProjectedCleanRate.projectionFlag, RepetitionRate.bernoulli, hprob] | |
| def outerLaw : FiniteDistribution (Outer parameters j input) := | |
| StoppedProjectedExperiment.outerLaw (Nat.succ_le_of_lt j.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| def baseLaw : FiniteDistribution (Base parameters i j) := | |
| StoppedProjectedExperiment.physicalBaseLaw (FixedRows.rows parameters.plan) hij | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) | |
| def record (o : Outer parameters j input) | |
| (pref : GeometricCutSplit.Prefix (FixedParameters.branch parameters) (j.val + 1) (i.val + 1)) | |
| (directions : CanonicalDirections.Tuple (FixedRows.rows parameters.plan) parameters.plan.depth) | |
| (arrays : Arrays (StoppedProjectedExperiment.nativeSlots (clauses input) o) | |
| (FixedRows.rows parameters.plan)) : | |
| StoppedSharedSampler.Record (clauses input) (FixedParameters.branch parameters) | |
| parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ) i.val | |
| (FixedRows.rows parameters.plan) := | |
| ⟨(o.1, (GeometricPrefixSplit.joinEquiv (Nat.succ_le_succ hij.le) | |
| (Nat.succ_le_of_lt j.isLt) (o.2, pref), directions)), arrays⟩ | |
| variable (designated : Fin (FixedParameters.branch parameters i.val) → | |
| Slots (FixedParameters.branch parameters) i.val) | |
| abbrev PhysicalSample := StoppedProjectedPhysicalLaw.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 | |
| abbrev OriginalSample := StoppedProjectedGlobalLaw.ExperimentSample | |
| (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 | |
| def physicalLaw : FiniteDistribution (PhysicalSample parameters i j hij input designated) := | |
| StoppedProjectedPhysicalLaw.law | |
| (clauses input) (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij designated | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i) | |
| def physicalRecord (sample : PhysicalSample parameters i j hij input designated) := | |
| record parameters i j hij input sample.1 sample.2.1.1 sample.2.1.2.2 | |
| (cast (congrArg (fun slots => Arrays slots (FixedRows.rows parameters.plan)) | |
| (StoppedProjectedArrays.nativeSlots_eq (clauses input) (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij designated sample.1 sample.2.1)) | |
| (StoppedProjectedArrays.physicalArrays (clauses input) (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated | |
| sample.1 sample.2)) | |
| def referenceMass (event : StoppedSharedSampler.Record (clauses input) | |
| (FixedParameters.branch parameters) parameters.plan.depth | |
| (FixedRows.sourceLength parameters.plan hδ) i.val (FixedRows.rows parameters.plan) → Bool) : ℝ := | |
| (outerLaw parameters j input).expectation (fun o => | |
| (baseLaw parameters i j hij).expectation (fun base => | |
| (WholeArraySampler.law (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (StoppedProjectedExperiment.stoppedPath (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij o | |
| (StoppedProjectedExperiment.baseTag (FixedRows.rows parameters.plan) base)) | |
| (StoppedProjectedExperiment.nativeSlots (clauses input) o)).probability | |
| (fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays)))) | |
| theorem physical_fiber_probability_lower | |
| (o : Outer parameters j input) (base : Base parameters i j) | |
| (event : StoppedSharedSampler.Record (clauses input) (FixedParameters.branch parameters) | |
| parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ) i.val | |
| (FixedRows.rows parameters.plan) → Bool) : | |
| (WholeArraySampler.law (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (StoppedProjectedExperiment.stoppedPath (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij o | |
| (StoppedProjectedExperiment.baseTag (FixedRows.rows parameters.plan) base)) | |
| (StoppedProjectedExperiment.nativeSlots (clauses input) o)).probability | |
| (fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays)) - | |
| parameters.accuracy ≤ | |
| (StoppedProjectedExperiment.physicalFiberLaw (clauses input) (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated | |
| (flags parameters i) o base).probability | |
| (fun raw => event (physicalRecord parameters i j hij input designated ⟨o, ⟨base, raw⟩⟩)) := by | |
| let p := StoppedProjectedExperiment.stoppedPath (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij o | |
| (StoppedProjectedExperiment.baseTag (FixedRows.rows parameters.plan) base) | |
| let outside := StoppedProjectedExperiment.nativeSlots (clauses input) o | |
| let placeholder := SourceProjectedTag.originalSlots (clauses input) | |
| (StoppedProjectedExperiment.cutQuestions (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij o | |
| (StoppedProjectedExperiment.baseTag (FixedRows.rows parameters.plan) base)) | |
| let sources := StoppedProjectedExperiment.physicalSources (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij designated o base | |
| have hslots : CutSlotAssembly.fill p outside | |
| (SourceChildKernel.parentLeftSlots (clauses input) designated sources) = outside := by | |
| simpa only [p, outside, sources, SourcePhysicalTapeLaw.nativeSlots] using | |
| (StoppedProjectedArrays.nativeSlots_eq (clauses input) (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij designated o base) | |
| let observed := fun arrays : SourcePhysicalStoppedComparison.NativeArrays | |
| (FixedRows.rows parameters.plan) p outside (clauses input) designated sources => | |
| event (record parameters i j hij input o base.1 base.2.2 | |
| (cast (congrArg (fun slots => Arrays slots (FixedRows.rows parameters.plan)) hslots) arrays)) | |
| have htv := SourcePhysicalStoppedComparison.conditional_variation | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) p outside placeholder | |
| (clauses input) designated (FixedParameters.branch_pos parameters i.val) sources | |
| (FixedParameters.projectionProbability parameters i.val : ℝ) | |
| (FixedPhysicalStoppedComparison.beta_nonneg parameters) | |
| (FixedPhysicalStoppedComparison.beta_le_one parameters) | |
| have herror : SourceChildProjectionComparison.error (FixedParameters.branch parameters) | |
| i.val (FixedRows.sourceLength parameters.plan hδ) | |
| (Fintype.card (WholeCutCalls.Index (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p)) (FixedRows.rows parameters.plan) | |
| (FixedParameters.projectionProbability parameters i.val : ℝ) < parameters.accuracy := by | |
| simpa only [SourceChildProjectionComparison.error] using | |
| FixedProjectionError.sparse_error_lt parameters | |
| (FixedPhysicalStoppedComparison.calls_le parameters p (le_refl _)) i.val | |
| have hp := FiniteDistribution.probability_le_add_totalVariation | |
| (SourcePhysicalStoppedComparison.stoppedLaw (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p outside (clauses input) designated sources) | |
| (SourcePhysicalStoppedComparison.modifiedLaw (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p outside placeholder (clauses input) designated | |
| (FixedParameters.projectionProbability parameters i.val : ℝ) | |
| (FixedPhysicalStoppedComparison.beta_nonneg parameters) | |
| (FixedPhysicalStoppedComparison.beta_le_one parameters) sources) observed | |
| rw [FiniteDistribution.totalVariation_comm] at hp | |
| have hreference : | |
| (SourcePhysicalStoppedComparison.stoppedLaw (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p outside (clauses input) designated sources).probability | |
| observed = | |
| (WholeArraySampler.law (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| p outside).probability | |
| (fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays)) := by | |
| simpa only [SourcePhysicalStoppedComparison.stoppedLaw, observed] using | |
| (probability_cast_array_law (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p | |
| (slots := CutSlotAssembly.fill p outside | |
| (SourceChildKernel.parentLeftSlots (clauses input) designated sources)) | |
| (slots' := outside) hslots | |
| (fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays))) | |
| have hstrict := htv.trans_lt herror | |
| have hbound : | |
| (SourcePhysicalStoppedComparison.stoppedLaw (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p outside (clauses input) designated sources).probability | |
| observed - parameters.accuracy ≤ | |
| (SourcePhysicalStoppedComparison.modifiedLaw (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p outside placeholder (clauses input) designated | |
| (FixedParameters.projectionProbability parameters i.val : ℝ) | |
| (FixedPhysicalStoppedComparison.beta_nonneg parameters) | |
| (FixedPhysicalStoppedComparison.beta_le_one parameters) sources).probability observed := | |
| sub_le_iff_le_add.mpr (hp.trans (add_le_add (le_refl _) hstrict.le)) | |
| rw [hreference, SourcePhysicalStoppedComparison.modifiedLaw, | |
| FiniteDistribution.probability_pushforward] at hbound | |
| change (WholeArraySampler.law (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p outside).probability | |
| (fun arrays => event (record parameters i j hij input o base.1 base.2.2 arrays)) - | |
| parameters.accuracy ≤ | |
| (SourcePhysicalWholeLaw.rawLaw (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p outside placeholder (clauses input) designated | |
| (fun _ => ProjectionPosterior.bernoulli | |
| (FixedParameters.projectionProbability parameters i.val : ℝ) | |
| (FixedPhysicalStoppedComparison.beta_nonneg parameters) | |
| (FixedPhysicalStoppedComparison.beta_le_one parameters)) sources).probability | |
| (fun raw => observed (SourcePhysicalWholeLaw.observeLeft (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) p outside placeholder (clauses input) designated sources raw)) | |
| exact hbound | |
| theorem physical_probability_lower | |
| (event : StoppedSharedSampler.Record (clauses input) (FixedParameters.branch parameters) | |
| parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ) i.val | |
| (FixedRows.rows parameters.plan) → Bool) : | |
| referenceMass parameters i j hij input event - parameters.accuracy ≤ | |
| (physicalLaw parameters i j hij input designated).probability | |
| (fun sample => event (physicalRecord parameters i j hij input designated sample)) := by | |
| have hmean := SmallBias.expectation_mono (outerLaw parameters j input) (fun o => | |
| SmallBias.expectation_mono (baseLaw parameters i j hij) (fun base => | |
| physical_fiber_probability_lower parameters i j hij input designated o base event)) | |
| simp only [DensityVariation.expectation_sub, SmallBias.expectation_const] at hmean | |
| simpa only [referenceMass, physicalLaw, StoppedProjectedPhysicalLaw.law, | |
| StoppedProjectedExperiment.physicalLaw, CompletionSoundness.sigmaLaw_probability, | |
| outerLaw, baseLaw] using hmean | |
| theorem referenceMass_eq_stopped | |
| (event : StoppedSharedSampler.Record (clauses input) | |
| (FixedParameters.branch parameters) parameters.plan.depth | |
| (FixedRows.sourceLength parameters.plan hδ) i.val | |
| (FixedRows.rows parameters.plan) → Bool) : | |
| referenceMass parameters i j hij input event = | |
| (StoppedSharedSampler.stoppedLaw (clauses input) | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).probability event := by | |
| simp only [referenceMass, outerLaw, baseLaw, | |
| StoppedProjectedExperiment.outerLaw, StoppedProjectedExperiment.physicalBaseLaw, | |
| StoppedSharedSampler.stoppedLaw, CompletionSoundness.sigmaLaw_probability, | |
| StoppedSharedSampler.contextLaw, FiniteDistribution.expectation_product] | |
| apply FiniteDistribution.expectation_congr | |
| intro questions | |
| simp only [record, StoppedProjectedExperiment.stoppedPath, | |
| StoppedProjectedExperiment.upperPath, StoppedProjectedExperiment.lowerPath, | |
| StoppedProjectedExperiment.baseTag, StoppedProjectedExperiment.withFlags, | |
| StoppedProjectedExperiment.nativeSlots, SmallBias.expectation_const] | |
| let value := fun | |
| (upper : GeometricCutSplit.Prefix (FixedParameters.branch parameters) | |
| parameters.plan.depth (j.val + 1)) | |
| (lower : GeometricCutSplit.Prefix (FixedParameters.branch parameters) | |
| (j.val + 1) (i.val + 1)) => | |
| (CanonicalDirections.law (FixedRows.rows parameters.plan) parameters.plan.depth | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).expectation | |
| (fun directions => | |
| (WholeArraySampler.law (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) | |
| ((GeometricCutSplit.prefixPath (Nat.succ_le_of_lt j.isLt) upper).append | |
| (GeometricCutSplit.prefixPath (Nat.succ_le_succ hij.le) lower)) | |
| (sourceSlots (clauses input) (PreliminarySampler.endpoints questions))).probability | |
| (fun arrays => event | |
| ⟨(questions, (GeometricPrefixSplit.joinEquiv (Nat.succ_le_succ hij.le) | |
| (Nat.succ_le_of_lt j.isLt) (upper, lower), directions)), arrays⟩)) | |
| calc | |
| _ = (GeometricCutSplit.prefixLaw (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k)).expectation | |
| (fun pref => value | |
| (GeometricPrefixSplit.splitEquiv (Nat.succ_le_succ hij.le) | |
| (Nat.succ_le_of_lt j.isLt) pref).1 | |
| (GeometricPrefixSplit.splitEquiv (Nat.succ_le_succ hij.le) | |
| (Nat.succ_le_of_lt j.isLt) pref).2) := | |
| (GeometricPrefixSplit.expectation_split (Nat.succ_le_succ hij.le) | |
| (Nat.succ_le_of_lt j.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k) value).symm | |
| _ = _ := by | |
| apply FiniteDistribution.expectation_congr | |
| intro pref | |
| dsimp only [value] | |
| rw [← GeometricPrefixSplit.prefixPath_split (Nat.succ_le_succ hij.le) | |
| (Nat.succ_le_of_lt j.isLt) pref, GeometricPrefixSplit.join_split] | |
| theorem stopped_probability_lower | |
| (event : StoppedSharedSampler.Record (clauses input) (FixedParameters.branch parameters) | |
| parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ) i.val | |
| (FixedRows.rows parameters.plan) → Bool) : | |
| (StoppedSharedSampler.stoppedLaw (clauses input) | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).probability event - | |
| parameters.accuracy ≤ | |
| (physicalLaw parameters i j hij input designated).probability | |
| (fun sample => event (physicalRecord parameters i j hij input designated sample)) := by | |
| rw [← referenceMass_eq_stopped parameters i j hij input event] | |
| exact physical_probability_lower parameters i j hij input designated event | |
| variable (strategy : FixedPreliminaryGame.Strategy parameters input) | |
| abbrev canonicalStrategy := PreliminaryStrategy.extend (clauses input) | |
| (FixedParameters.branch parameters) parameters.plan.depth | |
| (FixedRows.sourceLength parameters.plan hδ) (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) strategy | |
| abbrev experiment (o : Outer parameters j input) := | |
| StoppedProjectedGlobalLaw.experiment (clauses input) (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i) | |
| (canonicalStrategy parameters input strategy) o | |
| abbrev OriginalFiber (o : Outer parameters j input) := | |
| HierarchicalProjectedExperiment.Sample (rows := FixedRows.rows parameters.plan) | |
| (repeats := FixedRows.repeats parameters.plan) | |
| (StoppedProjectedExperiment.projectedSlots (clauses input) (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij designated o) | |
| (StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o) | |
| (StoppedProjectedExperiment.lower (FixedRows.rows parameters.plan) (Nat.succ_le_of_lt j.isLt) hij o) | |
| (StoppedProjectedExperiment.cut (FixedRows.rows parameters.plan) (Nat.succ_le_of_lt j.isLt) hij o) | |
| def originalLaw : FiniteDistribution (OriginalSample parameters i j hij input designated) := | |
| StoppedProjectedGlobalLaw.experimentLaw (clauses input) (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i) | |
| (canonicalStrategy parameters input strategy) | |
| def originalRecord (sample : OriginalSample parameters i j hij input designated) := | |
| record parameters i j hij input sample.1 sample.2.1.1 sample.2.1.2.2 | |
| ((experiment parameters i j hij input designated strategy sample.1).arrays sample.2) | |
| def upperEvent (o : Outer parameters j input) | |
| (x : OriginalFiber parameters i j hij input designated o) : Bool := | |
| PrefixTests.observeAtLevelNode (clauses input) (canonicalStrategy parameters input strategy) | |
| o.1 ((experiment parameters i j hij input designated strategy o).arrays x) j | |
| ⟨StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o, | |
| StoppedProjectedExperiment.upper_height (Nat.succ_le_of_lt j.isLt) o⟩ | |
| (x.1.2.2 j) | |
| theorem lowerEvent_eq_win (o : Outer parameters j input) | |
| (x : OriginalFiber parameters i j hij input designated o) : | |
| HierarchicalAdviceExperiment.lowerEvent | |
| (experiment parameters i j hij input designated strategy o) x = | |
| StoppedSharedEvents.win (clauses input) strategy (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k) i | |
| (originalRecord parameters i j hij input designated strategy ⟨o, x⟩) := by | |
| symm | |
| unfold StoppedSharedEvents.win StoppedSharedEvents.representativeLeaf originalRecord record | |
| dsimp only | |
| rw [GeometricPrefixSplit.prefixPath_join] | |
| exact (PrefixMatrixAcceptance.observe_endpoint (clauses input) | |
| (canonicalStrategy parameters input strategy) o.1 | |
| (StoppedProjectedExperiment.stoppedPath (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij o x.1) | |
| (StoppedSharedEvents.referenceSuffix (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k)) | |
| ((experiment parameters i j hij input designated strategy o).arrays x) | |
| (x.1.2.2 i)).trans | |
| (PrefixMatrixAcceptance.observe_eq_lower (clauses input) | |
| (canonicalStrategy parameters input strategy) o.1 | |
| ((experiment parameters i j hij input designated strategy o).arrays x) i | |
| ⟨StoppedProjectedExperiment.lower (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij o x.1, | |
| StoppedProjectedExperiment.lower_height (FixedRows.rows parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij o x.1⟩ (x.1.2.2 i) | |
| (StoppedProjectedExperiment.upper (Nat.succ_le_of_lt j.isLt) o)) | |
| theorem upperEvent_eq_win (o : Outer parameters j input) | |
| (x : OriginalFiber parameters i j hij input designated o) : | |
| upperEvent parameters i j hij input designated strategy o x = | |
| StoppedSharedEvents.win (clauses input) strategy (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k) j | |
| (originalRecord parameters i j hij input designated strategy ⟨o, x⟩) := by | |
| symm | |
| unfold StoppedSharedEvents.win StoppedSharedEvents.representativeLeaf originalRecord record | |
| dsimp only | |
| rw [GeometricPrefixSplit.prefixPath_join, Path.slotEmbedding_append] | |
| exact PrefixMatrixAcceptance.observe_endpoint (clauses input) | |
| (canonicalStrategy parameters input strategy) o.1 | |
| (StoppedProjectedExperiment.upperPath (Nat.succ_le_of_lt j.isLt) o) | |
| ((StoppedProjectedExperiment.lowerPath (FixedRows.rows parameters.plan) hij x.1).slotEmbedding | |
| (StoppedSharedEvents.referenceSuffix (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k))) | |
| ((experiment parameters i j hij input designated strategy o).arrays x) (x.1.2.2 j) | |
| theorem originalRecord_physicalRead | |
| (sample : PhysicalSample parameters i j hij input designated) : | |
| originalRecord parameters i j hij input designated strategy | |
| (StoppedProjectedPhysicalLaw.read (clauses input) (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) (Nat.succ_le_of_lt j.isLt) hij designated sample) = | |
| physicalRecord parameters i j hij input designated sample := by | |
| exact congrArg (record parameters i j hij input sample.1 sample.2.1.1 sample.2.1.2.2) | |
| (StoppedProjectedArrays.arrays_physicalExperimentRead_eq (clauses input) | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij designated | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i) | |
| (canonicalStrategy parameters input strategy) sample.1 sample.2) | |
| theorem joint_probability_eq_physical : | |
| (originalLaw parameters i j hij input designated strategy).probability | |
| (HierarchicalUsefulFamily.jointEvent | |
| (experiment parameters i j hij input designated strategy) | |
| (upperEvent parameters i j hij input designated strategy)) = | |
| (physicalLaw parameters i j hij input designated).probability (fun sample => | |
| StoppedSharedEvents.pair (clauses input) strategy (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k) i j | |
| (physicalRecord parameters i j hij input designated sample)) := by | |
| have hevent : HierarchicalUsefulFamily.jointEvent | |
| (experiment parameters i j hij input designated strategy) | |
| (upperEvent parameters i j hij input designated strategy) = | |
| (fun sample => StoppedSharedEvents.pair (clauses input) strategy | |
| (Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k) i j | |
| (originalRecord parameters i j hij input designated strategy sample)) := by | |
| funext sample | |
| exact congrArg₂ Bool.and | |
| (lowerEvent_eq_win parameters i j hij input designated strategy sample.1 sample.2) | |
| (upperEvent_eq_win parameters i j hij input designated strategy sample.1 sample.2) | |
| rw [hevent] | |
| have hread := StoppedProjectedPhysicalLaw.probability_read (clauses input) | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij designated | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) (flags parameters i) | |
| (canonicalStrategy parameters input strategy) | |
| (fun sample => StoppedSharedEvents.pair (clauses input) strategy | |
| (Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k) i j | |
| (originalRecord parameters i j hij input designated strategy sample)) | |
| simpa only [originalRecord_physicalRead, originalLaw, physicalLaw] using hread.symm | |
| theorem joint_probability_ge_of_reserve | |
| (hreserve : InitialParameters.simultaneous δ + 2 * parameters.accuracy ≤ | |
| (StoppedSharedSampler.stoppedLaw (clauses input) | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).probability | |
| (StoppedSharedEvents.pair (clauses input) strategy (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k) i j)) : | |
| InitialParameters.simultaneous δ ≤ | |
| (originalLaw parameters i j hij input designated strategy).probability | |
| (HierarchicalUsefulFamily.jointEvent | |
| (experiment parameters i j hij input designated strategy) | |
| (upperEvent parameters i j hij input designated strategy)) := by | |
| rw [joint_probability_eq_physical] | |
| have hcompare := stopped_probability_lower parameters i j hij input designated | |
| (StoppedSharedEvents.pair (clauses input) strategy (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k) i j) | |
| have haccuracy := parameters.accuracy_pos | |
| linarith | |
| theorem useful_probability_ge_of_reserve | |
| (hreserve : InitialParameters.simultaneous δ + 2 * parameters.accuracy ≤ | |
| (StoppedSharedSampler.stoppedLaw (clauses input) | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt i.isLt) (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1))).probability | |
| (StoppedSharedEvents.pair (clauses input) strategy (Nat.succ_le_of_lt i.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k) i j)) : | |
| InitialParameters.simultaneous δ / 2 ≤ | |
| (originalLaw parameters i j hij input designated strategy).probability | |
| (HierarchicalUsefulFamily.usefulEvent | |
| (experiment parameters i j hij input designated strategy) (InitialParameters.useful δ) | |
| (upperEvent parameters i j hij input designated strategy)) := by | |
| exact HierarchicalUsefulFamily.useful_probability_ge_half | |
| (experiment parameters i j hij input designated strategy) (outerLaw parameters j input) | |
| (upperEvent parameters i j hij input designated strategy) (InitialParameters.simultaneous δ) | |
| (InitialParameters.simultaneous_pos hδ).le | |
| (joint_probability_ge_of_reserve parameters i j hij input designated strategy hreserve) | |
| end | |
| end PerfectCompleteness.FixedStoppedUsefulMass | |
| end | |
| end OAI | |