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f18e7de | 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 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexMatrixTensor
import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexTensorRestrictionComposition
namespace OAI
/-! Finite coefficient tensors and their algebraic transformations. -/
noncomputable section
open MatrixMultiplication.Foundation
namespace MatrixMultiplication.TerminalRelabel
variable {K I A B C I' A' B' C' : Type*} [CommSemiring K]
def taggedPair (eI : I' ≃ I) (eA : A' ≃ A) (eB : B' ≃ B) :
I' × (A' × B') → I × (A × B) :=
fun x => (eI x.1, (eA x.2.1, eB x.2.2))
theorem directSum_matrix_pullback
[DecidableEq I] [DecidableEq A] [DecidableEq B] [DecidableEq C]
[DecidableEq I'] [DecidableEq A'] [DecidableEq B'] [DecidableEq C']
(eI : I' ≃ I) (eA : A' ≃ A) (eB : B' ≃ B) (eC : C' ≃ C) :
Tensor.pullback (taggedPair eI eA eB) (taggedPair eI eB eC) (taggedPair eI eC eA)
(Tensor.directSum (fun _ : I => Tensor.matrixCoefficients (K := K) A B C)) =
Tensor.directSum (fun _ : I' => Tensor.matrixCoefficients (K := K) A' B' C') := by
funext x y z
simp only [Tensor.pullback, taggedPair, Tensor.directSum, Tensor.matrixCoefficients,
eI.injective.eq_iff, eA.injective.eq_iff, eB.injective.eq_iff, eC.injective.eq_iff]
theorem directSum_matrix_restrict_equiv
[Fintype I] [Fintype A] [Fintype B] [Fintype C]
[DecidableEq I] [DecidableEq A] [DecidableEq B] [DecidableEq C]
[DecidableEq I'] [DecidableEq A'] [DecidableEq B'] [DecidableEq C']
(eI : I' ≃ I) (eA : A' ≃ A) (eB : B' ≃ B) (eC : C' ≃ C) :
Tensor.restrict
(fun x s => if s = taggedPair eI eA eB x then (1 : K) else 0)
(fun y s => if s = taggedPair eI eB eC y then (1 : K) else 0)
(fun z s => if s = taggedPair eI eC eA z then (1 : K) else 0)
(Tensor.directSum (fun _ : I => Tensor.matrixCoefficients (K := K) A B C)) =
Tensor.directSum (fun _ : I' => Tensor.matrixCoefficients (K := K) A' B' C') := by
rw [← Tensor.pullback_eq_restrict]
exact directSum_matrix_pullback eI eA eB eC
theorem exists_directSum_matrix_restrict_equiv
[Fintype I] [Fintype A] [Fintype B] [Fintype C]
[DecidableEq I] [DecidableEq A] [DecidableEq B] [DecidableEq C]
[DecidableEq I'] [DecidableEq A'] [DecidableEq B'] [DecidableEq C']
(eI : I' ≃ I) (eA : A' ≃ A) (eB : B' ≃ B) (eC : C' ≃ C) :
∃ (a : (I' × (A' × B')) → (I × (A × B)) → K)
(b : (I' × (B' × C')) → (I × (B × C)) → K)
(c : (I' × (C' × A')) → (I × (C × A)) → K),
Tensor.restrict a b c
(Tensor.directSum (fun _ : I => Tensor.matrixCoefficients (K := K) A B C)) =
Tensor.directSum (fun _ : I' => Tensor.matrixCoefficients (K := K) A' B' C') :=
⟨_, _, _, directSum_matrix_restrict_equiv eI eA eB eC⟩
theorem directSum_matrix_restrict_of_card_eq
(I A B C : Type*) [Fintype I] [Fintype A] [Fintype B] [Fintype C]
[DecidableEq I] [DecidableEq A] [DecidableEq B] [DecidableEq C]
{L a b c : ℕ} (hI : Fintype.card I = L) (hA : Fintype.card A = a)
(hB : Fintype.card B = b) (hC : Fintype.card C = c) :
∃ (aa : (Fin L × (Fin a × Fin b)) → (I × (A × B)) → K)
(bb : (Fin L × (Fin b × Fin c)) → (I × (B × C)) → K)
(cc : (Fin L × (Fin c × Fin a)) → (I × (C × A)) → K),
Tensor.restrict aa bb cc
(Tensor.directSum (fun _ : I => Tensor.matrixCoefficients (K := K) A B C)) =
Tensor.directSum (fun _ : Fin L =>
Tensor.matrixCoefficients (K := K) (Fin a) (Fin b) (Fin c)) :=
exists_directSum_matrix_restrict_equiv
(Fintype.equivFinOfCardEq hI).symm (Fintype.equivFinOfCardEq hA).symm
(Fintype.equivFinOfCardEq hB).symm (Fintype.equivFinOfCardEq hC).symm
theorem directSum_matrix_restrict_fin
(I A B C : Type*) [Fintype I] [Fintype A] [Fintype B] [Fintype C]
[DecidableEq I] [DecidableEq A] [DecidableEq B] [DecidableEq C] :
∃ (a : (Fin (Fintype.card I) × (Fin (Fintype.card A) × Fin (Fintype.card B))) →
(I × (A × B)) → K)
(b : (Fin (Fintype.card I) × (Fin (Fintype.card B) × Fin (Fintype.card C))) →
(I × (B × C)) → K)
(c : (Fin (Fintype.card I) × (Fin (Fintype.card C) × Fin (Fintype.card A))) →
(I × (C × A)) → K),
Tensor.restrict a b c
(Tensor.directSum (fun _ : I => Tensor.matrixCoefficients (K := K) A B C)) =
Tensor.directSum (fun _ : Fin (Fintype.card I) =>
Tensor.matrixCoefficients (K := K)
(Fin (Fintype.card A)) (Fin (Fintype.card B)) (Fin (Fintype.card C))) :=
directSum_matrix_restrict_of_card_eq I A B C rfl rfl rfl rfl
end MatrixMultiplication.TerminalRelabel
end
end OAI
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