Galois Aut comp
ArkLib.Lattices.CyclotomicModulus.galoisAut_comp
Plain-language statement
Composition law σ_i ∘ σ_j = σ_{ij} (for i, j odd, so the maps are genuine automorphisms). Proven on the semantic aeval side via the soundness bridge galoisAut_toQuotient and aeval_X_pow_aeval_X_pow.
Exact Lean statement
theorem galoisAut_comp (α i j : ℕ) (hi : Odd i) (hj : Odd j)
(a : Rq (powTwoCyclotomic (R := R) α)) :
galoisAut (powTwoCyclotomic α) i (galoisAut (powTwoCyclotomic α) j a)
= galoisAut (powTwoCyclotomic α) (i * j) aFormal artifact
Lean source
theorem galoisAut_comp (α i j : ℕ) (hi : Odd i) (hj : Odd j) (a : Rq (powTwoCyclotomic (R := R) α)) : galoisAut (powTwoCyclotomic α) i (galoisAut (powTwoCyclotomic α) j a) = galoisAut (powTwoCyclotomic α) (i * j) a := by apply Rq.toQuotient_injective (powTwoCyclotomic α) rw [galoisAut_toQuotient α i hi, galoisAut_toQuotient α j hj, galoisAut_toQuotient α (i * j) (hi.mul hj), galoisAutₛ_toQuotient α j hj, galoisAutₛ_mk, galoisAutₛ_toQuotient α (i * j) (hi.mul hj), aeval_X_pow_aeval_X_pow]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Galois/Group.lean:80-87
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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.
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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.