openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedSourceLowerCollision.lean
| import OAI.Computability.PerfectCompleteness.Decoding.FixedSourceChildCollision | |
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutScalarProjectionLemmas | |
| import OAI.Computability.PerfectCompleteness.Decoding.LowerCutUniformCoordinatesLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedSourceLowerCollision | |
| noncomputable section | |
| open scoped Classical BigOperators | |
| open FixedParameters FixedRows RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| variable {δ : ℚ} {hδ : 0 < δ} (parameters : Parameters δ hδ) | |
| {height v m : Nat} [NeZero m] | |
| (path : Path (branch parameters) parameters.plan.depth (height + 1)) | |
| (outside : Slots (branch parameters) parameters.plan.depth → | |
| Fin (sourceLength parameters.plan hδ) → MixedSupport.Slot) | |
| (placeholder : Slots (branch parameters) (height + 1) → | |
| Fin (sourceLength parameters.plan hδ) → MixedSupport.Slot) | |
| (clauses : Fin m → SourceClause.NormalizedClause v) | |
| (designated : Fin (branch parameters height) → Slots (branch parameters) height) | |
| (upper : Nodes (branch parameters) parameters.plan.depth) | |
| (d : HierarchicalFrozenTables.LowerNodes upper (height + 1)) | |
| (hnode : WholeArrayInteriorExterior.upperNode path = | |
| HierarchicalLeftDecoder.LowerNode upper (height + 1) d) | |
| {adviceRows : Nat} | |
| (A : ManyGoodRows.RowMap (Block (rows parameters.plan) upper) adviceRows) | |
| (exterior : CleanPhysicalReplay.Exterior (rows parameters.plan) (repeats parameters.plan) | |
| path outside placeholder) | |
| abbrev Questions := FixedSourceChildCollision.Questions | |
| (m := m) parameters (height := height) | |
| abbrev Table (q : Questions (m := m) parameters (height := height)) := | |
| SourceQuestionLowerForms.UpperTable (rows := rows parameters.plan) | |
| path outside clauses upper (height + 1) (adviceRows := adviceRows) q | |
| local instance rowSpaceFintype (q : Questions (m := m) parameters (height := height)) : | |
| Fintype (NodeEmbedding.RowSpace (SourceQuestionLowerForms.slots path outside clauses q) upper) := | |
| Fintype.ofFinite _ | |
| local instance upperAnswerFintype (q : Questions (m := m) parameters (height := height)) : | |
| Fintype (HierarchicalAllDecoderTables.UpperAnswer | |
| (SourceQuestionLowerForms.slots path outside clauses q) upper) := | |
| LeftDecoder.dualFintype | |
| (V := NodeEmbedding.RowSpace (SourceQuestionLowerForms.slots path outside clauses q) upper) | |
| local instance tableFintype (q : Questions (m := m) parameters (height := height)) : | |
| Fintype (Table parameters path outside clauses upper (adviceRows := adviceRows) q) := | |
| inferInstanceAs (Fintype (HierarchicalAllDecoderTables.Input (rows := rows parameters.plan) | |
| (SourceQuestionLowerForms.slots path outside clauses q) upper (height + 1) adviceRows → | |
| HierarchicalAllDecoderTables.UpperAnswer | |
| (SourceQuestionLowerForms.slots path outside clauses q) upper)) | |
| abbrev TableFamily := (q : Questions (m := m) parameters (height := height)) → | |
| Table parameters path outside clauses upper (adviceRows := adviceRows) q | |
| local instance directionNonempty : | |
| Nonempty (BucketSampler.Direction (rows parameters.plan (height + 1))) := | |
| ⟨BucketUniform.coordinateDirection ⟨0, lt_of_lt_of_le Nat.zero_lt_one | |
| (parameters.plan.rows_pos (parameters.plan.depth - (height + 1)))⟩⟩ | |
| def first (tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows)) := | |
| SourceQuestionLowerForms.first path outside placeholder clauses upper (height + 1) d hnode | |
| (FixedLowerRawCollision.branchPositive parameters) A exterior tables (cutoff parameters.plan hδ) | |
| def second (tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows)) | |
| (direction : BucketSampler.Direction (rows parameters.plan (height + 1))) := | |
| SourceQuestionLowerForms.second path outside placeholder clauses upper (height + 1) d hnode | |
| (FixedLowerRawCollision.branchPositive parameters) A exterior tables (cutoff parameters.plan hδ) | |
| direction | |
| def markedProbability | |
| (tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows)) | |
| (direction : BucketSampler.Direction (rows parameters.plan (height + 1))) : ℝ := | |
| (FixedSourceChildCollision.markedLaw parameters path clauses designated).probability | |
| (FixedSourceChildCollision.restrictedEvent parameters path clauses designated | |
| (first parameters path outside placeholder clauses upper d hnode A exterior tables) | |
| (second parameters path outside placeholder clauses upper d hnode A exterior tables direction)) | |
| def uniformProbability | |
| (tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows)) | |
| (direction : BucketSampler.Direction (rows parameters.plan (height + 1))) : ℝ := | |
| (FixedSourceChildCollision.uniformLaw parameters path clauses).probability | |
| (FixedSourceChildCollision.fullEvent parameters path clauses | |
| (first parameters path outside placeholder clauses upper d hnode A exterior tables) | |
| (second parameters path outside placeholder clauses upper d hnode A exterior tables direction)) | |
| variable | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount (branch parameters) parameters.plan.depth | |
| (sourceLength parameters.plan hδ))) | |
| (useful : (q : Questions (m := m) parameters (height := height)) → | |
| (bg : HierarchicalMatrixTable.Background (rows := rows parameters.plan) | |
| (SourceQuestionLowerForms.slots path outside clauses q) upper) → | |
| HierarchicalFrozenTables.QuotientMatrix | |
| (SourceQuestionLowerForms.slots path outside clauses q) upper (height + 1) bg → Prop) | |
| (density : ℝ) | |
| def tableLaw : FiniteDistribution | |
| (TableFamily parameters path outside clauses upper (adviceRows := adviceRows)) := | |
| FiniteProduct.law (fun q => HierarchicalAllDecoderTables.upperTableLaw | |
| (SourceQuestionLowerForms.slots path outside clauses q) upper (height + 1) | |
| σ (useful q) adviceRows density) | |
| omit [NeZero m] in | |
| private theorem supported_row | |
| (tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows)) | |
| (hsupport : (tableLaw parameters path outside clauses upper σ useful density).weight tables ≠ 0) | |
| (q : Questions (m := m) parameters (height := height)) : | |
| (HierarchicalAllDecoderTables.upperTableLaw | |
| (SourceQuestionLowerForms.slots path outside clauses q) upper (height + 1) | |
| σ (useful q) adviceRows density).weight (tables q) ≠ 0 := by | |
| intro hz | |
| apply hsupport | |
| exact Finset.prod_eq_zero (Finset.mem_univ q) hz | |
| theorem supported_uniform_le (hdensity : 0 < density) | |
| (tables : TableFamily parameters path outside clauses upper (adviceRows := adviceRows)) | |
| (hsupport : (tableLaw parameters path outside clauses upper σ useful density).weight tables ≠ 0) : | |
| (FiniteDistribution.uniform (BucketSampler.Direction (rows parameters.plan (height + 1)))).expectation | |
| (uniformProbability parameters path outside placeholder clauses upper d hnode A exterior tables) ≤ | |
| FixedRankContradiction.gamma parameters (Nodes.height upper) ^ 2 / 8 := by | |
| have hprob : | |
| uniformProbability parameters path outside placeholder clauses upper d hnode A exterior tables = | |
| fun direction => (PreliminarySampler.questionsLaw | |
| (branch := branch parameters) (n := height + 1) | |
| (t := sourceLength parameters.plan hδ) (m := m)).expectation (fun q => | |
| (CutChildGrouping.rawLaw (C := FixedSourceChildCollision.Calls parameters path) | |
| (SourceChildMarkedLaw.questionSlots clauses q) (rows parameters.plan)).probability | |
| (fun raw => BilinearCollisionTransfer.fullCollision | |
| (first parameters path outside placeholder clauses upper d hnode A exterior tables q raw) | |
| (second parameters path outside placeholder clauses upper d hnode A exterior tables | |
| direction q raw))) := by | |
| funext direction | |
| exact CompletionSoundness.sigmaLaw_probability _ _ _ | |
| rw [hprob, FiniteDistribution.expectation_comm] | |
| calc | |
| _ ≤ (PreliminarySampler.questionsLaw (branch := branch parameters) (n := height + 1) | |
| (t := sourceLength parameters.plan hδ) (m := m)).expectation | |
| (fun _ => FixedRankContradiction.gamma parameters (Nodes.height upper) ^ 2 / 8) := by | |
| apply SmallBias.expectation_mono | |
| intro q | |
| have heq : | |
| (FiniteDistribution.uniform (BucketSampler.Direction | |
| (rows parameters.plan (height + 1)))).expectation | |
| (fun direction => (CutChildGrouping.rawLaw | |
| (C := FixedSourceChildCollision.Calls parameters path) | |
| (SourceChildMarkedLaw.questionSlots clauses q) (rows parameters.plan)).probability | |
| (fun raw => BilinearCollisionTransfer.fullCollision | |
| (first parameters path outside placeholder clauses upper d hnode A exterior tables q raw) | |
| (second parameters path outside placeholder clauses upper d hnode A exterior tables | |
| direction q raw))) = | |
| FixedLowerRawCollision.fullCollisionProbability parameters path | |
| (SourceQuestionLowerForms.slots path outside clauses q) upper d hnode A | |
| (CleanPhysicalReplay.exteriorAt (rows parameters.plan) (repeats parameters.plan) | |
| path outside placeholder (SourceChildMarkedLaw.questionSlots clauses q) exterior) | |
| (tables q) := by | |
| apply FiniteDistribution.expectation_congr | |
| intro direction | |
| exact SourceQuestionLowerForms.fullCollision_probability path outside placeholder clauses upper | |
| (height + 1) d hnode (FixedLowerRawCollision.branchPositive parameters) A exterior tables | |
| (cutoff parameters.plan hδ) direction q | |
| rw [heq] | |
| exact FixedLowerRawCollision.supported_fullCollision_le parameters path | |
| (SourceQuestionLowerForms.slots path outside clauses q) upper d hnode A | |
| (CleanPhysicalReplay.exteriorAt (rows parameters.plan) (repeats parameters.plan) | |
| path outside placeholder (SourceChildMarkedLaw.questionSlots clauses q) exterior) | |
| σ (useful q) density hdensity (tables q) | |
| (supported_row parameters path outside clauses upper σ useful density tables hsupport q) | |
| _ = _ := SmallBias.expectation_const _ _ | |
| private theorem expectation_add_const {X : Type*} [Fintype X] | |
| (μ : FiniteDistribution X) (f : X → ℝ) (c : ℝ) : | |
| μ.expectation (fun x => f x + c) = μ.expectation f + c := by | |
| simp only [FiniteDistribution.expectation, mul_add, Finset.sum_add_distrib, ← Finset.sum_mul, μ.normalized, one_mul] | |
| theorem mean_marked_le (hdensity : 0 < density) : | |
| (tableLaw parameters path outside clauses upper σ useful density).expectation | |
| (fun tables => (FiniteDistribution.uniform | |
| (BucketSampler.Direction (rows parameters.plan (height + 1)))).expectation | |
| (markedProbability parameters path outside placeholder clauses designated upper d hnode | |
| A exterior tables)) ≤ | |
| FixedRankContradiction.gamma parameters (Nodes.height upper) ^ 2 / 8 + | |
| 2 * parameters.accuracy := by | |
| apply DecoderTableAdmissibility.expectation_le_of_support | |
| intro tables hsupport | |
| have hpoint : ∀ direction : BucketSampler.Direction (rows parameters.plan (height + 1)), | |
| markedProbability parameters path outside placeholder clauses designated upper d hnode | |
| A exterior tables direction ≤ | |
| uniformProbability parameters path outside placeholder clauses upper d hnode | |
| A exterior tables direction + 2 * parameters.accuracy := by | |
| intro direction | |
| exact FixedSourceChildCollision.restricted_probability_le parameters path clauses designated | |
| (first parameters path outside placeholder clauses upper d hnode A exterior tables) | |
| (second parameters path outside placeholder clauses upper d hnode A exterior tables direction) | |
| le_rfl | |
| have hmean := SmallBias.expectation_mono | |
| (FiniteDistribution.uniform (BucketSampler.Direction (rows parameters.plan (height + 1)))) hpoint | |
| rw [expectation_add_const] at hmean | |
| have huniform := supported_uniform_le parameters path outside placeholder clauses upper d hnode A exterior | |
| σ useful density hdensity tables hsupport | |
| linarith | |
| end | |
| end PerfectCompleteness.FixedSourceLowerCollision | |
| end | |
| end OAI | |