| 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 |
|
|
