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namespace OAI
/-! Polynomial tensor restrictions and exact coefficient extraction. -/
noncomputable section
open scoped BigOperators
namespace MatrixMultiplication.Foundation
namespace Tensor
section ConstantRestriction
variable {F X Y Z X' Y' Z' : Type*} [CommSemiring F]
variable [Fintype X] [Fintype Y] [Fintype Z]
def restrictPolynomial (A : X' → X → F) (B : Y' → Y → F)
(C : Z' → Z → F) (P : Tensor (Polynomial F) X Y Z) :
Tensor (Polynomial F) X' Y' Z' :=
restrict (fun x' x => Polynomial.C (A x' x))
(fun y' y => Polynomial.C (B y' y))
(fun z' z => Polynomial.C (C z' z)) P
@[simp] theorem restrictPolynomial_coeff (A : X' → X → F) (B : Y' → Y → F)
(C : Z' → Z → F) (P : Tensor (Polynomial F) X Y Z) (k : ℕ)
(x' : X') (y' : Y') (z' : Z') :
(restrictPolynomial A B C P x' y' z').coeff k =
restrict A B C (fun x y z => (P x y z).coeff k) x' y' z' := by
simp only [restrictPolynomial, restrict, Polynomial.finsetSum_coeff,
mul_assoc, Polynomial.coeff_C_mul]
theorem RankAtMost.restrictPolynomial {P : Tensor (Polynomial F) X Y Z} {r : ℕ}
(hP : RankAtMost P r) (A : X' → X → F) (B : Y' → Y → F)
(C : Z' → Z → F) : RankAtMost (restrictPolynomial A B C P) r :=
hP.restrict (fun x' x => Polynomial.C (A x' x))
(fun y' y => Polynomial.C (B y' y))
(fun z' z => Polynomial.C (C z' z))
theorem restrictPolynomial_degree_le (A : X' → X → F) (B : Y' → Y → F)
(C : Z' → Z → F) (P : Tensor (Polynomial F) X Y Z) (D : ℕ)
(hdegree : ∀ x y z, (P x y z).degree ≤ D)
(x' : X') (y' : Y') (z' : Z') :
(restrictPolynomial A B C P x' y' z').degree ≤ D := by
apply (Polynomial.degree_le_iff_coeff_zero _ _).2
intro k hk
rw [restrictPolynomial_coeff]
have hzero (x : X) (y : Y) (z : Z) : (P x y z).coeff k = 0 :=
(Polynomial.degree_le_iff_coeff_zero _ _).1 (hdegree x y z) k hk
simp only [restrict, hzero, mul_zero, Finset.sum_const_zero]
end ConstantRestriction
namespace PolynomialApproximation
variable {F X Y Z X' Y' Z' : Type*} [Field F]
def ofRankAtMost (T : Tensor F X Y Z) (r : ℕ) (hT : RankAtMost T r) :
PolynomialApproximation T r 0 0 where
polynomial := fun x y z => Polynomial.C (T x y z)
rank_bound := hT.map Polynomial.C
vanishes := by
intro x y z k hk
exact (Nat.not_lt_zero k hk).elim
leading := by
intro x y z
exact Polynomial.coeff_C_zero
degree_bound := by
intro x y z
simpa only [Nat.cast_zero] using (Polynomial.degree_C_le (a := T x y z))
@[simp] theorem ofRankAtMost_polynomial (T : Tensor F X Y Z) (r : ℕ)
(hT : RankAtMost T r) :
(ofRankAtMost T r hT).polynomial = (fun x y z => Polynomial.C (T x y z)) := rfl
variable [Fintype X] [Fintype Y] [Fintype Z]
variable {T : Tensor F X Y Z} {r d D : ℕ}
def restrict (W : PolynomialApproximation T r d D)
(A : X' → X → F) (B : Y' → Y → F) (C : Z' → Z → F) :
PolynomialApproximation (Tensor.restrict A B C T) r d D where
polynomial := restrictPolynomial A B C W.polynomial
rank_bound := W.rank_bound.restrictPolynomial A B C
vanishes := by
intro x' y' z' k hk
rw [restrictPolynomial_coeff]
have hzero (x : X) (y : Y) (z : Z) : (W.polynomial x y z).coeff k = 0 :=
W.vanishes x y z k hk
simp only [Tensor.restrict, hzero, mul_zero, Finset.sum_const_zero]
leading := by
intro x' y' z'
rw [restrictPolynomial_coeff]
simp only [W.leading]
degree_bound := restrictPolynomial_degree_le A B C W.polynomial D W.degree_bound
@[simp] theorem restrict_polynomial (W : PolynomialApproximation T r d D)
(A : X' → X → F) (B : Y' → Y → F) (C : Z' → Z → F) :
(W.restrict A B C).polynomial = restrictPolynomial A B C W.polynomial := rfl
end PolynomialApproximation
end Tensor
end MatrixMultiplication.Foundation
end
end OAI
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