openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /CoppersmithWinograd /CWStrands.lean
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWCompatibilityTransfer | |
| namespace OAI | |
| /-! Coppersmith–Winograd tensors, tensor powers and local restrictions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CWStrands | |
| open MatrixMultiplication.Foundation | |
| open scoped BigOperators | |
| variable {F : Type*} [CommRing F] | |
| private theorem site_weight_zero_support (x y z : Fin 7) (hx : x.val = 0) | |
| (h : FieldCW.tensor F 5 x y z ≠ 0) : | |
| CWLeafStatistics.weight x + CWLeafStatistics.weight y + CWLeafStatistics.weight z = 2 := by | |
| rw [FieldCW.tensor_zero_side F 5 x y z hx] at h | |
| have hs : (y.val = 0 ∧ z.val = 6) ∨ (y.val = 6 ∧ z.val = 0) ∨ | |
| (1 ≤ y.val ∧ y.val ≤ 5 ∧ y = z) := by | |
| by_contra hs | |
| exact h (ite_eq_right hs) | |
| rcases hs with ⟨hy, hz⟩ | ⟨hy, hz⟩ | ⟨hylo, hyhi, hyz⟩ | |
| · simp [CWLeafStatistics.weight, hx, hy, hz] | |
| · simp [CWLeafStatistics.weight, hx, hy, hz] | |
| · subst z | |
| have hy0 : y.val ≠ 0 := by omega | |
| have hy6 : y.val ≠ 6 := by omega | |
| simp [CWLeafStatistics.weight, hx, hy0, hy6] | |
| theorem site_weight_support (x y z : Fin 7) | |
| (h : FieldCW.tensor F 5 x y z ≠ 0) : | |
| CWLeafStatistics.weight x + CWLeafStatistics.weight y + CWLeafStatistics.weight z = 2 := by | |
| by_cases hx : x.val = 0 | |
| · exact site_weight_zero_support x y z hx h | |
| by_cases hy : y.val = 0 | |
| · rw [FieldCW.tensor_cyclic F 5 x y z] at h | |
| have hw := site_weight_zero_support y z x hy h | |
| omega | |
| by_cases hz : z.val = 0 | |
| · rw [FieldCW.tensor_cyclic F 5 x y z, FieldCW.tensor_cyclic F 5 y z x] at h | |
| have hw := site_weight_zero_support z x y hz h | |
| omega | |
| · exact False.elim (h (by simp [FieldCW.tensor, FieldCW.zeroForm, hx, hy, hz])) | |
| def weight {P : Type*} [Fintype P] (w : P → Fin 7) : ℕ := | |
| ∑ i, CWLeafStatistics.weight (w i) | |
| def strand (P : Type*) [Fintype P] : Tensor F (P → Fin 7) (P → Fin 7) (P → Fin 7) := | |
| fun x y z => ∏ i, FieldCW.tensor F 5 (x i) (y i) (z i) | |
| theorem strand_support {P : Type*} [Fintype P] | |
| (x y z : P → Fin 7) (h : strand (F := F) P x y z ≠ 0) : | |
| weight x + weight y + weight z = 2 * Fintype.card P := by | |
| have hi (i : P) : FieldCW.tensor F 5 (x i) (y i) (z i) ≠ 0 := by | |
| intro he | |
| exact h (Finset.prod_eq_zero (Finset.mem_univ i) he) | |
| calc | |
| weight x + weight y + weight z = | |
| ∑ i, (CWLeafStatistics.weight (x i) + CWLeafStatistics.weight (y i) + | |
| CWLeafStatistics.weight (z i)) := by simp [weight, Finset.sum_add_distrib] | |
| _ = ∑ _i : P, 2 := Finset.sum_congr rfl (fun i _ => site_weight_support _ _ _ (hi i)) | |
| _ = 2 * Fintype.card P := by simp [Nat.mul_comm] | |
| def shapeTensor (P : Type*) [Fintype P] (g : Fin 3 → ℕ) : | |
| Tensor F (P → Fin 7) (P → Fin 7) (P → Fin 7) := | |
| fun x y z => if weight x = g 0 ∧ weight y = g 1 ∧ weight z = g 2 | |
| then strand P x y z else 0 | |
| theorem shapeTensor_support {P : Type*} [Fintype P] | |
| (g : Fin 3 → ℕ) (x y z : P → Fin 7) | |
| (h : shapeTensor (F := F) P g x y z ≠ 0) : | |
| weight x = g 0 ∧ weight y = g 1 ∧ weight z = g 2 := by | |
| classical | |
| by_contra hn | |
| exact h (ite_eq_right hn) | |
| theorem weight_sum_elim {P Q : Type*} [Fintype P] [Fintype Q] | |
| (x : P → Fin 7) (y : Q → Fin 7) : | |
| weight (Sum.elim x y) = weight x + weight y := by | |
| simp [weight, Fintype.sum_sum_type] | |
| theorem strand_sum_elim {P Q : Type*} [Fintype P] [Fintype Q] | |
| (x : (P → Fin 7) × (Q → Fin 7)) | |
| (y : (P → Fin 7) × (Q → Fin 7)) | |
| (z : (P → Fin 7) × (Q → Fin 7)) : | |
| strand (F := F) (P ⊕ Q) (Sum.elim x.1 x.2) (Sum.elim y.1 y.2) (Sum.elim z.1 z.2) = | |
| Tensor.product (strand P) (strand Q) x y z := by | |
| simp [strand, Tensor.product, Fintype.prod_sum_type] | |
| theorem shape_split {P Q : Type*} [Fintype P] [Fintype Q] | |
| (u v : Fin 3 → ℕ) | |
| (x : (P → Fin 7) × (Q → Fin 7)) | |
| (y : (P → Fin 7) × (Q → Fin 7)) | |
| (z : (P → Fin 7) × (Q → Fin 7)) : | |
| (if weight x.1 = u 0 ∧ weight y.1 = u 1 ∧ weight z.1 = u 2 then | |
| shapeTensor (F := F) (P ⊕ Q) (u + v) (Sum.elim x.1 x.2) (Sum.elim y.1 y.2) | |
| (Sum.elim z.1 z.2) else 0) = | |
| Tensor.product (shapeTensor P u) (shapeTensor Q v) x y z := by | |
| classical | |
| simp only [shapeTensor, weight_sum_elim, Pi.add_apply, strand_sum_elim, | |
| Tensor.product] | |
| by_cases hx : weight x.1 = u 0 | |
| · by_cases hy : weight y.1 = u 1 | |
| · by_cases hz : weight z.1 = u 2 | |
| · simp [hx, hy, hz, mul_ite] | |
| · simp [hz] | |
| · simp [hy] | |
| · simp [hx] | |
| theorem shape_split_sub {P Q : Type*} [Fintype P] [Fintype Q] | |
| (g u : Fin 3 → ℕ) (hu : ∀ s, u s ≤ g s) | |
| (x : (P → Fin 7) × (Q → Fin 7)) | |
| (y : (P → Fin 7) × (Q → Fin 7)) | |
| (z : (P → Fin 7) × (Q → Fin 7)) : | |
| (if weight x.1 = u 0 ∧ weight y.1 = u 1 ∧ weight z.1 = u 2 then | |
| shapeTensor (F := F) (P ⊕ Q) g (Sum.elim x.1 x.2) (Sum.elim y.1 y.2) | |
| (Sum.elim z.1 z.2) else 0) = | |
| Tensor.product (shapeTensor P u) (shapeTensor Q (g - u)) x y z := by | |
| have hg : u + (g - u) = g := by | |
| funext s | |
| exact Nat.add_sub_of_le (hu s) | |
| simpa only [hg] using shape_split u (g - u) x y z | |
| def splitMatrix {P Q : Type*} [Fintype P] (u : ℕ) : | |
| ((P → Fin 7) × (Q → Fin 7)) → ((P ⊕ Q) → Fin 7) → F := by | |
| classical | |
| exact fun x w => if w = Sum.elim x.1 x.2 ∧ weight x.1 = u then 1 else 0 | |
| theorem shape_split_restrict {P Q : Type*} [Fintype P] [Fintype Q] | |
| [DecidableEq P] [DecidableEq Q] (u v : Fin 3 → ℕ) : | |
| Tensor.restrict (splitMatrix (F := F) (u 0)) (splitMatrix (u 1)) (splitMatrix (u 2)) | |
| (shapeTensor (P ⊕ Q) (u + v)) = | |
| Tensor.product (shapeTensor P u) (shapeTensor Q v) := by | |
| classical | |
| funext x y z | |
| have hcollapse : Tensor.restrict (splitMatrix (F := F) (u 0)) | |
| (splitMatrix (u 1)) (splitMatrix (u 2)) (shapeTensor (P ⊕ Q) (u + v)) x y z = | |
| if weight x.1 = u 0 ∧ weight y.1 = u 1 ∧ weight z.1 = u 2 then | |
| shapeTensor (P ⊕ Q) (u + v) (Sum.elim x.1 x.2) (Sum.elim y.1 y.2) | |
| (Sum.elim z.1 z.2) else 0 := by | |
| simp [Tensor.restrict, splitMatrix, ite_and, ite_mul, mul_ite] | |
| split_ifs <;> simp_all | |
| rw [hcollapse] | |
| exact shape_split u v x y z | |
| end MatrixMultiplication.CWStrands | |
| end | |
| end OAI | |