openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /FiniteConvolutionAlgebra.lean
| import Mathlib.Algebra.Polynomial.Degree.TrailingDegree | |
| import Mathlib.Algebra.Polynomial.OfFn | |
| import Mathlib.Basic.Complex.Basic | |
| namespace OAI | |
| namespace InvariantProjection | |
| open Polynomial | |
| theorem exists_nonzero_product_coeff_le | |
| {R : Type*} [Semiring R] [NoZeroDivisors R] | |
| (A B : R[X]) (k l : ℕ) | |
| (hA : A.coeff k ≠ 0) (hB : B.coeff l ≠ 0) : | |
| ∃ n ≤ k + l, (A * B).coeff n ≠ 0 := by | |
| have hA0 : A ≠ 0 := by | |
| intro h | |
| simp [h] at hA | |
| have hB0 : B ≠ 0 := by | |
| intro h | |
| simp [h] at hB | |
| refine ⟨(A * B).natTrailingDegree, ?_, ?_⟩ | |
| · rw [natTrailingDegree_mul hA0 hB0] | |
| exact Nat.add_le_add | |
| (natTrailingDegree_le_of_ne_zero hA) | |
| (natTrailingDegree_le_of_ne_zero hB) | |
| · exact coeff_natTrailingDegree_ne_zero.mpr (mul_ne_zero hA0 hB0) | |
| theorem exists_nonzero_product_coeff_le_two_mul | |
| (A B : ℂ[X]) (K : ℕ) | |
| (hA : A.coeff K ≠ 0) (hB : B.coeff K ≠ 0) : | |
| ∃ n ≤ 2 * K, (A * B).coeff n ≠ 0 := by | |
| simpa [two_mul] using exists_nonzero_product_coeff_le A B K K hA hB | |
| end InvariantProjection | |
| end OAI | |