openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixFields /Construction /ComplexPolynomialWitnessProduct.lean
| import Mathlib | |
| 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 | |