Mk aeval X pow periodic
ArkLib.Lattices.CyclotomicModulus.mk_aeval_X_pow_periodic
Plain-language statement
(C-3 helper) aeval (X^n) and aeval (X^{n mod 2^{α+1}}) agree in the quotient.
Exact Lean statement
theorem mk_aeval_X_pow_periodic (α n : ℕ) (p : Polynomial R) :
Ideal.Quotient.mk (powTwoCyclotomic (R := R) α).modIdeal
(Polynomial.aeval (Polynomial.X ^ n : Polynomial R) p)
= Ideal.Quotient.mk _
(Polynomial.aeval (Polynomial.X ^ (n % 2 ^ (α + 1)) : Polynomial R) p)Formal artifact
Lean source
theorem mk_aeval_X_pow_periodic (α n : ℕ) (p : Polynomial R) : Ideal.Quotient.mk (powTwoCyclotomic (R := R) α).modIdeal (Polynomial.aeval (Polynomial.X ^ n : Polynomial R) p) = Ideal.Quotient.mk _ (Polynomial.aeval (Polynomial.X ^ (n % 2 ^ (α + 1)) : Polynomial R) p) := by have e : ∀ j : ℕ, Ideal.Quotient.mk (powTwoCyclotomic (R := R) α).modIdeal (aeval (Polynomial.X ^ j) p) = aeval (Ideal.Quotient.mk (powTwoCyclotomic (R := R) α).modIdeal (Polynomial.X ^ j)) p := by intro j have h := aeval_algHom_apply (Ideal.Quotient.mkₐ R (powTwoCyclotomic (R := R) α).modIdeal) (Polynomial.X ^ j) p simp only [Ideal.Quotient.mkₐ_eq_mk] at h exact h.symm rw [e, e, mk_X_pow_periodic]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Galois/Automorphism.lean:228-242
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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.