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import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialExecutionSymmetry
namespace OAI
/-! Finite coefficient tensors and their algebraic transformations. -/
noncomputable section
namespace MatrixMultiplication.MatrixBalancing
open MatrixMultiplication.Foundation
open MatrixMultiplication.Foundation.Tensor
variable {K X Y Z : Type*} [Field K]
variable {r d D : ℕ}
def swapXYApproximation {T : Tensor K X Y Z}
(P : PolynomialApproximation T r d D) :
PolynomialApproximation (Tensor.swapXY T) r d D where
polynomial := Tensor.swapXY P.polynomial
rank_bound := P.rank_bound.swapXY
vanishes := fun y x z k hk => P.vanishes x y z k hk
leading := fun y x z => P.leading x y z
degree_bound := fun y x z => P.degree_bound x y z
private theorem swapXY_directSum {ι : Type*} [DecidableEq ι]
(T : ι → Tensor K X Y Z) :
Tensor.swapXY (Tensor.directSum T) =
Tensor.directSum (fun i => Tensor.swapXY (T i)) := by
funext x y z
have hguard : (y.1 = x.1 ∧ y.1 = z.1) ↔ (x.1 = y.1 ∧ x.1 = z.1) := by
constructor
· rintro ⟨hyx, hyz⟩
exact ⟨hyx.symm, hyx.symm.trans hyz⟩
· rintro ⟨hxy, hxz⟩
exact ⟨hxy.symm, hxy.symm.trans hxz⟩
simp only [Tensor.swapXY, Tensor.directSum, hguard]
split_ifs with h
· rw [h.1]
· rfl
def directSum_swapXY {ι : Type*} [DecidableEq ι]
{T : ι → Tensor K X Y Z}
(P : PolynomialApproximation (Tensor.directSum T) r d D) :
PolynomialApproximation (Tensor.directSum (fun i => Tensor.swapXY (T i))) r d D := by
simpa only [swapXY_directSum] using swapXYApproximation P
section Matrix
variable {ι A B C : Type*} [DecidableEq ι]
variable [DecidableEq A] [DecidableEq B] [DecidableEq C]
theorem matrix_swapYZ :
Tensor.pullback (Prod.swap : B × A → A × B)
(Prod.swap : A × C → C × A) (Prod.swap : C × B → B × C)
(Tensor.cyclic (Tensor.swapXY (matrixCoefficients (K := K) A B C))) =
matrixCoefficients (K := K) B A C := by
funext x y z
simp only [Tensor.pullback, Tensor.cyclic, Tensor.swapXY, Prod.swap,
matrixCoefficients]
simp only [eq_comm, and_comm, and_left_comm]
theorem matrix_swapXZ :
Tensor.pullback (Prod.swap : A × C → C × A)
(Prod.swap : C × B → B × C) (Prod.swap : B × A → A × B)
(Tensor.swapXY (Tensor.cyclic (matrixCoefficients (K := K) A B C))) =
matrixCoefficients (K := K) A C B := by
funext x y z
simp only [Tensor.pullback, Tensor.cyclic, Tensor.swapXY, Prod.swap,
matrixCoefficients]
simp only [eq_comm, and_comm, and_left_comm, and_assoc]
def matrixDirectSum_swapYZ
(P : PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := K) A B C)) r d D) :
PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := K) B A C)) r d D := by
have h := (directSum_swapXY P).directSum_cyclic.directSum_pullback
(Prod.swap : B × A → A × B) (Prod.swap : A × C → C × A)
(Prod.swap : C × B → B × C)
simpa only [matrix_swapYZ] using h
def matrixDirectSum_swapXZ
(P : PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := K) A B C)) r d D) :
PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := K) A C B)) r d D := by
have h := (directSum_swapXY P.directSum_cyclic).directSum_pullback
(Prod.swap : A × C → C × A) (Prod.swap : C × B → B × C)
(Prod.swap : B × A → A × B)
simpa only [matrix_swapXZ] using h
def matrixDirectSum_reindex {κ A' B' C' : Type*}
[DecidableEq κ] [DecidableEq A'] [DecidableEq B'] [DecidableEq C']
(P : PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := K) A B C)) r d D)
(e : κ → ι) (he : Function.Injective e)
(f : A' → A) (hf : Function.Injective f)
(g : B' → B) (hg : Function.Injective g)
(j : C' → C) (hj : Function.Injective j) :
PolynomialApproximation
(Tensor.directSum (fun _ : κ => matrixCoefficients (K := K) A' B' C')) r d D := by
have h := P.pullback
(fun x : κ × (A' × B') => (e x.1, (f x.2.1, g x.2.2)))
(fun y : κ × (B' × C') => (e y.1, (g y.2.1, j y.2.2)))
(fun z : κ × (C' × A') => (e z.1, (j z.2.1, f z.2.2)))
convert h using 1
funext x y z
simp only [Tensor.pullback, Tensor.directSum, matrixCoefficients,
he.eq_iff, hf.eq_iff, hg.eq_iff, hj.eq_iff]
def matrixDirectSum_product {κ A' B' C' : Type*}
[DecidableEq κ] [DecidableEq A'] [DecidableEq B'] [DecidableEq C']
{s e E : ℕ}
(P : PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := K) A B C)) r d D)
(Q : PolynomialApproximation
(Tensor.directSum (fun _ : κ => matrixCoefficients (K := K) A' B' C')) s e E) :
PolynomialApproximation
(Tensor.directSum (fun _ : ι × κ =>
matrixCoefficients (K := K) (A × A') (B × B') (C × C')))
(r * s) (d + e) (D + E) := by
have h := (P.directSum_product Q).directSum_pullback
(fun x : (A × A') × (B × B') => ((x.1.1, x.2.1), (x.1.2, x.2.2)))
(fun y : (B × B') × (C × C') => ((y.1.1, y.2.1), (y.1.2, y.2.2)))
(fun z : (C × C') × (A × A') => ((z.1.1, z.2.1), (z.1.2, z.2.2)))
convert h using 1
funext x y z
simp only [Tensor.directSum, Tensor.pullback, matrixCoefficients_product]
def matrixDirectSum_balanced
(P : PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := K) A B C)) r d D) :
PolynomialApproximation
(Tensor.directSum (fun _ : ι × ι =>
matrixCoefficients (K := K) (A × B) (C × C) (A × B)))
(r ^ 2) (2 * d) (2 * D) := by
have h := matrixDirectSum_swapXZ (matrixDirectSum_product P (matrixDirectSum_swapYZ P))
have h' := matrixDirectSum_reindex h
(Equiv.refl (ι × ι)) (Equiv.refl _).injective
(Equiv.refl (A × B)) (Equiv.refl _).injective
(Equiv.refl (C × C)) (Equiv.refl _).injective
(Equiv.prodComm A B) (Equiv.prodComm A B).injective
simpa only [pow_two, two_mul] using h'
end Matrix
def finiteBalanced {L a b c : ℕ}
(P : PolynomialApproximation
(Tensor.directSum (fun _ : Fin L =>
matrixCoefficients (K := K) (Fin a) (Fin b) (Fin c))) r d D) :
PolynomialApproximation
(Tensor.directSum (fun _ : Fin (L ^ 2) =>
matrixCoefficients (K := K) (Fin (a * b)) (Fin (c ^ 2)) (Fin (a * b))))
(r ^ 2) (2 * d) (2 * D) := by
let tags : Fin (L ^ 2) ≃ Fin L × Fin L := Fintype.equivOfCardEq (by simp [pow_two])
let outer : Fin (a * b) ≃ Fin a × Fin b := Fintype.equivOfCardEq (by simp)
let inner : Fin (c ^ 2) ≃ Fin c × Fin c := Fintype.equivOfCardEq (by simp [pow_two])
exact matrixDirectSum_reindex (matrixDirectSum_balanced P)
tags tags.injective outer outer.injective inner inner.injective outer outer.injective
end MatrixMultiplication.MatrixBalancing
end
end OAI
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