openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /CoppersmithWinograd /ComplexCW75InitialExecution.lean
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.ComplexCW75Primitive | |
| import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialKernelExecution | |
| namespace OAI | |
| /-! Coppersmith–Winograd tensors, tensor powers and local restrictions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.Foundation.CW75InitialExecution | |
| open Polynomial | |
| open scoped BigOperators Classical | |
| abbrev Word := CW75Primitive.Word | |
| abbrev Blocks (N : ℕ) := Fin N → Word | |
| def fullSource (N : ℕ) : Tensor ℂ (Blocks N) (Blocks N) (Blocks N) := | |
| Tensor.power (Tensor.power CoppersmithWinograd.tensor 3) N | |
| def support {N : ℕ} (x y z : Blocks N) : Prop := fullSource N x y z ≠ 0 | |
| theorem support_iff {N : ℕ} (x y z : Blocks N) : | |
| support x y z ↔ ∀ i j, (x i j).val + (y i j).val + (z i j).val = 2 := by | |
| simp [support, fullSource, Tensor.power, Finset.prod_ne_zero_iff, | |
| CWOriginalWords.tensor_ne_zero_iff] | |
| def boundaryPolynomial (x y z : Word) : Polynomial ℂ := | |
| ∏ i, CWBoundary.scalingPolynomial (x i) (y i) (z i) | |
| def boundaryNormalized (x y z : Word) : Polynomial ℂ := | |
| ∏ i, (C (CWBoundary.tensor (x i) (y i) (z i)) + | |
| X ^ 2 * C (CWBoundary.deleted (x i) (y i) (z i))) | |
| theorem boundaryPolynomial_factor (x y z : Word) : | |
| boundaryPolynomial x y z = X ^ 3 * boundaryNormalized x y z := by | |
| have hp (i : Fin 3) : CWBoundary.scalingPolynomial (x i) (y i) (z i) = | |
| X * (C (CWBoundary.tensor (x i) (y i) (z i)) + | |
| X ^ 2 * C (CWBoundary.deleted (x i) (y i) (z i))) := by | |
| rw [CWBoundary.scalingPolynomial_apply] | |
| ring | |
| simp [boundaryPolynomial, boundaryNormalized, hp, Finset.prod_mul_distrib] | |
| theorem boundaryPolynomial_vanishes (x y z : Word) (j : ℕ) (hj : j < 3) : | |
| (boundaryPolynomial x y z).coeff j = 0 := by | |
| rw [boundaryPolynomial_factor] | |
| simp only [Polynomial.coeff_X_pow_mul', Nat.not_le_of_lt hj, ite_false] | |
| theorem boundaryPolynomial_leading (x y z : Word) : | |
| (boundaryPolynomial x y z).coeff 3 = CW75Primitive.source x y z := by | |
| rw [boundaryPolynomial_factor] | |
| simp [Polynomial.coeff_X_pow_mul', boundaryNormalized, | |
| Polynomial.coeff_zero_prod, CW75Primitive.source, Tensor.power] | |
| def completionPolynomial (x y z : Word) : Polynomial ℂ := | |
| (if CW75Primitive.hasB x then X else 1) * | |
| (if CW75Primitive.hasC y then X else 1) * | |
| (if CW75Primitive.allA z then 0 else 1) | |
| private theorem boundary_one (x y z : Fin 3) (h : CWBoundary.tensor x y z ≠ 0) : | |
| CWBoundary.tensor x y z = 1 := by | |
| by_cases hd : x = 1 ∧ y = 0 ∧ z = 1 | |
| · simp [CWBoundary.tensor, hd] at h | |
| · by_cases hs : x.val + y.val + z.val = 2 | |
| · simp [CWBoundary.tensor, CoppersmithWinograd.tensor, hd, hs] | |
| · simp [CWBoundary.tensor, CoppersmithWinograd.tensor, hd, hs] at h | |
| private theorem boundarySource_one (x y z : Word) (h : CW75Primitive.source x y z ≠ 0) : | |
| CW75Primitive.source x y z = 1 := by | |
| unfold CW75Primitive.source Tensor.power | |
| apply Finset.prod_eq_one | |
| intro i hi | |
| exact boundary_one _ _ _ (CW75Primitive.source_factors x y z h i) | |
| theorem retained_coefficient_one (x y z : Word) (h : CW75Primitive.tensor x y z ≠ 0) : | |
| CW75Primitive.tensor x y z = 1 := by | |
| by_cases hr : CW75Primitive.retained x y z | |
| · rw [CW75Primitive.tensor, ite_eq_left hr] at h ⊢ | |
| exact boundarySource_one x y z h | |
| · simp [CW75Primitive.tensor, hr] at h | |
| private theorem completion_eq (x y z : Word) (h : CW75Primitive.source x y z ≠ 0) : | |
| completionPolynomial x y z = CW75Primitive.polynomial x y z := by | |
| have hone := boundarySource_one x y z h | |
| rw [CW75Primitive.polynomial_apply] | |
| by_cases hz : CW75Primitive.allA z <;> simp [completionPolynomial, hone, hz] | |
| private theorem source_mul_retained (x y z : Word) : | |
| CW75Primitive.source x y z * CW75Primitive.tensor x y z = | |
| CW75Primitive.tensor x y z := by | |
| by_cases hs : CW75Primitive.source x y z = 0 | |
| · simp [CW75Primitive.tensor, hs] | |
| · rw [boundarySource_one x y z hs, one_mul] | |
| def leftBlock (x : Word) : Polynomial ℂ := | |
| Polynomial.expand ℂ 2 (∏ i, CWBoundary.outerWeight X (x i)) * | |
| (if CW75Primitive.hasB x then X else 1) | |
| def middleBlock (y : Word) : Polynomial ℂ := | |
| Polynomial.expand ℂ 2 (∏ i, CWBoundary.middleWeight X (y i)) * | |
| (if CW75Primitive.hasC y then X else 1) | |
| def rightBlock (z : Word) : Polynomial ℂ := | |
| Polynomial.expand ℂ 2 (∏ i, CWBoundary.outerWeight X (z i)) * | |
| (if CW75Primitive.allA z then 0 else 1) | |
| def blockKernel (x y z : Word) : Polynomial ℂ := leftBlock x * middleBlock y * rightBlock z | |
| theorem blockKernel_factor (x y z : Word) | |
| (hs : ∀ i, (x i).val + (y i).val + (z i).val = 2) : | |
| blockKernel x y z = Polynomial.expand ℂ 2 (boundaryPolynomial x y z) * | |
| completionPolynomial x y z := by | |
| have hpoint (i : Fin 3) : | |
| CWBoundary.outerWeight X (x i) * CWBoundary.middleWeight X (y i) * | |
| CWBoundary.outerWeight X (z i) = CWBoundary.scalingPolynomial (x i) (y i) (z i) := by | |
| unfold CWBoundary.scalingPolynomial | |
| rw [CWBoundary.outerMap, CWBoundary.middleMap, CWBoundary.restrict_diagonal_apply] | |
| simp [CoppersmithWinograd.tensor, hs i] | |
| have hprod : (∏ i, CWBoundary.outerWeight X (x i)) * | |
| (∏ i, CWBoundary.middleWeight X (y i)) * (∏ i, CWBoundary.outerWeight X (z i)) = | |
| boundaryPolynomial x y z := by | |
| simp only [boundaryPolynomial, ← hpoint, Finset.prod_mul_distrib] | |
| calc | |
| blockKernel x y z = Polynomial.expand ℂ 2 | |
| ((∏ i, CWBoundary.outerWeight X (x i)) * | |
| (∏ i, CWBoundary.middleWeight X (y i)) * (∏ i, CWBoundary.outerWeight X (z i))) * | |
| completionPolynomial x y z := by | |
| simp only [blockKernel, leftBlock, middleBlock, rightBlock, completionPolynomial, map_mul] | |
| ring | |
| _ = _ := by rw [hprod] | |
| theorem blockKernel_vanishes (x y z : Word) | |
| (hs : ∀ i, (x i).val + (y i).val + (z i).val = 2) | |
| (j : ℕ) (hj : j < 7) : (blockKernel x y z).coeff j = 0 := by | |
| rw [blockKernel_factor x y z hs] | |
| apply PolynomialControlledKernel.expanded_mul_vanishes_of_leading_support | |
| (boundaryPolynomial x y z) (completionPolynomial x y z) 3 1 | |
| (CW75Primitive.source x y z) | |
| (boundaryPolynomial_vanishes x y z) (boundaryPolynomial_leading x y z) _ j hj | |
| intro hn k hk | |
| rw [completion_eq x y z hn] | |
| have hkzero : k = 0 := by omega | |
| subst k | |
| simp [CW75Primitive.polynomial_coeff] | |
| theorem blockKernel_leading (x y z : Word) | |
| (hs : ∀ i, (x i).val + (y i).val + (z i).val = 2) : | |
| (blockKernel x y z).coeff 7 = CW75Primitive.tensor x y z := by | |
| rw [blockKernel_factor x y z hs] | |
| have h := PolynomialControlledKernel.expanded_mul_leading_of_leading_support | |
| (boundaryPolynomial x y z) (completionPolynomial x y z) 3 1 | |
| (CW75Primitive.source x y z) (CW75Primitive.tensor x y z) | |
| (boundaryPolynomial_vanishes x y z) (boundaryPolynomial_leading x y z) (by | |
| intro hn | |
| rw [completion_eq x y z hn] | |
| simp [CW75Primitive.polynomial_coeff]) | |
| exact h.trans (source_mul_retained x y z) | |
| private theorem degree_product {n D : ℕ} (f : Fin n → Polynomial ℂ) | |
| (h : ∀ i, (f i).degree ≤ D) : (∏ i, f i).degree ≤ D * n := by | |
| apply Polynomial.degree_le_of_natDegree_le | |
| calc | |
| (∏ i, f i).natDegree ≤ ∑ i, (f i).natDegree := Polynomial.natDegree_prod_le _ _ | |
| _ ≤ ∑ _i : Fin n, D := by | |
| apply Finset.sum_le_sum | |
| intro i hi | |
| exact Polynomial.natDegree_le_of_degree_le (h i) | |
| _ = D * n := by simp [Nat.mul_comm] | |
| private theorem degree_expanded_product (P Q : Polynomial ℂ) {d e : ℕ} | |
| (hP : P.degree ≤ d) (hQ : Q.degree ≤ e) : | |
| (Polynomial.expand ℂ 2 P * Q).degree ≤ d * 2 + e := by | |
| apply Polynomial.degree_le_of_natDegree_le | |
| apply Polynomial.natDegree_mul_le_of_le | |
| · rw [Polynomial.natDegree_expand] | |
| exact Nat.mul_le_mul_right 2 (Polynomial.natDegree_le_of_degree_le hP) | |
| · exact Polynomial.natDegree_le_of_degree_le hQ | |
| private theorem outerWeight_degree (a : Fin 3) : | |
| (CWBoundary.outerWeight (X : Polynomial ℂ) a).degree ≤ 1 := by | |
| by_cases h : a = 1 <;> simp [CWBoundary.outerWeight, h] | |
| private theorem middleWeight_degree (a : Fin 3) : | |
| (CWBoundary.middleWeight (X : Polynomial ℂ) a).degree ≤ 1 := by | |
| by_cases h : a = 1 <;> simp [CWBoundary.middleWeight, h] | |
| theorem leftBlock_degree (x : Word) : (leftBlock x).degree ≤ 7 := by | |
| apply degree_expanded_product (d := 3) (e := 1) | |
| (∏ i, CWBoundary.outerWeight X (x i)) (if CW75Primitive.hasB x then X else 1) | |
| (degree_product _ (fun i => outerWeight_degree (x i))) | |
| by_cases h : CW75Primitive.hasB x <;> simp [h] | |
| theorem middleBlock_degree (y : Word) : (middleBlock y).degree ≤ 7 := by | |
| apply degree_expanded_product (d := 3) (e := 1) | |
| (∏ i, CWBoundary.middleWeight X (y i)) (if CW75Primitive.hasC y then X else 1) | |
| (degree_product _ (fun i => middleWeight_degree (y i))) | |
| by_cases h : CW75Primitive.hasC y <;> simp [h] | |
| theorem rightBlock_degree (z : Word) : (rightBlock z).degree ≤ 6 := by | |
| apply degree_expanded_product (d := 3) (e := 0) | |
| (∏ i, CWBoundary.outerWeight X (z i)) (if CW75Primitive.allA z then 0 else 1) | |
| (degree_product _ (fun i => outerWeight_degree (z i))) | |
| by_cases h : CW75Primitive.allA z <;> simp [h] | |
| def auxiliary : Tensor ℂ PUnit.{1} PUnit.{1} PUnit.{1} := fun _ _ _ => 1 | |
| theorem auxiliary_rank : Tensor.RankAtMost auxiliary 1 := by | |
| refine ⟨fun _ _ => 1, fun _ _ => 1, fun _ _ => 1, ?_⟩ | |
| funext x y z | |
| simp [auxiliary, Tensor.rankOne] | |
| def leftMap {N : ℕ} (x : Blocks N) (_ : PUnit.{1} × PUnit.{1}) (_ : PUnit.{1}) : Polynomial ℂ := | |
| ∏ i, leftBlock (x i) | |
| def middleMap {N : ℕ} (y : Blocks N) (_ : PUnit.{1} × PUnit.{1}) (_ : PUnit.{1}) : Polynomial ℂ := | |
| ∏ i, middleBlock (y i) | |
| def rightMap {N : ℕ} (z : Blocks N) (_ : PUnit.{1} × PUnit.{1}) (_ : PUnit.{1}) : Polynomial ℂ := | |
| ∏ i, rightBlock (z i) | |
| def value {N : ℕ} : Tensor ℂ (Blocks N × (PUnit.{1} × PUnit.{1})) | |
| (Blocks N × (PUnit.{1} × PUnit.{1})) (Blocks N × (PUnit.{1} × PUnit.{1})) := | |
| fun x y z => ∏ i, CW75Primitive.tensor (x.1 i) (y.1 i) (z.1 i) | |
| theorem kernel_eq {N : ℕ} (x y z : Blocks N × (PUnit.{1} × PUnit.{1})) : | |
| PolynomialLocalConstruction.kernel auxiliary leftMap middleMap rightMap x y z = | |
| ∏ i, blockKernel (x.1 i) (y.1 i) (z.1 i) := by | |
| simp [PolynomialLocalConstruction.kernel, LocalMaps.fiberTransform, | |
| auxiliary, leftMap, middleMap, rightMap, blockKernel, Finset.prod_mul_distrib] | |
| theorem kernel_coefficients {N : ℕ} (x y z : Blocks N × (PUnit.{1} × PUnit.{1})) | |
| (hs : support x.1 y.1 z.1) : | |
| (∀ j, j < 7 * N → | |
| (PolynomialLocalConstruction.kernel auxiliary leftMap middleMap rightMap x y z).coeff j = 0) ∧ | |
| (PolynomialLocalConstruction.kernel auxiliary leftMap middleMap rightMap x y z).coeff (7 * N) = | |
| value x y z := by | |
| have hsites := (support_iff x.1 y.1 z.1).mp hs | |
| have hd (i : Fin N) : X ^ 7 ∣ blockKernel (x.1 i) (y.1 i) (z.1 i) := | |
| Polynomial.X_pow_dvd_iff.mpr (blockKernel_vanishes _ _ _ (hsites i)) | |
| let Q : Fin N → Polynomial ℂ := fun i => Classical.choose (hd i) | |
| have hQ (i : Fin N) : blockKernel (x.1 i) (y.1 i) (z.1 i) = X ^ 7 * Q i := | |
| Classical.choose_spec (hd i) | |
| have hQzero (i : Fin N) : (Q i).coeff 0 = CW75Primitive.tensor (x.1 i) (y.1 i) (z.1 i) := by | |
| have h := blockKernel_leading _ _ _ (hsites i) | |
| rw [hQ] at h | |
| simpa only [Polynomial.coeff_X_pow_mul', le_refl, ite_true, Nat.sub_self] using h | |
| have hfactor : (∏ i, blockKernel (x.1 i) (y.1 i) (z.1 i)) = | |
| X ^ (7 * N) * ∏ i, Q i := by | |
| simp only [hQ, Finset.prod_mul_distrib, Finset.prod_const, | |
| Finset.card_univ, Fintype.card_fin, ← pow_mul] | |
| simp only [kernel_eq, hfactor] | |
| constructor | |
| · intro j hj | |
| simp only [Polynomial.coeff_X_pow_mul', Nat.not_le_of_lt hj, ite_false] | |
| · simp only [Polynomial.coeff_X_pow_mul', le_refl, ite_true, Nat.sub_self, | |
| Polynomial.coeff_zero_prod, hQzero, value] | |
| def execution (N : ℕ) : PolynomialKernelExecution.Execution | |
| (Blocks N) (Blocks N) (Blocks N) PUnit.{1} PUnit.{1} PUnit.{1} PUnit.{1} PUnit.{1} PUnit.{1} PUnit.{1} support where | |
| auxiliary := auxiliary | |
| rankBound := 1 | |
| auxiliary_rank := auxiliary_rank | |
| order := 7 * N | |
| leftDegree := 7 * N | |
| middleDegree := 7 * N | |
| rightDegree := 6 * N | |
| leftMap := leftMap | |
| middleMap := middleMap | |
| rightMap := rightMap | |
| left_degree := fun x _ _ => degree_product _ (fun i => leftBlock_degree (x i)) | |
| middle_degree := fun y _ _ => degree_product _ (fun i => middleBlock_degree (y i)) | |
| right_degree := fun z _ _ => degree_product _ (fun i => rightBlock_degree (z i)) | |
| value := value | |
| vanishes := fun x y z hs => (kernel_coefficients x y z hs).1 | |
| leading := fun x y z hs => (kernel_coefficients x y z hs).2 | |
| synchronized := fun _ _ _ _ _ => ⟨Subsingleton.elim _ _, Subsingleton.elim _ _⟩ | |
| @[simp] theorem execution_value (N : ℕ) : (execution N).value = value := rfl | |
| theorem value_ne_zero_iff {N : ℕ} (x y z : Blocks N × (PUnit.{1} × PUnit.{1})) : | |
| (execution N).value x y z ≠ 0 ↔ | |
| ∀ i, CW75Primitive.tensor (x.1 i) (y.1 i) (z.1 i) ≠ 0 := by | |
| simp [execution_value, value, Finset.prod_ne_zero_iff] | |
| theorem value_retained {N : ℕ} (x y z : Blocks N × (PUnit.{1} × PUnit.{1})) : | |
| (execution N).value x y z = | |
| if ∀ i, CW75Primitive.tensor (x.1 i) (y.1 i) (z.1 i) ≠ 0 then 1 else 0 := by | |
| by_cases h : ∀ i, CW75Primitive.tensor (x.1 i) (y.1 i) (z.1 i) ≠ 0 | |
| · simp only [ite_eq_left h, execution_value, value] | |
| apply Finset.prod_eq_one | |
| intro i hi | |
| exact retained_coefficient_one _ _ _ (h i) | |
| · rw [ite_eq_right h] | |
| by_contra hn | |
| exact h ((value_ne_zero_iff x y z).mp hn) | |
| end MatrixMultiplication.Foundation.CW75InitialExecution | |
| end | |
| end OAI | |