To Polynomial degree le
GuruswamiSudan.toPolynomial_degree_le
Plain-language statement
The polynomial corresponding to a codeword has degree at most k-1.
Exact Lean statement
lemma toPolynomial_degree_le (hk : k + 1 ≤ n) (p : code ωs k) :
(toPolynomial p).natDegree ≤ k - 1Formal artifact
Lean source
lemma toPolynomial_degree_le (hk : k + 1 ≤ n) (p : code ωs k) : (toPolynomial p).natDegree ≤ k - 1 := by rw [toPolynomial_def] obtain ⟨q, hq, hp⟩ := p.2 have h_interpolate : (Lagrange.interpolate Finset.univ ωs.toFun) (evalOnPoints ωs q) = q := by simpa [evalOnPoints, Function.Embedding.toFun_eq_coe] using interpolate_eq_of_degree_lt q (lt_of_le_of_lt (natDegree_le_of_degree_le <| mem_degreeLT.mp hq |> le_of_lt) hk) rcases k <;> simp_all only [Function.Embedding.toFun_eq_coe, Lagrange.interpolate_apply, zero_add, degreeLT, ge_iff_le, zero_le, iInf_pos, Submodule.coe_iInf, Set.mem_iInter, SetLike.mem_coe, LinearMap.mem_ker, lcoeff_apply, zero_tsub, nonpos_iff_eq_zero] · rw [show q = 0 from Polynomial.ext hq]; norm_num · exact natDegree_le_iff_coeff_eq_zero.mpr hq- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/GuruswamiSudan/Basic.lean:855-867
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Affine gaps lifted to interleaved codes
affine_gaps_lifted_to_interleaved_codes
Project documentation
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.
Gadget Decompose coeff
ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
Plain-language statement
The k-th coefficient (k < deg φ) of a gadget-decomposition block is exactly the corresponding digit of the corresponding input coefficient.
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose lawful
ArkLib.Lattices.Ajtai.gadgetDecompose_lawful
Plain-language statement
The base-b gadget decomposition is a lawful gadget decomposition.
Source project: ArkLib
Person-level attribution pending.