Xpow coeff of lt
ArkLib.Lattices.CyclotomicModulus.Xpow_coeff_of_lt
Plain-language statement
For i < 2^α (below the modulus degree), X^i is already reduced, so its j-th coefficient is the Kronecker delta [j = i].
Exact Lean statement
theorem Xpow_coeff_of_lt (α : ℕ) {i : ℕ} (hi : i < 2 ^ α) (j : ℕ) :
(Xpow (powTwoCyclotomic (R := R) α) i).1.coeff j = if j = i then (1 : R) else 0Formal artifact
Lean source
theorem Xpow_coeff_of_lt (α : ℕ) {i : ℕ} (hi : i < 2 ^ α) (j : ℕ) : (Xpow (powTwoCyclotomic (R := R) α) i).1.coeff j = if j = i then (1 : R) else 0 := by have h2 : (0 : ℕ) < 2 ^ α := pow_pos (by norm_num) α have hself : (powTwoCyclotomic (R := R) α).reduce (CompPoly.CPolynomial.monomial i 1) = CompPoly.CPolynomial.monomial i 1 := by refine CyclotomicModulus.reduce_eq_self_of_degree_lt _ ?_ rw [toPoly_monomial, Polynomial.degree_monomial i (one_ne_zero), powTwoCyclotomic_toPoly, ← Polynomial.C_1, Polynomial.degree_X_pow_add_C h2 (1 : R)] exact_mod_cast hi change ((powTwoCyclotomic (R := R) α).reduce (CompPoly.CPolynomial.monomial i 1)).coeff j = _ rw [hself, CPolynomial.coeff_monomial]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Subfield/Basis.lean:49-59
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.