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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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | 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
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