| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexTensorRestrictionComposition |
| import Mathlib.Tactic.SplitIfs |
|
|
| namespace OAI |
|
|
| /-! Finite coefficient tensors and their algebraic transformations. -/ |
|
|
| open scoped BigOperators |
| open MatrixMultiplication.Foundation |
|
|
| namespace MatrixMultiplication.Replication |
|
|
| variable {K I R X Y Z X' Y' Z' : Type*} [CommSemiring K] |
| |
| theorem restrict_directSum |
| [Fintype I] [DecidableEq I] [Fintype X] [Fintype Y] [Fintype Z] |
| (T : I → Tensor K X Y Z) |
| (a : I → X' → X → K) (b : I → Y' → Y → K) (c : I → Z' → Z → K) : |
| Tensor.restrict |
| (fun x s => if x.1 = s.1 then a x.1 x.2 s.2 else 0) |
| (fun y s => if y.1 = s.1 then b y.1 y.2 s.2 else 0) |
| (fun z s => if z.1 = s.1 then c z.1 z.2 s.2 else 0) |
| (Tensor.directSum T) = Tensor.directSum (fun i => Tensor.restrict (a i) (b i) (c i) (T i)) := by |
| classical |
| funext x y z |
| rcases x with ⟨i, x⟩ |
| rcases y with ⟨j, y⟩ |
| rcases z with ⟨k, z⟩ |
| by_cases hij : i = j |
| · subst j |
| by_cases hik : i = k |
| · subst k |
| simp [Tensor.restrict, Tensor.directSum, Fintype.sum_prod_type, |
| ite_and, ite_mul, mul_ite] |
| · simp [Tensor.restrict, Tensor.directSum, Fintype.sum_prod_type, |
| ite_and, ite_mul, mul_ite, hik] |
| · simp [Tensor.restrict, Tensor.directSum, Fintype.sum_prod_type, |
| ite_and, ite_mul, mul_ite, hij] |
| intro hjk hik |
| exact (hij (hik.trans hjk.symm)).elim |
|
|
| theorem swap_replication [DecidableEq I] [DecidableEq R] |
| (T : I → Tensor K X Y Z) : |
| Tensor.pullback |
| (fun x : I × (R × X) => (x.2.1, (x.1, x.2.2))) |
| (fun y : I × (R × Y) => (y.2.1, (y.1, y.2.2))) |
| (fun z : I × (R × Z) => (z.2.1, (z.1, z.2.2))) |
| (Tensor.directSum (fun _ : R => Tensor.directSum T)) = |
| Tensor.directSum (fun i => Tensor.directSum (fun _ : R => T i)) := by |
| funext x y z |
| simp only [Tensor.pullback, Tensor.directSum] |
| split_ifs <;> simp_all |
|
|
| theorem repair_branches |
| [Fintype I] [DecidableEq I] [Fintype R] [DecidableEq R] |
| [Fintype X] [Fintype Y] [Fintype Z] |
| (T : I → Tensor K X Y Z) (Q : I → Tensor K X' Y' Z') |
| (a : I → X' → (R × X) → K) (b : I → Y' → (R × Y) → K) |
| (c : I → Z' → (R × Z) → K) |
| (h : ∀ i, Tensor.restrict (a i) (b i) (c i) |
| (Tensor.directSum (fun _ : R => T i)) = Q i) : |
| ∃ (A : (I × X') → (R × (I × X)) → K) |
| (B : (I × Y') → (R × (I × Y)) → K) |
| (C : (I × Z') → (R × (I × Z)) → K), |
| Tensor.restrict A B C (Tensor.directSum (fun _ : R => Tensor.directSum T)) = |
| Tensor.directSum Q := by |
| classical |
| have hb := restrict_directSum (fun i => Tensor.directSum (fun _ : R => T i)) a b c |
| simp_rw [h] at hb |
| rw [← swap_replication T, Tensor.pullback_eq_restrict, Tensor.restrict_restrict] at hb |
| exact ⟨_, _, _, hb⟩ |
| |
| end MatrixMultiplication.Replication |
| |
| end OAI |
| |
