File size: 2,894 Bytes
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 | 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
|