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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.FiniteFamily
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldInitialSource
open MatrixMultiplication.Foundation AllFieldFiniteFamily CommonDimensions
open scoped BigOperators Classical
abbrev Sites (K N : ℕ) := Fin K × (Fin N × Fin 8)
abbrev Words (K N : ℕ) := Fin K → Fin N → Fin 8 → Fin 7
def sitesEquiv (K N : ℕ) : Sites K N ≃ Fin (8 * K * N) :=
Fintype.equivOfCardEq (by simp only [Sites, Fintype.card_prod, Fintype.card_fin]; ring)
def flatten (K N : ℕ) (w : Words K N) : AllFieldSource.Word K N :=
fun i => let p := (sitesEquiv K N).symm i; w p.1 p.2.1 p.2.2
def groupedSource (F : Type*) [Field F] (K N : ℕ) :
Tensor F (Words K N) (Words K N) (Words K N) :=
fun x y z => ∏ p : Sites K N,
FieldCW.tensor F 5 (x p.1 p.2.1 p.2.2) (y p.1 p.2.1 p.2.2) (z p.1 p.2.1 p.2.2)
theorem groupedSource_eq_lots (F : Type*) [Field F] (K N : ℕ) :
groupedSource F K N =
familyProduct (fun _ : Fin K => Tensor.power (Tensor.power (FieldCW.tensor F 5) 8) N) := by
funext x y z
simp only [groupedSource, Sites, familyProduct, Tensor.power, Fintype.prod_prod_type]
def groupingMap (F : Type*) [Field F] (K N : ℕ) :
LocalMap (cwSource F K N) (groupedSource F K N) where
x := fun x s => if s = flatten K N x then 1 else 0
y := fun y s => if s = flatten K N y then 1 else 0
z := fun z s => if s = flatten K N z then 1 else 0
coefficient := by
rw [← Tensor.pullback_eq_restrict]
funext x y z
simp only [Tensor.pullback, cwSource, Tensor.power, groupedSource, flatten]
apply Fintype.prod_equiv (sitesEquiv K N).symm
intro i
rfl
def initialExecution (F : Type*) [Field F] (K N : ℕ) :
Execution (cwSource F K N) (groupedSource F K N) :=
(startCW F K N).restrict _ (groupingMap F K N)
@[simp] theorem initialExecution_copies (F : Type*) [Field F] (K N : ℕ) :
Fintype.card (initialExecution F K N).Copies = 1 := rfl
end MatrixMultiplication.AllFieldInitialSource
end
end OAI
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