openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /CoppersmithWinograd /CWStrandCompatibility.lean
| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.CWStrands | |
| namespace OAI | |
| /-! Coppersmith–Winograd tensors, tensor powers and local restrictions. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.CWStrandCompatibility | |
| open MatrixMultiplication.Foundation CWStrands CWCompatibilityTransfer InheritedMasks | |
| open scoped BigOperators | |
| variable {F : Type*} [CommRing F] | |
| theorem shape_nonzero_power {n : ℕ} (g : Fin 3 → ℕ) | |
| (x y z : Fin n → Fin 7) (h : shapeTensor (F := F) (Fin n) g x y z ≠ 0) : | |
| Tensor.power (FieldCW.tensor F 5) n x y z ≠ 0 := by | |
| have hm := shapeTensor_support g x y z h | |
| simpa only [shapeTensor, ite_eq_left hm, strand, Tensor.power] using h | |
| section Classes | |
| variable {P A : Type*} [Fintype P] [DecidableEq P] | |
| [Fintype A] [DecidableEq A] {n : ℕ} | |
| (g : P → Fin 3 → ℕ) (x y z : P → (Fin n → Fin 7)) | |
| (h : (∏ i, shapeTensor (F := F) (Fin n) (g i) (x i) (y i) (z i)) ≠ 0) | |
| (statistic : (Fin n → Fin 7) → A) (κ : A ≃ A) | |
| (hstatistic : ∀ w, statistic (wordComplement w) = κ (statistic w)) | |
| (ν : A → ℝ) (η : ℝ) | |
| include h | |
| omit [DecidableEq P] in | |
| theorem class_coefficient_nonzero (i : P) : | |
| shapeTensor (F := F) (Fin n) (g i) (x i) (y i) (z i) ≠ 0 := by | |
| intro hi | |
| exact h (Finset.prod_eq_zero (Finset.mem_univ i) hi) | |
| include hstatistic | |
| theorem zeroZ_transfer (hg : ∀ i, g i 2 = 0) | |
| (hx : typeWindow ν η (fun i => statistic (x i))) : | |
| typeWindow (ν ∘ κ.symm) η (fun i => statistic (y i)) := by | |
| apply transfer_window statistic κ hstatistic x y _ ν η hx | |
| intro i | |
| have hi := class_coefficient_nonzero g x y z h i | |
| have hm := shapeTensor_support (g i) (x i) (y i) (z i) hi | |
| have hz : wordWeight (z i) = 0 := by | |
| change weight (z i) = 0 | |
| exact hm.2.2.trans (hg i) | |
| exact zeroZ_word_match (x i) (y i) (z i) | |
| (shape_nonzero_power (g i) (x i) (y i) (z i) hi) hz | |
| theorem zeroY_transfer (hg : ∀ i, g i 1 = 0) | |
| (hx : typeWindow ν η (fun i => statistic (x i))) : | |
| typeWindow (ν ∘ κ.symm) η (fun i => statistic (z i)) := by | |
| apply transfer_window statistic κ hstatistic x z _ ν η hx | |
| intro i | |
| have hi := class_coefficient_nonzero g x y z h i | |
| have hm := shapeTensor_support (g i) (x i) (y i) (z i) hi | |
| have hy : wordWeight (y i) = 0 := by | |
| change weight (y i) = 0 | |
| exact hm.2.1.trans (hg i) | |
| exact zeroY_word_match (x i) (y i) (z i) | |
| (shape_nonzero_power (g i) (x i) (y i) (z i) hi) hy | |
| theorem zeroX_transfer (hg : ∀ i, g i 0 = 0) | |
| (hy : typeWindow ν η (fun i => statistic (y i))) : | |
| typeWindow (ν ∘ κ.symm) η (fun i => statistic (z i)) := by | |
| apply transfer_window statistic κ hstatistic y z _ ν η hy | |
| intro i | |
| have hi := class_coefficient_nonzero g x y z h i | |
| have hm := shapeTensor_support (g i) (x i) (y i) (z i) hi | |
| have hx : wordWeight (x i) = 0 := by | |
| change weight (x i) = 0 | |
| exact hm.1.trans (hg i) | |
| exact zeroX_word_match (x i) (y i) (z i) | |
| (shape_nonzero_power (g i) (x i) (y i) (z i) hi) hx | |
| end Classes | |
| end MatrixMultiplication.CWStrandCompatibility | |
| end | |
| end OAI | |