Mk monomial coeff lt
ArkLib.Lattices.CyclotomicModulus.mk_monomial_coeff_lt
Plain-language statement
The j-th coefficient of the reduced c·X^i (for i < d) is c·[j = i].
Exact Lean statement
theorem mk_monomial_coeff_lt (α : ℕ) {i : ℕ} (hi : i < 2 ^ α) (c : R) (j : ℕ) :
(Rq.mk (powTwoCyclotomic α) (CompPoly.CPolynomial.monomial i c)).1.coeff j
= if j = i then c else 0Formal artifact
Lean source
theorem mk_monomial_coeff_lt (α : ℕ) {i : ℕ} (hi : i < 2 ^ α) (c : R) (j : ℕ) : (Rq.mk (powTwoCyclotomic α) (CompPoly.CPolynomial.monomial i c)).1.coeff j = if j = i then c else 0 := by have hself : (powTwoCyclotomic (R := R) α).reduce (CompPoly.CPolynomial.monomial i c) = CompPoly.CPolynomial.monomial i c := by refine CyclotomicModulus.reduce_eq_self_of_degree_lt _ ?_ rw [toPoly_monomial] calc (Polynomial.monomial i c).degree ≤ (i : WithBot ℕ) := Polynomial.degree_monomial_le i c _ < (powTwoCyclotomic α).φ.toPoly.degree := by rw [powTwoCyclotomic_toPoly_degree]; exact_mod_cast hi change ((powTwoCyclotomic (R := R) α).reduce (CompPoly.CPolynomial.monomial i c)).coeff j = _ rw [hself, CPolynomial.coeff_monomial]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Subfield/Basis.lean:382-393
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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.