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import OAI.LinearAlgebra.MatrixFields.Construction.ComplexRectangularASI
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
noncomputable section
namespace MatrixMultiplication.Foundation
namespace Tensor
namespace PolynomialApproximation
variable {F X Y Z U V W X' Y' Z' : Type*} [Field F]
variable {T : Tensor F X Y Z} {S : Tensor F U V W}
variable {r s d e D E : ℕ}
theorem product_polynomial_eq (A : PolynomialApproximation T r d D)
(B : PolynomialApproximation S s e E) (x : X × U) (y : Y × V) (z : Z × W) :
Tensor.product A.polynomial B.polynomial x y z =
Polynomial.X ^ (d + e) * Tensor.product A.normalized B.normalized x y z := by
dsimp only [Tensor.product]
rw [A.polynomial_eq, B.polynomial_eq, pow_add]
ring
def product (A : PolynomialApproximation T r d D)
(B : PolynomialApproximation S s e E) :
PolynomialApproximation (Tensor.product T S) (r * s) (d + e) (D + E) where
polynomial := Tensor.product A.polynomial B.polynomial
rank_bound := A.rank_bound.product B.rank_bound
vanishes := by
intro x y z k hk
rw [A.product_polynomial_eq B]
simp only [Polynomial.coeff_X_pow_mul', Nat.not_le_of_lt hk, ite_false]
leading := by
intro x y z
rw [A.product_polynomial_eq B]
simp only [Polynomial.coeff_X_pow_mul', le_refl, ite_true, Nat.sub_self,
Tensor.product, Polynomial.mul_coeff_zero, A.normalized_constant, B.normalized_constant]
degree_bound := by
intro x y z
apply Polynomial.degree_le_of_natDegree_le
exact Polynomial.natDegree_mul_le_of_le
(Polynomial.natDegree_le_of_degree_le (A.degree_bound x.1 y.1 z.1))
(Polynomial.natDegree_le_of_degree_le (B.degree_bound x.2 y.2 z.2))
@[simp] theorem product_polynomial (A : PolynomialApproximation T r d D)
(B : PolynomialApproximation S s e E) :
(A.product B).polynomial = Tensor.product A.polynomial B.polynomial := rfl
def productExact (A : PolynomialApproximation T r d D)
(S : Tensor F U V W) (s : ℕ) (hS : RankAtMost S s) :
PolynomialApproximation (Tensor.product T S) (r * s) d D := by
simpa only [Nat.add_zero] using A.product (ofRankAtMost S s hS)
def pullback (A : PolynomialApproximation T r d D)
(fx : X' → X) (fy : Y' → Y) (fz : Z' → Z) :
PolynomialApproximation (Tensor.pullback fx fy fz T) r d D where
polynomial := Tensor.pullback fx fy fz A.polynomial
rank_bound := A.rank_bound.pullback fx fy fz
vanishes := fun x y z k hk => A.vanishes (fx x) (fy y) (fz z) k hk
leading := fun x y z => A.leading (fx x) (fy y) (fz z)
degree_bound := fun x y z => A.degree_bound (fx x) (fy y) (fz z)
@[simp] theorem pullback_polynomial (A : PolynomialApproximation T r d D)
(fx : X' → X) (fy : Y' → Y) (fz : Z' → Z) :
(A.pullback fx fy fz).polynomial = Tensor.pullback fx fy fz A.polynomial := rfl
def cyclic (A : PolynomialApproximation T r d D) :
PolynomialApproximation (Tensor.cyclic T) r d D where
polynomial := Tensor.cyclic A.polynomial
rank_bound := A.rank_bound.cyclic
vanishes := fun y z x k hk => A.vanishes x y z k hk
leading := fun y z x => A.leading x y z
degree_bound := fun y z x => A.degree_bound x y z
@[simp] theorem cyclic_polynomial (A : PolynomialApproximation T r d D) :
A.cyclic.polynomial = Tensor.cyclic A.polynomial := rfl
def directSum_product {ι κ : Type*} [DecidableEq ι] [DecidableEq κ]
{T : ι → Tensor F X Y Z} {S : κ → Tensor F U V W}
(A : PolynomialApproximation (Tensor.directSum T) r d D)
(B : PolynomialApproximation (Tensor.directSum S) s e E) :
PolynomialApproximation
(Tensor.directSum (fun i : ι × κ => Tensor.product (T i.1) (S i.2)))
(r * s) (d + e) (D + E) := by
have h := (A.product B).pullback (directSumProductEquiv ι κ X U)
(directSumProductEquiv ι κ Y V) (directSumProductEquiv ι κ Z W)
simpa only [Tensor.directSum_product] using h
def directSum_pullback {ι : Type*} [DecidableEq ι]
{T : ι → Tensor F X Y Z}
(A : PolynomialApproximation (Tensor.directSum T) r d D)
(fx : X' → X) (fy : Y' → Y) (fz : Z' → Z) :
PolynomialApproximation
(Tensor.directSum (fun i => Tensor.pullback fx fy fz (T i))) r d D :=
A.pullback (fun x : ι × X' => (x.1, fx x.2))
(fun y : ι × Y' => (y.1, fy y.2)) (fun z : ι × Z' => (z.1, fz z.2))
def directSum_cyclic {ι : Type*} [DecidableEq ι]
{T : ι → Tensor F X Y Z}
(A : PolynomialApproximation (Tensor.directSum T) r d D) :
PolynomialApproximation (Tensor.directSum (fun i => Tensor.cyclic (T i))) r d D := by
simpa only [Tensor.cyclic_directSum] using A.cyclic
variable {A B C : Type*} [DecidableEq A] [DecidableEq B] [DecidableEq C]
def matrixCoefficients_symmetrized
(P : PolynomialApproximation (matrixCoefficients (K := F) A B C) r d D) :
PolynomialApproximation
(matrixCoefficients (K := F) (A × B × C) (A × B × C) (A × B × C))
(r ^ 3) (3 * d) (3 * D) := by
have h := ((P.product P.cyclic).product P.cyclic.cyclic).pullback
(rectangularSquareX A B C) (rectangularSquareY A B C) (rectangularSquareZ A B C)
have hd : d + d + d = 3 * d := by ring
have hD : D + D + D = 3 * D := by ring
simpa only [Tensor.cyclic_matrixCoefficients, Tensor.matrixCoefficients_cyclic_product_square,
pow_succ, pow_zero, one_mul, hd, hD] using h
def matrixCoefficients_directSum_symmetrized {ι : Type*} [DecidableEq ι]
(P : PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := F) A B C)) r d D) :
PolynomialApproximation
(Tensor.directSum (fun _ : (ι × ι) × ι =>
matrixCoefficients (K := F) (A × B × C) (A × B × C) (A × B × C)))
(r ^ 3) (3 * d) (3 * D) := by
have P₁ : PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := F) B C A)) r d D := by
simpa only [Tensor.cyclic_matrixCoefficients] using P.directSum_cyclic
have P₂ : PolynomialApproximation
(Tensor.directSum (fun _ : ι => matrixCoefficients (K := F) C A B)) r d D := by
simpa only [Tensor.cyclic_matrixCoefficients] using P₁.directSum_cyclic
have h := ((P.directSum_product P₁).directSum_product P₂).directSum_pullback
(rectangularSquareX A B C) (rectangularSquareY A B C) (rectangularSquareZ A B C)
have hd : d + d + d = 3 * d := by ring
have hD : D + D + D = 3 * D := by ring
simpa only [Tensor.matrixCoefficients_cyclic_product_square,
pow_succ, pow_zero, one_mul, hd, hD] using h
end PolynomialApproximation
end Tensor
end MatrixMultiplication.Foundation
end
end MatrixAllFields
end OAI
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