openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Algebra /TensorAffineHeavyWitness.lean
| import OAI.Computability.PerfectCompleteness.Algebra.AffineLinearHeavyWitness | |
| import OAI.Computability.PerfectCompleteness.Algebra.BilinearGramCompressionLemmas | |
| import OAI.Computability.PerfectCompleteness.Algebra.TensorBucketEvaluationLemmas | |
| import OAI.Computability.PerfectCompleteness.Algebra.TensorSliceDimension | |
| import OAI.Computability.PerfectCompleteness.Foundations.OutputAnnihilator | |
| import OAI.Computability.PerfectCompleteness.Sampling.OwnLawHeavyList | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.TensorAffineHeavyWitness | |
| open scoped TensorProduct Classical | |
| open QuarterBalance (F2) | |
| open UniqueGamesTheorem.Foundations.Games | |
| noncomputable section | |
| variable {ℓ : Nat} {H : Type*} [AddCommGroup H] [Module F2 H] | |
| [Finite H] [FiniteDimensional F2 H] | |
| local instance dualFintype {V : Type*} [AddCommGroup V] [Module F2 V] [Fintype V] : | |
| Fintype (Module.Dual F2 V) := | |
| Fintype.ofInjective (fun f : Module.Dual F2 V => (f : V → F2)) DFunLike.coe_injective | |
| def heavyList (W : Submodule F2 (Fin ℓ → F2)) | |
| (μ : FiniteDistribution (W ⊗[F2] H)) (G : (W ⊗[F2] H) → (Fin ℓ → F2)) | |
| (τ : ℝ) (σ : Module.Dual F2 (Fin ℓ → F2)) : | |
| Finset (Module.Dual F2 (W ⊗[F2] H)) := | |
| OwnLawHeavyList.heavyList μ (fun X => QuarterBalance.sign (σ (G X))) τ | |
| omit [FiniteDimensional F2 H] in | |
| theorem heavyList_card_le (W : Submodule F2 (Fin ℓ → F2)) | |
| (μ : FiniteDistribution (W ⊗[F2] H)) (G : (W ⊗[F2] H) → (Fin ℓ → F2)) | |
| (τ biasBound : ℝ) (hτ : 0 < τ) (hbiasBound : 0 ≤ biasBound) (hsmall : biasBound ≤ τ ^ 2 / 2) | |
| (hbias : ∀ Φ : Module.Dual F2 (W ⊗[F2] H), Φ ≠ 0 → | |
| |μ.expectation (fun X => QuarterBalance.sign (Φ X))| ≤ biasBound) | |
| (σ : Module.Dual F2 (Fin ℓ → F2)) : | |
| ((heavyList W μ G τ σ).card : ℝ) ≤ 2 / τ ^ 2 := by | |
| apply OwnLawHeavyList.heavyList_card_le μ _ τ biasBound hτ hbiasBound hsmall | |
| · intro X | |
| exact (OwnLawHeavyList.character_sq σ (G X)).le | |
| · exact hbias | |
| def conditionalMatch (W : Submodule F2 (Fin ℓ → F2)) | |
| (μ : FiniteDistribution (W ⊗[F2] H)) (G : (W ⊗[F2] H) → (Fin ℓ → F2)) | |
| (Q : Submodule F2 (Module.Dual F2 H)) (target : Q →ₗ[F2] W) | |
| (z : Module.Dual F2 H) (offset : Fin ℓ → F2) : ℝ := | |
| AffineLinearHeavyWitness.conditionalMatch μ (TensorRestriction.restrictionMap Q) | |
| target G (TensorBucketEvaluation.tensorOutputLinear W z) offset | |
| theorem exists_heavy_candidate (W : Submodule F2 (Fin ℓ → F2)) | |
| (μ : FiniteDistribution (W ⊗[F2] H)) (G : (W ⊗[F2] H) → (Fin ℓ → F2)) | |
| (ρ h₀ biasBound : ℝ) (hρ : 0 < ρ) (hh₀ : 0 < h₀) (hbiasBound : 0 ≤ biasBound) | |
| (hsmall : biasBound ≤ h₀ / 2) | |
| (hbias : ∀ Φ : Module.Dual F2 (W ⊗[F2] H), Φ ≠ 0 → | |
| |μ.expectation (fun X => QuarterBalance.sign (Φ X))| ≤ biasBound) | |
| (hexcluded : 1 / (2 : ℝ) ^ Module.finrank F2 W < ρ / 8) | |
| (Q : Submodule F2 (Module.Dual F2 H)) | |
| (hsize : h₀ ≤ 1 / (2 : ℝ) ^ Module.finrank F2 (Q →ₗ[F2] W)) | |
| (target : Q →ₗ[F2] W) (z : Module.Dual F2 H) (offset : Fin ℓ → F2) | |
| (hmatch : ρ ≤ conditionalMatch W μ G Q target z offset) : | |
| ∃ σ : Module.Dual F2 (Fin ℓ → F2), σ.comp W.subtype ≠ 0 ∧ | |
| ∃ Φ ∈ heavyList W μ G (ρ * h₀ / 4) σ, | |
| ∀ w : W, σ w = 1 → TensorCosetEvaluation.scalarRestriction Φ w - z ∈ Q := by | |
| let : Module.Free F2 (Q →ₗ[F2] W) := Module.Free.of_divisionRing F2 (Q →ₗ[F2] W) | |
| let : Fintype (Module.Dual F2 (W ⊗[F2] H)) := | |
| Fintype.ofInjective (fun functional : Module.Dual F2 (W ⊗[F2] H) => | |
| (functional : (W ⊗[F2] H) → F2)) DFunLike.coe_injective | |
| let b := Module.finBasis F2 (Q →ₗ[F2] W) | |
| let c := Module.finBasis F2 (Fin ℓ → F2) | |
| have hsize' : h₀ ≤ 1 / (Fintype.card | |
| (Fin (Module.finrank F2 (Q →ₗ[F2] W)) → Bool) : ℝ) := by | |
| simpa only [Fintype.card_fun, Fintype.card_bool, Fintype.card_fin, | |
| Nat.cast_pow, Nat.cast_ofNat] using hsize | |
| have hbad : ((OutputAnnihilator.bad W).card : ℝ) / | |
| Fintype.card (Module.Dual F2 (Fin ℓ → F2)) < ρ / 8 := by | |
| rw [OutputAnnihilator.bad_fraction] | |
| exact hexcluded | |
| unfold conditionalMatch at hmatch | |
| have candidateBasis := | |
| AffineLinearHeavyWitness.exists_heavy_in_dual_coset | |
| (U := W ⊗[F2] H) (V := Q →ₗ[F2] W) (K := Fin ℓ → F2) μ b c | |
| have candidateRestriction := candidateBasis | |
| (TensorRestriction.restrictionMap Q) | |
| (TensorRestriction.restrictionMap_surjective (W := W) Q) | |
| have candidateData := candidateRestriction | |
| target G (TensorBucketEvaluation.tensorOutputLinear W z) offset | |
| (OutputAnnihilator.bad W) ρ h₀ biasBound | |
| have candidateBounds := candidateData hρ hh₀ hsize' hbiasBound hsmall hbias hbad | |
| have candidateWitness := candidateBounds hmatch | |
| obtain ⟨σ, hσ, ψ, hψ, hheavy⟩ := candidateWitness | |
| refine ⟨σ, (OutputAnnihilator.not_mem_bad_iff W σ).mp hσ, | |
| σ.comp (TensorBucketEvaluation.tensorOutputLinear W z) + ψ, ?_, ?_⟩ | |
| · simp only [heavyList, OwnLawHeavyList.heavyList, SmallBias.heavySet, | |
| Finset.mem_filter, Finset.mem_univ, true_and] at hheavy ⊢ | |
| simpa only [OwnLawHeavyList.character, BinaryDualCoordinates.character] using hheavy | |
| · intro w hw | |
| have hψ' : TensorCosetEvaluation.scalarRestriction ψ w ∈ Q := | |
| TensorCosetEvaluation.scalarRestriction_mem_of_vanishes Q ψ | |
| (TensorRestriction.dualImage_vanishes Q ψ hψ) w | |
| rw [TensorTargetCharacter.comp_tensorOutputLinear, | |
| TensorCosetEvaluation.scalarRestriction_add, | |
| TensorCosetEvaluation.scalarRestriction_pure] | |
| have hw' : (σ.comp W.subtype) w = 1 := hw | |
| rw [hw', one_smul, add_sub_cancel_left] | |
| exact hψ' | |
| theorem lemma53 (W : Submodule F2 (Fin ℓ → F2)) | |
| (μ : FiniteDistribution (W ⊗[F2] H)) (G : (W ⊗[F2] H) → (Fin ℓ → F2)) | |
| (r : Nat) (hcodim : Module.finrank F2 ((Fin ℓ → F2) ⧸ W) ≤ r) | |
| (ρ biasBound : ℝ) (hρ : 0 < ρ) (hbiasBound : 0 ≤ biasBound) | |
| (hgap : 1 / (2 : ℝ) ^ (ℓ - r) < ρ / 8) | |
| (hsmall : biasBound ≤ min ((1 / (2 : ℝ) ^ (r * ℓ)) / 2) | |
| ((ρ * (1 / (2 : ℝ) ^ (r * ℓ)) / 4) ^ 2 / 2)) | |
| (hbias : ∀ Φ : Module.Dual F2 (W ⊗[F2] H), Φ ≠ 0 → | |
| |μ.expectation (fun X => QuarterBalance.sign (Φ X))| ≤ biasBound) : | |
| (∀ σ : Module.Dual F2 (Fin ℓ → F2), | |
| ((heavyList W μ G (ρ * (1 / (2 : ℝ) ^ (r * ℓ)) / 4) σ).card : ℝ) ≤ | |
| 2 / (ρ * (1 / (2 : ℝ) ^ (r * ℓ)) / 4) ^ 2) ∧ | |
| ∀ (Q : Submodule F2 (Module.Dual F2 H)), Module.finrank F2 Q ≤ r → | |
| ∀ (target : Q →ₗ[F2] W) (z : Module.Dual F2 H) (offset : Fin ℓ → F2), | |
| ρ ≤ conditionalMatch W μ G Q target z offset → | |
| ∃ σ : Module.Dual F2 (Fin ℓ → F2), σ.comp W.subtype ≠ 0 ∧ | |
| ∃ Φ ∈ heavyList W μ G (ρ * (1 / (2 : ℝ) ^ (r * ℓ)) / 4) σ, | |
| ∀ w : W, σ w = 1 → | |
| TensorCosetEvaluation.scalarRestriction Φ w - z ∈ Q := by | |
| have hh₀ : 0 < 1 / (2 : ℝ) ^ (r * ℓ) := by positivity | |
| have hτ : 0 < ρ * (1 / (2 : ℝ) ^ (r * ℓ)) / 4 := by positivity | |
| have hsmall' := le_min_iff.mp hsmall | |
| constructor | |
| · intro σ | |
| exact heavyList_card_le W μ G _ biasBound hτ hbiasBound hsmall'.2 hbias σ | |
| · intro Q hQ target z offset hmatch | |
| have hexcluded : 1 / (2 : ℝ) ^ Module.finrank F2 W < ρ / 8 := by | |
| rw [← OutputAnnihilator.bad_fraction W] | |
| apply OutputAnnihilator.bad_fraction_lt_of_codim_le W r ρ hcodim | |
| simpa only [Module.finrank_fin_fun] using hgap | |
| have hsize : 1 / (2 : ℝ) ^ (r * ℓ) ≤ | |
| 1 / (2 : ℝ) ^ Module.finrank F2 (Q →ₗ[F2] W) := by | |
| simpa only [TensorSliceDimension.mask_card, Nat.cast_pow, Nat.cast_ofNat] using | |
| TensorSliceDimension.reciprocal_equation_mask_card_ge W Q hQ | |
| exact exists_heavy_candidate W μ G ρ _ biasBound hρ hh₀ hbiasBound hsmall'.1 hbias hexcluded | |
| Q hsize target z offset hmatch | |
| end | |
| end PerfectCompleteness.TensorAffineHeavyWitness | |
| end | |
| end OAI | |