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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 | 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
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