File size: 3,736 Bytes
cdec422 | 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 | import OAI.LinearAlgebra.MatrixMultiplication.Completion.Labels
import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialRestriction
namespace OAI
/-! Readable tensor completion and its finite arithmetic realization. -/
noncomputable section
namespace MatrixMultiplication.CompletionTermination
open MatrixMultiplication.Foundation RecursiveCompletion CompletionLabels
attribute [local instance] Classical.propDecidable Classical.decEq
universe uCoord
variable {X Y Z : Type uCoord}
def tensor (S : FlaggedTensor X Y Z) (deleted : Color) : Tensor ℂ X Y Z :=
fun x y z => if ownerFlag S deleted (coordinate (x, y, z) deleted) then 0
else S.tensor x y z
theorem support_iff (S : FlaggedTensor X Y Z) (deleted : Color) (x y z) :
tensor S deleted x y z ≠ 0 ↔ S.tensor x y z ≠ 0 ∧
¬ ownerFlag S deleted (coordinate (x, y, z) deleted) := by
by_cases h : ownerFlag S deleted (coordinate (x, y, z) deleted) <;>
simp [tensor, h]
def leftMap (S : FlaggedTensor X Y Z) (deleted : Color) : X → X → ℂ :=
diagonalMap (fun x => if deleted = .B ∧ S.flagB x then 0 else 1)
def middleMap (S : FlaggedTensor X Y Z) (deleted : Color) : Y → Y → ℂ :=
diagonalMap (fun y => if deleted = .C ∧ S.flagC y then 0 else 1)
def rightMap (S : FlaggedTensor X Y Z) (deleted : Color) : Z → Z → ℂ :=
diagonalMap (fun z => if deleted = .A ∧ S.flagA z then 0 else 1)
variable [Fintype X] [Fintype Y] [Fintype Z]
theorem restriction (S : FlaggedTensor X Y Z) (deleted : Color) :
Tensor.restrict (leftMap S deleted) (middleMap S deleted) (rightMap S deleted)
S.tensor = tensor S deleted := by
funext x y z
rw [leftMap, middleMap, rightMap, restrict_diagonal]
cases deleted <;> simp [tensor, ownerFlag, coordinate, ite_mul, reduceCtorEq] <;> rfl
def approximation (S : FlaggedTensor X Y Z) (deleted : Color) {R d D : ℕ}
(W : Tensor.PolynomialApproximation S.tensor R d D) :
Tensor.PolynomialApproximation (tensor S deleted) R d D := by
simpa only [restriction] using
W.restrict (leftMap S deleted) (middleMap S deleted) (rightMap S deleted)
def Leaf (S : FlaggedTensor X Y Z) (deleted : Color) :=
{p : X × Y × Z // tensor S deleted p.1 p.2.1 p.2.2 ≠ 0}
instance leafFintype (S : FlaggedTensor X Y Z) (deleted : Color) :
Fintype (Leaf S deleted) :=
inferInstanceAs (Fintype {p : X × Y × Z // tensor S deleted p.1 p.2.1 p.2.2 ≠ 0})
def leafEquiv (S : FlaggedTensor X Y Z) (deleted : Color) :
Leaf S deleted ≃ {a : CompletionLabels.Leaf S // leafColor S a ≠ deleted} where
toFun a :=
⟨⟨a.val, ((support_iff S deleted _ _ _).mp a.property).1⟩,
fun h => ((support_iff S deleted _ _ _).mp a.property).2
((leafColor_eq_iff S _ deleted).mp h)⟩
invFun a :=
⟨a.val.val, (support_iff S deleted _ _ _).mpr
⟨a.val.property, fun h => a.property ((leafColor_eq_iff S _ deleted).mpr h)⟩⟩
left_inv a := by apply Subtype.ext; rfl
right_inv a := by apply Subtype.ext; apply Subtype.ext; rfl
def retainedLeafEquiv (S : FlaggedTensor X Y Z) (deleted first second : Color)
(colors : ∀ c : Color, c ≠ deleted ↔ c = first ∨ c = second) :
Leaf S deleted ≃ RetainedLeaf S first second :=
(leafEquiv S deleted).trans
(Equiv.subtypeEquivRight (fun a => colors (leafColor S a)))
omit [Fintype X] [Fintype Y] [Fintype Z] in
@[simp] theorem retainedLeafEquiv_coordinates
(S : FlaggedTensor X Y Z) (deleted first second : Color)
(colors : ∀ c : Color, c ≠ deleted ↔ c = first ∨ c = second)
(a : Leaf S deleted) :
((retainedLeafEquiv S deleted first second colors a).val).val = a.val := rfl
end MatrixMultiplication.CompletionTermination
end
end OAI
|