openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /CoppersmithWinograd /CWCompatibilityTransfer.lean
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWLeafStatistics | |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMasks | |
| namespace OAI | |
| /-! Coppersmith–Winograd tensors, tensor powers and local restrictions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CWCompatibilityTransfer | |
| open MatrixMultiplication.Foundation CWLeafRestrictions InheritedMasks | |
| open scoped BigOperators | |
| def wordWeight {n : ℕ} (w : Fin n → Fin 7) : ℕ := ∑ i, CWLeafStatistics.weight (w i) | |
| def wordComplement {n : ℕ} (w : Fin n → Fin 7) : Fin n → Fin 7 := | |
| fun i => complement 5 (w i) | |
| @[simp] theorem wordComplement_involution {n : ℕ} (w : Fin n → Fin 7) : | |
| wordComplement (wordComplement w) = w := by | |
| funext i | |
| exact complement_involution 5 (w i) | |
| theorem weight_eq_zero_iff : ∀ x : Fin 7, CWLeafStatistics.weight x = 0 ↔ x = 0 := by | |
| decide +kernel | |
| theorem wordWeight_zero {n : ℕ} (w : Fin n → Fin 7) (h : wordWeight w = 0) : | |
| ∀ i, w i = 0 := by | |
| intro i | |
| have hle : CWLeafStatistics.weight (w i) ≤ wordWeight w := by | |
| unfold wordWeight | |
| exact Finset.single_le_sum (fun j _ => Nat.zero_le (CWLeafStatistics.weight (w j))) | |
| (Finset.mem_univ i) | |
| apply (weight_eq_zero_iff (w i)).mp | |
| omega | |
| theorem site_nonzero {F : Type*} [CommRing F] {n : ℕ} | |
| (x y z : Fin n → Fin 7) | |
| (h : Tensor.power (FieldCW.tensor F 5) n x y z ≠ 0) (i : Fin n) : | |
| FieldCW.tensor F 5 (x i) (y i) (z i) ≠ 0 := by | |
| intro hi | |
| apply h | |
| exact Finset.prod_eq_zero (Finset.mem_univ i) hi | |
| theorem zeroX_word_match {F : Type*} [CommRing F] {n : ℕ} | |
| (x y z : Fin n → Fin 7) | |
| (h : Tensor.power (FieldCW.tensor F 5) n x y z ≠ 0) | |
| (hx : wordWeight x = 0) : z = wordComplement y := by | |
| funext i | |
| have hi := site_nonzero x y z h i | |
| rw [wordWeight_zero x hx i] at hi | |
| have heq : FieldCW.tensor F 5 0 (y i) (z i) = | |
| if y i = complement 5 (z i) then 1 else 0 := by | |
| simpa only [complement_involution] using | |
| tensor_zero_matching F 5 (y i) (complement 5 (z i)) | |
| have hyz : y i = complement 5 (z i) := by | |
| by_contra hn | |
| rw [heq, ite_eq_right hn] at hi | |
| exact hi rfl | |
| have hh := congrArg (complement 5) hyz | |
| simpa only [wordComplement, complement_involution] using hh.symm | |
| theorem zeroY_word_match {F : Type*} [CommRing F] {n : ℕ} | |
| (x y z : Fin n → Fin 7) | |
| (h : Tensor.power (FieldCW.tensor F 5) n x y z ≠ 0) | |
| (hy : wordWeight y = 0) : z = wordComplement x := by | |
| funext i | |
| have hi := site_nonzero x y z h i | |
| rw [wordWeight_zero y hy i] at hi | |
| have heq : FieldCW.tensor F 5 (x i) 0 (z i) = | |
| if x i = complement 5 (z i) then 1 else 0 := by | |
| simpa only [complement_involution] using | |
| tensor_middle_zero_matching F 5 (x i) (complement 5 (z i)) | |
| have hxz : x i = complement 5 (z i) := by | |
| by_contra hn | |
| rw [heq, ite_eq_right hn] at hi | |
| exact hi rfl | |
| have hh := congrArg (complement 5) hxz | |
| simpa only [wordComplement, complement_involution] using hh.symm | |
| theorem zeroZ_word_match {F : Type*} [CommRing F] {n : ℕ} | |
| (x y z : Fin n → Fin 7) | |
| (h : Tensor.power (FieldCW.tensor F 5) n x y z ≠ 0) | |
| (hz : wordWeight z = 0) : y = wordComplement x := by | |
| funext i | |
| have hi := site_nonzero x y z h i | |
| rw [wordWeight_zero z hz i] at hi | |
| have heq : FieldCW.tensor F 5 (x i) (y i) 0 = | |
| if x i = complement 5 (y i) then 1 else 0 := by | |
| simpa only [complement_involution] using | |
| tensor_last_zero_matching F 5 (x i) (complement 5 (y i)) | |
| have hxy : x i = complement 5 (y i) := by | |
| by_contra hn | |
| rw [heq, ite_eq_right hn] at hi | |
| exact hi rfl | |
| have hh := congrArg (complement 5) hxy | |
| simpa only [wordComplement, complement_involution] using hh.symm | |
| theorem transfer_window {P A : Type*} [Fintype P] [DecidableEq P] | |
| [Fintype A] [DecidableEq A] {n : ℕ} | |
| (statistic : (Fin n → Fin 7) → A) (κ : A ≃ A) | |
| (hstatistic : ∀ w, statistic (wordComplement w) = κ (statistic w)) | |
| (x y : P → (Fin n → Fin 7)) (hmatch : ∀ i, y i = wordComplement (x i)) | |
| (ν : A → ℝ) (η : ℝ) | |
| (hx : typeWindow ν η (fun i => statistic (x i))) : | |
| typeWindow (ν ∘ κ.symm) η (fun i => statistic (y i)) := by | |
| have heq : (fun i => statistic (y i)) = κ ∘ (fun i => statistic (x i)) := by | |
| funext i | |
| rw [hmatch i, hstatistic] | |
| rfl | |
| rw [heq] | |
| exact (typeWindow_equiv ν η (fun i => statistic (x i)) κ).mpr hx | |
| end MatrixMultiplication.CWCompatibilityTransfer | |
| end | |
| end OAI | |