openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedStoppedDecoderLaw.lean
| import OAI.Computability.PerfectCompleteness.Decoding.DecoderFamilyLawLemmas | |
| import OAI.Computability.PerfectCompleteness.Decoding.FixedStoppedDecoderContext | |
| import OAI.Computability.PerfectCompleteness.Foundations.SlotProjectedExperiment | |
| import OAI.Computability.PerfectCompleteness.Reduction.FixedPhysicalStoppedComparison | |
| import OAI.Computability.PerfectCompleteness.Reduction.PreliminaryOutputLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedStoppedDecoderLaw | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| private abbrev familySampleFintype | |
| {branch rows repeats : Nat → Nat} {n h t v m : Nat} [NeZero m] | |
| {O : Type*} [Fintype O] | |
| {upper : O → Nodes branch n} | |
| {lower : (o : O) → HierarchicalFrozenTables.LowerNodes (upper o) (h + 1)} | |
| (S : (o : O) → CleanDecoderRate.Setup branch rows repeats n h t v m (upper o) (lower o)) : | |
| Fintype (DecoderFamilyLaw.Sample S) := inferInstance | |
| attribute [local instance] linearMapFintype | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ) | |
| (i j : Fin parameters.plan.depth) (hij : i < j) (input : List Bool) | |
| abbrev Context := FixedStoppedDecoderContext.Context parameters i j hij input | |
| def labeling (strategy : FixedPreliminaryGame.Strategy parameters input) : | |
| KeyStrategy.Strategy (TreeCanonical.locationCount (FixedParameters.branch parameters) | |
| parameters.plan.depth (FixedRows.sourceLength parameters.plan hδ)) := | |
| PreliminaryStrategy.extend (PCPSource.clauseFamily (BinaryLanguage.totalRename input)) | |
| (FixedParameters.branch parameters) parameters.plan.depth | |
| (FixedRows.sourceLength parameters.plan hδ) (FixedRows.rows parameters.plan) | |
| (FixedRows.repeats parameters.plan) strategy | |
| def flag : 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) | |
| def markFamily (strategy : FixedPreliminaryGame.Strategy parameters input) | |
| (pref : FixedStoppedDecoderContext.Prefixes parameters i j) := | |
| SlotProjectedExperiment.family | |
| (FixedRows.rows parameters.plan) (FixedRows.repeats parameters.plan) | |
| (Nat.succ_le_of_lt j.isLt) hij pref.1 | |
| (fun k _ => FixedParameters.branch_pos parameters k) | |
| (fun k => FixedPreliminaryGame.rows_pos parameters (k + 1)) | |
| (flag parameters i) (labeling parameters input strategy) | |
| abbrev setups := | |
| FixedDecoderFamilyRate.setups parameters | |
| (FixedStoppedDecoderContext.geometry parameters i j hij input) input | |
| def useful (strategy : FixedPreliminaryGame.Strategy parameters input) | |
| (context : Context parameters i j hij input) | |
| (record : CleanDecoderRate.Record (setups parameters i j hij input context)) : | |
| CleanLeftDecoder.Useful | |
| (CleanDecoderContext.leftExposed | |
| (CleanDecoderRate.context (setups parameters i j hij input context) record)) | |
| (FixedStoppedDecoderContext.upperNode parameters i j hij input context) (i.val + 1) := | |
| OriginalDecoderMark.useful | |
| (markFamily parameters i j hij input strategy context.1) | |
| (FixedStoppedDecoderContext.upperNode parameters i j hij input context) (i.val + 1) | |
| (InitialParameters.useful δ) | |
| (CleanDecoderContext.leftExposed | |
| (CleanDecoderRate.context (setups parameters i j hij input context) record)) | |
| def threshold : ℝ := | |
| UpperParameterScalars.tau (InitialParameters.useful δ) parameters.plan.density | |
| parameters.plan.order (FixedRows.rows parameters.plan (j.val + 1)) | |
| abbrev Sample := DecoderFamilyLaw.Sample (setups parameters i j hij input) | |
| local instance sampleFintype : Fintype (Sample parameters i j hij input) := | |
| familySampleFintype (setups parameters i j hij input) | |
| def law (strategy : FixedPreliminaryGame.Strategy parameters input) : | |
| FiniteDistribution (Sample parameters i j hij input) := | |
| DecoderFamilyLaw.law (setups parameters i j hij input) | |
| (labeling parameters input strategy) (useful parameters i j hij input strategy) | |
| parameters.plan.order parameters.plan.density | |
| (FixedStoppedDecoderContext.advice parameters i j hij input) (FixedStoppedDecoderContext.direction parameters i j hij input) | |
| (threshold parameters j) (FixedStoppedDecoderContext.law parameters i j hij input) | |
| (parameters.cubeSize i.val) (FixedDecoderCleanRate.cubePositive parameters i.val) | |
| abbrev meeting : Sample parameters i j hij input → Bool := | |
| DecoderFamilyLaw.meeting (setups parameters i j hij input) | |
| abbrev lowAgreement : Sample parameters i j hij input → Bool := | |
| DecoderFamilyLaw.lowAgreement (setups parameters i j hij input) | |
| (FixedDecoderCleanRate.branchPositive parameters) | |
| (FixedRows.cutoff parameters.plan hδ) | |
| abbrev highAgreement : Sample parameters i j hij input → Bool := | |
| DecoderFamilyLaw.highAgreement (setups parameters i j hij input) | |
| (FixedDecoderCleanRate.branchPositive parameters) | |
| (FixedRows.cutoff parameters.plan hδ) | |
| theorem probability_split (strategy : FixedPreliminaryGame.Strategy parameters input) : | |
| (law parameters i j hij input strategy).probability (meeting parameters i j hij input) = | |
| (law parameters i j hij input strategy).probability | |
| (lowAgreement parameters i j hij input) + | |
| (law parameters i j hij input strategy).probability | |
| (highAgreement parameters i j hij input) := | |
| DecoderFamilyLaw.probability_split (setups parameters i j hij input) | |
| (FixedDecoderCleanRate.branchPositive parameters) | |
| (labeling parameters input strategy) (useful parameters i j hij input strategy) | |
| parameters.plan.order parameters.plan.density | |
| (FixedStoppedDecoderContext.advice parameters i j hij input) (FixedStoppedDecoderContext.direction parameters i j hij input) | |
| (threshold parameters j) (FixedStoppedDecoderContext.law parameters i j hij input) | |
| (FixedRows.cutoff parameters.plan hδ) (parameters.cubeSize i.val) | |
| (FixedDecoderCleanRate.cubePositive parameters i.val) | |
| theorem low_probability_lt_accuracy | |
| (strategy : FixedPreliminaryGame.Strategy parameters input) | |
| (unsat : ¬ BinaryLanguage.language input) : | |
| (law parameters i j hij input strategy).probability | |
| (lowAgreement parameters i j hij input) < parameters.accuracy := | |
| FixedDecoderFamilyRate.probability_lt_accuracy parameters | |
| (FixedStoppedDecoderContext.geometry parameters i j hij input) input | |
| (labeling parameters input strategy) (useful parameters i j hij input strategy) | |
| parameters.plan.order parameters.plan.density | |
| (FixedStoppedDecoderContext.advice parameters i j hij input) (FixedStoppedDecoderContext.direction parameters i j hij input) | |
| (threshold parameters j) (FixedStoppedDecoderContext.law parameters i j hij input) | |
| unsat (le_refl _) parameters.plan.density_pos | |
| end | |
| end PerfectCompleteness.FixedStoppedDecoderLaw | |
| end | |
| end OAI | |