| 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 |
|
|
| instance leafFintype (S : FlaggedTensor X Y Z) (deleted : Color) : |
| Fintype (Leaf S deleted) := |
| inferInstanceAs (Fintype {p : X × Y × Z |
|
|
| def leafEquiv (S : FlaggedTensor X Y Z) (deleted : Color) : |
| Leaf S deleted ≃ {a : CompletionLabels.Leaf S |
| 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 |
|
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| end OAI |
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|
