openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Polynomial /ComplexPolynomialRestriction.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialApproximation | |
| 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 | |