File size: 2,897 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 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexLocalMaps
import Mathlib.Logic.Equiv.Basic
namespace OAI
/-! Finite coefficient tensors and their algebraic transformations. -/
open scoped BigOperators
namespace MatrixMultiplication.Foundation.LabelHierarchySeparation
variable {K I : Type*} [CommSemiring K] [Fintype I]
variable {X Y Z : I → Type*}
def bankTensor (T : ∀ i, Tensor K (X i) (Y i) (Z i)) :
Tensor K (∀ i, X i) (∀ i, Y i) (∀ i, Z i) :=
fun x y z => ∏ i, T i (x i) (y i) (z i)
theorem bankTensor_rankAtMost (T : ∀ i, Tensor K (X i) (Y i) (Z i))
(r : I → ℕ) (h : ∀ i, Tensor.RankAtMost (T i) (r i)) :
Tensor.RankAtMost (bankTensor T) (∏ i, r i) := by
classical
choose a b c hc using h
let aa : (∀ i, Fin (r i)) → (∀ i, X i) → K :=
fun f x => ∏ i, a i (f i) (x i)
let bb : (∀ i, Fin (r i)) → (∀ i, Y i) → K :=
fun f y => ∏ i, b i (f i) (y i)
let cc : (∀ i, Fin (r i)) → (∀ i, Z i) → K :=
fun f z => ∏ i, c i (f i) (z i)
have heq : bankTensor T =
fun x y z => ∑ f, Tensor.rankOne (aa f) (bb f) (cc f) x y z := by
funext x y z
simp only [bankTensor, hc, Fintype.prod_sum]
apply Finset.sum_congr rfl
intro f hf
simp only [Tensor.rankOne, aa, bb, cc, Finset.prod_mul_distrib]
rw [heq]
simpa only [Fintype.card_pi, Fintype.card_fin] using
Tensor.rankAtMost_sum_rankOne aa bb cc
theorem bankTensor_reindex {J : Type*} [Fintype J]
{X' Y' Z' : J → Type*} (e : I ≃ J)
(ex : ∀ i, X i ≃ X' (e i)) (ey : ∀ i, Y i ≃ Y' (e i))
(ez : ∀ i, Z i ≃ Z' (e i))
(T : ∀ i, Tensor K (X i) (Y i) (Z i))
(S : ∀ j, Tensor K (X' j) (Y' j) (Z' j))
(compatible : ∀ i x y z, S (e i) (ex i x) (ey i y) (ez i z) = T i x y z) :
Tensor.pullback (e.piCongr ex) (e.piCongr ey) (e.piCongr ez)
(bankTensor S) = bankTensor T := by
funext x y z
change (∏ j, S j ((e.piCongr ex) x j) ((e.piCongr ey) y j)
((e.piCongr ez) z j)) = ∏ i, T i (x i) (y i) (z i)
rw [← e.prod_comp (fun j => S j ((e.piCongr ex) x j)
((e.piCongr ey) y j) ((e.piCongr ez) z j))]
apply Finset.prod_congr rfl
intro i hi
simpa only [Equiv.piCongr_apply_apply] using compatible i (x i) (y i) (z i)
theorem bankTensor_reindex_rankAtMost {J : Type*} [Fintype J]
{X' Y' Z' : J → Type*} (e : I ≃ J)
(ex : ∀ i, X i ≃ X' (e i)) (ey : ∀ i, Y i ≃ Y' (e i))
(ez : ∀ i, Z i ≃ Z' (e i))
(T : ∀ i, Tensor K (X i) (Y i) (Z i))
(S : ∀ j, Tensor K (X' j) (Y' j) (Z' j))
(compatible : ∀ i x y z, S (e i) (ex i x) (ey i y) (ez i z) = T i x y z)
{r : ℕ} (hS : Tensor.RankAtMost (bankTensor S) r) :
Tensor.RankAtMost (bankTensor T) r := by
rw [← bankTensor_reindex e ex ey ez T S compatible]
exact hS.pullback _ _ _
end MatrixMultiplication.Foundation.LabelHierarchySeparation
end OAI
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