File size: 3,511 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.FiniteFamily
import OAI.LinearAlgebra.MatrixMultiplication.Tensor.TerminalRelabel
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldTerminal
open MatrixMultiplication.Foundation AllFieldFiniteFamily CommonDimensions
open scoped BigOperators Classical
variable {F : Type*} {H : Type} [Field F] [Fintype H]
{X Y Z : H → Type} [∀ h, Fintype (X h)] [∀ h, Fintype (Y h)] [∀ h, Fintype (Z h)]
(T : ∀ h, Tensor F (X h) (Y h) (Z h)) (A B C : H → Type)
[∀ h, Fintype (A h)] [∀ h, Fintype (B h)] [∀ h, Fintype (C h)]
(maps : ∀ h, LocalMap (T h) (Tensor.matrixCoefficients (A h) (B h) (C h)))
def terminalProductMap :
LocalMap (familyProduct T) (Tensor.matrixCoefficients (∀ h, A h) (∀ h, B h) (∀ h, C h)) :=
LocalMap.ofExists (TerminalProducts.matrix_restriction_of_history_restrictions
T A B C (fun h => (maps h).x) (fun h => (maps h).y) (fun h => (maps h).z)
(fun h => (maps h).coefficient))
variable {I : Type} [Fintype I] {K N : ℕ}
def terminalExecution
(E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T))) :
Execution (cwSource F K N)
(AllFieldSource.matrixTarget F (Fintype.card (∀ h, A h))
(Fintype.card (∀ h, B h)) (Fintype.card (∀ h, C h)) (Fintype.card I)) :=
(E.restrict _ ((terminalProductMap T A B C maps).parallelCopies I)).restrict _
(LocalMap.ofExists (TerminalRelabel.directSum_matrix_restrict_fin (K := F)
I (∀ h, A h) (∀ h, B h) (∀ h, C h)))
@[simp] theorem terminalExecution_copies
(E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T))) :
Fintype.card (terminalExecution T A B C maps E).Copies = Fintype.card E.Copies := rfl
def finish
(E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T)))
(hA : 0 < Fintype.card (∀ h, A h)) (hB : 0 < Fintype.card (∀ h, B h))
(hC : 0 < Fintype.card (∀ h, C h)) (hI : 0 < Fintype.card I) : AllFieldWitness F :=
(terminalExecution T A B C maps E).toWitness hA hB hC hI
@[simp] theorem finish_multiplicity
(E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T)))
(hA : 0 < Fintype.card (∀ h, A h)) (hB : 0 < Fintype.card (∀ h, B h))
(hC : 0 < Fintype.card (∀ h, C h)) (hI : 0 < Fintype.card I) :
(finish T A B C maps E hA hB hC hI).multiplicity = Fintype.card I := rfl
theorem finish_volume
(E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T)))
(hA : 0 < Fintype.card (∀ h, A h)) (hB : 0 < Fintype.card (∀ h, B h))
(hC : 0 < Fintype.card (∀ h, C h)) (hI : 0 < Fintype.card I) :
(finish T A B C maps E hA hB hC hI).volume =
∏ h, Fintype.card (A h) * Fintype.card (B h) * Fintype.card (C h) := by
change Fintype.card (∀ h, A h) * Fintype.card (∀ h, B h) *
Fintype.card (∀ h, C h) = _
simp only [Fintype.card_pi, Finset.prod_mul_distrib]
@[simp] theorem finish_rankBound
(E : Execution (cwSource F K N) (Tensor.directSum (fun _ : I => familyProduct T)))
(hA : 0 < Fintype.card (∀ h, A h)) (hB : 0 < Fintype.card (∀ h, B h))
(hC : 0 < Fintype.card (∀ h, C h)) (hI : 0 < Fintype.card I) :
(finish T A B C maps E hA hB hC hI).rankBound =
AllFieldSource.rankBudget E.Copies K N := rfl
end MatrixMultiplication.AllFieldTerminal
end
end OAI
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