openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedStoppedDecoderContext.lean
| import OAI.Computability.PerfectCompleteness.Decoding.DecoderFamilyLawLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.CanonicalDirections | |
| import OAI.Computability.PerfectCompleteness.Foundations.GeometricPrefixSplit | |
| import OAI.Computability.PerfectCompleteness.Repetition.StoppedCleanGeometry | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedStoppedDecoderContext | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) | |
| attribute [local instance] linearMapFintype | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : FixedParameters.Parameters δ hδ) | |
| (i j : Fin parameters.plan.depth) (hij : i < j) | |
| abbrev Prefixes := | |
| GeometricCutSplit.Prefix (FixedParameters.branch parameters) parameters.plan.depth (j.val + 1) × | |
| GeometricCutSplit.Prefix (FixedParameters.branch parameters) (j.val + 1) (i.val + 1) | |
| def upperPath (pref : Prefixes parameters i j) : | |
| Path (FixedParameters.branch parameters) parameters.plan.depth (j.val + 1) := | |
| GeometricCutSplit.prefixPath (Nat.succ_le_of_lt j.isLt) pref.1 | |
| def lowerPath (pref : Prefixes parameters i j) : | |
| Path (FixedParameters.branch parameters) (j.val + 1) (i.val + 1) := | |
| GeometricCutSplit.prefixPath (Nat.succ_le_succ (Nat.le_of_lt hij)) pref.2 | |
| def upper (pref : Prefixes parameters i j) : Nodes (FixedParameters.branch parameters) parameters.plan.depth := | |
| WholeArrayInteriorExterior.upperNode (upperPath parameters i j pref) | |
| def lower (pref : Prefixes parameters i j) : | |
| HierarchicalFrozenTables.LowerNodes (upper parameters i j pref) (i.val + 1) := | |
| StoppedCleanGeometry.descendant (upperPath parameters i j pref) | |
| (lowerPath parameters i j hij pref) hij | |
| @[simp] theorem upper_height (pref : Prefixes parameters i j) : | |
| Nodes.height (upper parameters i j pref) = j.val + 1 := | |
| WholeArrayInteriorExterior.upperNode_height _ | |
| def prefixLaw : FiniteDistribution (Prefixes parameters i j) := | |
| (GeometricCutSplit.prefixLaw (Nat.succ_le_of_lt j.isLt) | |
| (fun k _ => FixedParameters.branch_pos parameters k)).product | |
| (GeometricCutSplit.prefixLaw (Nat.succ_le_succ (Nat.le_of_lt hij)) | |
| (fun k _ => FixedParameters.branch_pos parameters k)) | |
| variable (input : List Bool) | |
| abbrev OutsideQuestions (pref : Prefixes parameters i j) := | |
| StoppedCleanGeometry.OutsideQuestions parameters input | |
| (upperPath parameters i j pref) (lowerPath parameters i j hij pref) | |
| abbrev Exterior (pref : Prefixes parameters i j) | |
| (external : OutsideQuestions parameters i j hij input pref) := | |
| StoppedCleanGeometry.Exterior parameters input | |
| (upperPath parameters i j pref) (lowerPath parameters i j hij pref) external | |
| abbrev Advice (pref : Prefixes parameters i j) := | |
| ManyGoodRows.RowMap (Block (FixedRows.rows parameters.plan) (upper parameters i j pref)) | |
| parameters.plan.order | |
| abbrev Directions := CanonicalDirections.Tuple (FixedRows.rows parameters.plan) parameters.plan.depth | |
| abbrev Fiber (pref : Prefixes parameters i j) | |
| (external : OutsideQuestions parameters i j hij input pref) := | |
| Advice parameters i j pref × | |
| (Directions parameters × Exterior parameters i j hij input pref external) | |
| abbrev Context := | |
| (pref : Prefixes parameters i j) × | |
| (external : OutsideQuestions parameters i j hij input pref) × | |
| Fiber parameters i j hij input pref external | |
| def directionLaw : FiniteDistribution (Directions parameters) := | |
| CanonicalDirections.law (FixedRows.rows parameters.plan) parameters.plan.depth | |
| (fun k => parameters.plan.rows_pos (parameters.plan.depth - (k + 1))) | |
| def fiberLaw (pref : Prefixes parameters i j) | |
| (external : OutsideQuestions parameters i j hij input pref) : | |
| FiniteDistribution (Fiber parameters i j hij input pref external) := | |
| (FiniteDistribution.uniform (Advice parameters i j pref)).product | |
| ((directionLaw parameters).product | |
| (FiniteDistribution.uniform (Exterior parameters i j hij input pref external))) | |
| def law : FiniteDistribution (Context parameters i j hij input) := | |
| CompletionSoundness.sigmaLaw (prefixLaw parameters i j hij) (fun pref => | |
| CompletionSoundness.sigmaLaw | |
| (FiniteDistribution.uniform (OutsideQuestions parameters i j hij input pref)) | |
| (fiberLaw parameters i j hij input pref)) | |
| def upperNode (context : Context parameters i j hij input) := upper parameters i j context.1 | |
| def lowerIndex (context : Context parameters i j hij input) := lower parameters i j hij context.1 | |
| def geometry (context : Context parameters i j hij input) : | |
| FixedDecoderCleanRate.Geometry parameters parameters.plan.depth i.val | |
| (upperNode parameters i j hij input context) (lowerIndex parameters i j hij input context) := | |
| StoppedCleanGeometry.geometry parameters input | |
| (upperPath parameters i j context.1) (lowerPath parameters i j hij context.1) hij | |
| context.2.1 context.2.2.2.2 | |
| def advice (context : Context parameters i j hij input) : | |
| ManyGoodRows.RowMap | |
| (Block (FixedRows.rows parameters.plan) (upperNode parameters i j hij input context)) | |
| parameters.plan.order := context.2.2.1 | |
| def direction (context : Context parameters i j hij input) : | |
| OwnInputReference.LowerVector (FixedRows.rows parameters.plan) | |
| (HierarchicalLeftDecoder.LowerNode (upperNode parameters i j hij input context) | |
| (i.val + 1) (lowerIndex parameters i j hij input context)) := | |
| (PrefixTests.nodeDirection (FixedRows.rows parameters.plan) i | |
| ⟨HierarchicalLeftDecoder.LowerNode (upperNode parameters i j hij input context) | |
| (i.val + 1) (lowerIndex parameters i j hij input context), | |
| CleanDecoderContext.lowerHeight (upperNode parameters i j hij input context) | |
| (lowerIndex parameters i j hij input context)⟩ (context.2.2.2.1 i)).val | |
| theorem row_count (context : Context parameters i j hij input) : | |
| FixedRows.rows parameters.plan (Nodes.height (upperNode parameters i j hij input context)) = | |
| FixedRows.rows parameters.plan (j.val + 1) := by | |
| rw [upperNode, upper_height] | |
| theorem advice_marginal (pref : Prefixes parameters i j) | |
| (external : OutsideQuestions parameters i j hij input pref) : | |
| (fiberLaw parameters i j hij input pref external).pushforward Prod.fst = | |
| FiniteDistribution.uniform (Advice parameters i j pref) := | |
| HiddenBucketBias.product_fst _ _ | |
| end | |
| end PerfectCompleteness.FixedStoppedDecoderContext | |
| end | |
| end OAI | |