openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedDecoderCleanRateLemmas.lean
| import OAI.Computability.PerfectCompleteness.Decoding.CleanDecoderRateLemmas | |
| import OAI.Computability.PerfectCompleteness.Repetition.FixedCleanRate | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedDecoderCleanRate | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| abbrev F2 := ZMod 2 | |
| structure Geometry {δ : ℚ} {hδ : 0 < δ} (p : FixedParameters.Parameters δ hδ) | |
| (n h : Nat) (upper : Nodes (FixedParameters.branch p) n) | |
| (d : HierarchicalFrozenTables.LowerNodes upper (h + 1)) where | |
| designated : Fin (FixedParameters.branch p h) → Slots (FixedParameters.branch p) h | |
| cut : OwnInputReference.Cut upper (HierarchicalLeftDecoder.LowerNode upper (h + 1) d) | |
| outside : Slots (FixedParameters.branch p) n → | |
| Fin (FixedRows.sourceLength p.plan hδ) → MixedSupport.Slot | |
| placeholder : Slots (FixedParameters.branch p) (h + 1) → | |
| Fin (FixedRows.sourceLength p.plan hδ) → MixedSupport.Slot | |
| exterior : CleanPhysicalReplay.Exterior (FixedRows.rows p.plan) (FixedRows.repeats p.plan) | |
| (CleanRecordDecoderInput.path upper d cut) outside placeholder | |
| variable {δ : ℚ} {hδ : 0 < δ} (p : FixedParameters.Parameters δ hδ) | |
| {n h : Nat} {upper : Nodes (FixedParameters.branch p) n} | |
| {d : HierarchicalFrozenTables.LowerNodes upper (h + 1)} | |
| (G : Geometry p n h upper d) | |
| def setup (input : List Bool) : | |
| CleanDecoderRate.Setup (FixedParameters.branch p) (FixedRows.rows p.plan) | |
| (FixedRows.repeats p.plan) n h (FixedRows.sourceLength p.plan hδ) | |
| (PCPSource.normalizedFormula (BinaryLanguage.totalRename input)).variables | |
| (PCPSource.normalizedFormula (BinaryLanguage.totalRename input)).clauses.length | |
| upper d where | |
| clauses := PCPSource.clauseFamily (BinaryLanguage.totalRename input) | |
| designated := G.designated | |
| cut := G.cut | |
| outside := G.outside | |
| placeholder := G.placeholder | |
| exterior := G.exterior | |
| theorem calls_le (input : List Bool) (hn : n ≤ p.plan.depth) : | |
| Fintype.card (CleanDecoderRate.Calls (setup p G input)) ≤ FixedParameters.calls p.plan := by | |
| have hactual := FixedCallBudget.modified_calls_le (FixedRows.rows p.plan) | |
| (FixedRows.repeats p.plan) (CleanDecoderRate.path (setup p G input)) hn | |
| apply hactual.trans | |
| unfold FixedParameters.calls | |
| omega | |
| theorem cubePositive (height : Nat) : 0 < p.cubeSize height := by | |
| have hsize := (p.cube_errors height).1 | |
| omega | |
| theorem branchPositive : ∀ k < n, 0 < FixedParameters.branch p k := | |
| fun k _ => FixedParameters.branch_pos p k | |
| variable (input : List Bool) | |
| (σ : KeyStrategy.Strategy | |
| (TreeCanonical.locationCount (FixedParameters.branch p) n | |
| (FixedRows.sourceLength p.plan hδ))) | |
| (useful : (record : CleanDecoderRate.Record (setup p G input)) → | |
| CleanLeftDecoder.Useful | |
| (CleanDecoderContext.leftExposed (CleanDecoderRate.context (setup p G input) record)) | |
| upper (h + 1)) | |
| (r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block (FixedRows.rows p.plan) upper) r) | |
| (W : Submodule F2 (OwnInputReference.UpperVector (FixedRows.rows p.plan) upper)) | |
| (a : OwnInputReference.LowerVector (FixedRows.rows p.plan) | |
| (HierarchicalLeftDecoder.LowerNode upper (h + 1) d)) | |
| (threshold : ℝ) | |
| theorem probability_lt_accuracy (unsat : ¬ BinaryLanguage.language input) | |
| (hn : n ≤ p.plan.depth) (hρ : 0 < ρ) : | |
| (CleanDecoderRate.law (setup p G input) σ useful r ρ A W a threshold | |
| (p.cubeSize h) (cubePositive p h)).probability | |
| (fun z => CleanDecoderRate.agrees (setup p G input) (branchPositive p) | |
| (FixedRows.cutoff p.plan hδ) z.1 z.2) < p.accuracy := by | |
| have hbound := CleanDecoderRate.bound (setup p G input) σ useful r ρ A W a threshold | |
| (branchPositive p) (cubePositive p h) hρ (FixedParameters.source_rate p input unsat) | |
| apply hbound.trans_lt | |
| simpa only [FixedParameters.branch] using | |
| FixedCleanRate.actual_clean_error p (calls_le p G input hn) h | |
| end | |
| end PerfectCompleteness.FixedDecoderCleanRate | |
| end | |
| section | |
| namespace PerfectCompleteness.FixedDecoderPairRate | |
| noncomputable section | |
| open scoped Classical | |
| open TreeSourceSpaces HierarchicalArrays FixedDecoderCleanRate | |
| abbrev F2 := ZMod 2 | |
| variable {δ : ℚ} {hδ : 0 < δ} (p : FixedParameters.Parameters δ hδ) | |
| {n h : Nat} {upper : Nodes (FixedParameters.branch p) n} | |
| {d : HierarchicalFrozenTables.LowerNodes upper (h + 1)} | |
| (G : Geometry p n h upper d) | |
| (input : List Bool) | |
| (σ : KeyStrategy.Strategy | |
| (TreeCanonical.locationCount (FixedParameters.branch p) n | |
| (FixedRows.sourceLength p.plan hδ))) | |
| (useful : (record : CleanDecoderRate.Record (setup p G input)) → | |
| CleanLeftDecoder.Useful | |
| (CleanDecoderContext.leftExposed (CleanDecoderRate.context (setup p G input) record)) | |
| upper (h + 1)) | |
| (r : Nat) (ρ : ℝ) (A : ManyGoodRows.RowMap (Block (FixedRows.rows p.plan) upper) r) | |
| (W : Submodule F2 (OwnInputReference.UpperVector (FixedRows.rows p.plan) upper)) | |
| (a : OwnInputReference.LowerVector (FixedRows.rows p.plan) | |
| (HierarchicalLeftDecoder.LowerNode upper (h + 1) d)) | |
| (threshold : ℝ) | |
| theorem probability_lt_accuracy (unsat : ¬ BinaryLanguage.language input) | |
| (hn : n ≤ p.plan.depth) (hρ : 0 < ρ) : | |
| (CleanDecoderPairLaw.pairLaw (setup p G input) σ useful r ρ A W a threshold | |
| (p.cubeSize h) (cubePositive p h)).probability | |
| (fun z => CleanDecoderPairLaw.pairAgrees (setup p G input) (branchPositive p) | |
| (FixedRows.cutoff p.plan hδ) z.1 z.2) < p.accuracy := by | |
| rw [CleanDecoderPairLaw.pair_probability_eq] | |
| exact FixedDecoderCleanRate.probability_lt_accuracy p G input σ useful r ρ A W a threshold | |
| unsat hn hρ | |
| end | |
| end PerfectCompleteness.FixedDecoderPairRate | |
| end | |
| end OAI | |