openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /RestrictionCrossLaw.lean
| import OAI.Computability.PerfectCompleteness.Algebra.LowerCrossRank | |
| import OAI.Computability.PerfectCompleteness.Foundations.DualQuotientRowsLemmas | |
| import OAI.Computability.PerfectCompleteness.Sampling.RestrictionRowsLaw | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.RestrictionCrossLaw | |
| noncomputable section | |
| open UniqueGamesTheorem.Integration.BinaryLinear (F2) | |
| open UniqueGamesTheorem.Fourier.MatrixRestrictions | |
| open UniqueGamesTheorem.Foundations.Games | |
| open RestrictionRowsLaw DualQuotientRows | |
| variable {E C I : Type*} [AddCommGroup E] [Module F2 E] | |
| [AddCommGroup C] [Module F2 C] [Fintype I] | |
| (W : Submodule F2 E) (C' : Submodule F2 C) (b : Module.Basis I F2 C') | |
| def rowPairEquiv (m : Nat) : | |
| ((Fin m ⊕ Fin m) → Module.Dual F2 (E ⧸ W)) ≃ₗ[F2] | |
| ((Fin m → W.dualAnnihilator) × (Fin m → W.dualAnnihilator)) := | |
| (LinearEquiv.piCongrRight | |
| (fun _ : Fin m ⊕ Fin m => quotientRowEquiv W)).trans | |
| (LinearEquiv.sumArrowLequivProdArrow | |
| (Fin m) (Fin m) F2 W.dualAnnihilator) | |
| variable {m : Nat} (sel : (Fin m ⊕ Fin m) → I) | |
| def selectedPair : Parameter W C' →ₗ[F2] | |
| ((Fin m → W.dualAnnihilator) × (Fin m → W.dualAnnihilator)) := | |
| (rowPairEquiv W m).toLinearMap.comp (selectedRowsLinear W C' b sel) | |
| omit [Fintype I] in | |
| @[simp] theorem selectedPair_fst (A : Parameter W C') (i : Fin m) : | |
| (selectedPair W C' b sel A).1 i = | |
| quotientRowEquiv W (rowsLinear W C' b A (sel (Sum.inl i))) := rfl | |
| omit [Fintype I] in | |
| @[simp] theorem selectedPair_snd (A : Parameter W C') (j : Fin m) : | |
| (selectedPair W C' b sel A).2 j = | |
| quotientRowEquiv W (rowsLinear W C' b A (sel (Sum.inr j))) := rfl | |
| theorem selectedPair_surjective (hsel : Function.Injective sel) : | |
| Function.Surjective (selectedPair W C' b sel) := | |
| (rowPairEquiv W m).surjective.comp (selectedRowsLinear_surjective W C' b sel hsel) | |
| private theorem uniform_product {A B : Type*} [Fintype A] [Fintype B] | |
| [Nonempty A] [Nonempty B] : | |
| (FiniteDistribution.uniform A).product (FiniteDistribution.uniform B) = | |
| FiniteDistribution.uniform (A × B) := by | |
| apply FiniteDistribution.eq_of_weight_eq | |
| intro p | |
| change (1 / (Fintype.card A : ℝ)) * (1 / (Fintype.card B : ℝ)) = | |
| 1 / (Fintype.card (A × B) : ℝ) | |
| simp only [Fintype.card_prod, Nat.cast_mul, one_div, mul_inv] | |
| theorem selectedPair_law [Fintype (Parameter W C')] [Fintype W.dualAnnihilator] | |
| (hsel : Function.Injective sel) : | |
| (FiniteDistribution.uniform (Parameter W C')).pushforward (selectedPair W C' b sel) = | |
| (FiniteDistribution.uniform (Fin m → W.dualAnnihilator)).product | |
| (FiniteDistribution.uniform (Fin m → W.dualAnnihilator)) := by | |
| rw [uniform_product] | |
| exact UniformLinearImage.uniform_pushforward_linearMap | |
| (selectedPair W C' b sel) (selectedPair_surjective W C' b sel hsel) | |
| omit [Fintype I] in | |
| theorem coordinate_translate_left (coordinates : I → Module.Dual F2 C) | |
| (hcoordinates : ∀ i, (coordinates i).comp C'.subtype = b.coord i) | |
| (X : E →ₗ[F2] C) (A : Parameter W C') (i : Fin m) : | |
| (coordinates (sel (Sum.inl i))).comp (translate W C' X A) = | |
| (coordinates (sel (Sum.inl i))).comp X + | |
| ((selectedPair W C' b sel A).1 i : Module.Dual F2 E) := by | |
| simpa only [affineRow, selectedPair_fst, quotientRowEquiv_val] using | |
| coordinate_translate W C' b coordinates hcoordinates X A (sel (Sum.inl i)) | |
| omit [Fintype I] in | |
| theorem coordinate_translate_right (coordinates : I → Module.Dual F2 C) | |
| (hcoordinates : ∀ i, (coordinates i).comp C'.subtype = b.coord i) | |
| (X : E →ₗ[F2] C) (A : Parameter W C') (j : Fin m) : | |
| (coordinates (sel (Sum.inr j))).comp (translate W C' X A) = | |
| (coordinates (sel (Sum.inr j))).comp X + | |
| ((selectedPair W C' b sel A).2 j : Module.Dual F2 E) := by | |
| simpa only [affineRow, selectedPair_snd, quotientRowEquiv_val] using | |
| coordinate_translate W C' b coordinates hcoordinates X A (sel (Sum.inr j)) | |
| theorem crossMatrix_law [Fintype (Parameter W C')] [Fintype W.dualAnnihilator] | |
| (hsel : Function.Injective sel) | |
| (coordinates : I → Module.Dual F2 C) | |
| (hcoordinates : ∀ i, (coordinates i).comp C'.subtype = b.coord i) | |
| (X : E →ₗ[F2] C) | |
| (F : Module.Dual F2 E →ₗ[F2] Module.Dual F2 (Module.Dual F2 E)) : | |
| (FiniteDistribution.uniform (Parameter W C')).pushforward | |
| (Γ := Matrix (Fin m) (Fin m) F2) | |
| (fun A i j => | |
| F ((coordinates (sel (Sum.inl i))).comp (translate W C' X A)) | |
| ((coordinates (sel (Sum.inr j))).comp (translate W C' X A))) = | |
| ((FiniteDistribution.uniform (Fin m → W.dualAnnihilator)).product | |
| (FiniteDistribution.uniform (Fin m → W.dualAnnihilator))).pushforward | |
| (Γ := Matrix (Fin m) (Fin m) F2) | |
| (fun p => LowerCrossRank.crossMatrix F W.dualAnnihilator | |
| (fun i => (coordinates (sel (Sum.inl i))).comp X) | |
| (fun j => (coordinates (sel (Sum.inr j))).comp X) p.1 p.2) := by | |
| have hcross : | |
| (fun (A : Parameter W C') => | |
| (fun i j => F ((coordinates (sel (Sum.inl i))).comp (translate W C' X A)) | |
| ((coordinates (sel (Sum.inr j))).comp (translate W C' X A)) : | |
| Matrix (Fin m) (Fin m) F2)) = | |
| (fun A => LowerCrossRank.crossMatrix F W.dualAnnihilator | |
| (fun i => (coordinates (sel (Sum.inl i))).comp X) | |
| (fun j => (coordinates (sel (Sum.inr j))).comp X) | |
| (selectedPair W C' b sel A).1 (selectedPair W C' b sel A).2) := by | |
| funext A i j | |
| rw [coordinate_translate_left W C' b sel coordinates hcoordinates X A i, | |
| coordinate_translate_right W C' b sel coordinates hcoordinates X A j] | |
| rfl | |
| rw [hcross, ← FiniteDistribution.pushforward_comp | |
| (FiniteDistribution.uniform (Parameter W C')) (selectedPair W C' b sel) | |
| (fun p => LowerCrossRank.crossMatrix F W.dualAnnihilator | |
| (fun i => (coordinates (sel (Sum.inl i))).comp X) | |
| (fun j => (coordinates (sel (Sum.inr j))).comp X) p.1 p.2), | |
| selectedPair_law W C' b sel hsel] | |
| end | |
| end PerfectCompleteness.RestrictionCrossLaw | |
| end | |
| end OAI | |