File size: 3,090 Bytes
cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | 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
|