openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /FixedLowerUniformCollision.lean
| import OAI.Computability.PerfectCompleteness.Construction.BucketFullAgreement | |
| import OAI.Computability.PerfectCompleteness.Decoding.HierarchicalLowerInverseLemmas | |
| import OAI.Computability.PerfectCompleteness.Foundations.MultiplicationFormInjectiveLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.FixedLowerUniformCollision | |
| noncomputable section | |
| open scoped BigOperators Classical | |
| open TreeSourceSpaces HierarchicalArrays | |
| open UniqueGamesTheorem.Foundations.Games | |
| attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype | |
| private theorem expectation_uniform_eq_expect {A : Type*} [Fintype A] [Nonempty A] | |
| (f : A → ℝ) : | |
| (FiniteDistribution.uniform A).expectation f = 𝔼 x : A, f x := by | |
| rw [FiniteDistribution.expectation_uniform, Fintype.expect_eq_sum_div_card] | |
| section Uniform | |
| variable {Ω Y : Type*} [Fintype Y] | |
| (V : Submodule F2 (Ω → F2)) [Fintype V] [FiniteDimensional F2 V] | |
| (ℓ : Nat) [Nonempty (BucketSampler.Direction ℓ)] | |
| def uniformLaw : FiniteDistribution | |
| (BucketSampler.Direction ℓ × (BucketMatrixResampling.Matrix V ℓ × V)) := | |
| (FiniteDistribution.uniform (BucketSampler.Direction ℓ)).product | |
| ((FiniteDistribution.uniform (BucketMatrixResampling.Matrix V ℓ)).product | |
| (FiniteDistribution.uniform V)) | |
| def uniformCollision (f : BucketMatrixResampling.Matrix V ℓ → Option Y) | |
| (x : BucketSampler.Direction ℓ × (BucketMatrixResampling.Matrix V ℓ × V)) : Bool := | |
| TwoResponseCollision.collision f | |
| (x.2.1, EvaluationMatrix.shift V x.2.1 x.1.val x.2.2) | |
| theorem uniform_collision_eq (f : BucketMatrixResampling.Matrix V ℓ → Option Y) : | |
| (uniformLaw V ℓ).probability (uniformCollision V ℓ f) = | |
| PartialTableInverse.nonzeroAgreement f := by | |
| unfold uniformLaw uniformCollision | |
| rw [BucketQuotientAgreement.probability_collision_eq_expectation] | |
| simp only [FiniteDistribution.expectation_product, expectation_uniform_eq_expect] | |
| calc | |
| _ = FiniteRangeInverse.agreement f := by | |
| unfold FiniteRangeInverse.agreement | |
| refine Finset.expect_congr (ι := BucketSampler.Direction ℓ) ?_ ?_ | |
| · ext a | |
| simp only [Finset.mem_univ] | |
| intro a _ | |
| rfl | |
| _ = _ := FiniteRangeInverse.agreement_eq_nonzeroAgreement f | |
| end Uniform | |
| variable {δ : ℚ} (plan : FixedRows.Plan δ) (hδ : 0 < δ) | |
| {branch : Nat → Nat} {t : Nat} | |
| (slots : RecursiveSpaces.Slots branch plan.depth → Fin t → MixedSupport.Slot) | |
| (upper : Nodes branch plan.depth) (lowerLevel : Nat) | |
| (background : HierarchicalMatrixTable.Background (rows := FixedRows.rows plan) slots upper) | |
| local instance lowerHFintype (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| Fintype (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) := Fintype.ofFinite _ | |
| local instance upperDualFintype : | |
| Fintype (HierarchicalAllDecoderTables.UpperAnswer slots upper) := | |
| HierarchicalAllDecoderTables.upperDualFintype slots upper | |
| variable (hbranch : ∀ k < plan.depth, 0 < branch k) | |
| (σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch plan.depth t)) | |
| (useful : (bg : HierarchicalMatrixTable.Background | |
| (rows := FixedRows.rows plan) slots upper) → | |
| HierarchicalFrozenTables.QuotientMatrix slots upper lowerLevel bg → Prop) | |
| include hbranch σ useful | |
| local instance directionNonempty (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) : | |
| Nonempty (BucketSampler.Direction | |
| (FixedRows.rows plan (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))) := | |
| ⟨BucketUniform.coordinateDirection ⟨0, | |
| lt_of_lt_of_le Nat.zero_lt_one | |
| (plan.rows_pos (plan.depth - | |
| Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))⟩⟩ | |
| theorem supported_collision_lt {adviceRows : Nat} | |
| (density : ℝ) (hdensity : 0 < density) | |
| (d : HierarchicalFrozenTables.LowerNodes upper lowerLevel) | |
| (upperMatrix : HierarchicalMatrixTable.Matrix (rows := FixedRows.rows plan) slots upper) | |
| (A : ManyGoodRows.RowMap (Block (FixedRows.rows plan) upper) adviceRows) | |
| (table : HierarchicalAllDecoderTables.Input | |
| (rows := FixedRows.rows plan) slots upper lowerLevel adviceRows → | |
| HierarchicalAllDecoderTables.UpperAnswer slots upper) | |
| (hsupport : (HierarchicalAllDecoderTables.upperTableLaw slots upper lowerLevel σ useful | |
| adviceRows density).weight table ≠ 0) : | |
| (uniformLaw (HierarchicalDecoderTables.LowerH slots upper lowerLevel d) | |
| (FixedRows.rows plan (Nodes.height (HierarchicalLeftDecoder.LowerNode upper lowerLevel d)))).probability | |
| (uniformCollision _ _ | |
| (HierarchicalLowerInverse.matrixTable slots upper lowerLevel background d upperMatrix A | |
| (HierarchicalAllDecoderTables.decodeTable slots upper lowerLevel hbranch adviceRows d | |
| (FixedRows.cutoff plan hδ) table))) < | |
| UpperParameterScalars.gamma (InitialParameters.useful δ) plan.density | |
| (InitialParameters.inverse δ) plan.order | |
| (FixedRows.rows plan (Nodes.height upper)) ^ 2 / 8 := by | |
| rw [uniform_collision_eq] | |
| exact HierarchicalFixedLowerInverse.supported_upper_matrixTable_agreement_lt | |
| plan hδ slots upper lowerLevel background hbranch σ useful density hdensity | |
| d upperMatrix A table hsupport | |
| end | |
| end PerfectCompleteness.FixedLowerUniformCollision | |
| end | |
| end OAI | |