Ps exists p of degree x eq zero nat degree y eq zero
ps_exists_p_of_degree_x_eq_zero_nat_degree_y_eq_zero
Plain-language statement
A bivariate polynomial of degree zero in both variables divides every polynomial.
Exact Lean statement
lemma ps_exists_p_of_degree_x_eq_zero_nat_degree_y_eq_zero {F : Type} [Field F]
{A B : F[X][Y]} (hA0 : A ≠ 0)
(hdegX : degreeX A = 0) (hdegY : natDegreeY A = 0) :
∃ P : F[X][Y], B = P * AFormal artifact
Lean source
lemma ps_exists_p_of_degree_x_eq_zero_nat_degree_y_eq_zero {F : Type} [Field F] {A B : F[X][Y]} (hA0 : A ≠ 0) (hdegX : degreeX A = 0) (hdegY : natDegreeY A = 0) : ∃ P : F[X][Y], B = P * A := by classical rcases natDegree_eq_zero.1 (by simpa [natDegreeY] using hdegY) with ⟨a0, ha0⟩ subst ha0; simp_all only [ne_eq, C_eq_zero] rcases natDegree_eq_zero.1 (by simpa [degreeX, support_C hA0] using hdegX) with ⟨a, ha⟩ subst ha; simp_all only [map_eq_zero] exact ⟨B * C (C a⁻¹), by ext n m : 2; simp [coeff_mul_C, inv_mul_cancel_right₀ hA0]⟩- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/PolishchukSpielman/Existence.lean:33-42
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This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...
Source project: ArkLib
Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.