Rq l2Norm Sq le nat Degree mul l Infty Norm sq
ArkLib.Lattices.CyclotomicModulus.Rq.l2NormSq_le_natDegree_mul_lInftyNorm_sq
Plain-language statement
ℓ∞ → ℓ₂² bridge (ring element): ‖x‖₂² ≤ deg φ · ‖x‖∞² , each of the deg φ centered coefficients contributes at most ‖x‖∞².
Exact Lean statement
theorem Rq.l2NormSq_le_natDegree_mul_lInftyNorm_sq (x : Rq Φ) :
‖x‖₂² ≤ Φ.φ.natDegree * (Rq.lInftyNorm Φ x) ^ 2Formal artifact
Lean source
theorem Rq.l2NormSq_le_natDegree_mul_lInftyNorm_sq (x : Rq Φ) : ‖x‖₂² ≤ Φ.φ.natDegree * (Rq.lInftyNorm Φ x) ^ 2 := by unfold Rq.l2NormSq calc ∑ k ∈ Finset.range Φ.φ.natDegree, (x.1.coeff k).valMinAbs.natAbs ^ 2 ≤ ∑ _k ∈ Finset.range Φ.φ.natDegree, (Rq.lInftyNorm Φ x) ^ 2 := Finset.sum_le_sum fun k hk => Nat.pow_le_pow_left (Finset.le_sup (f := fun k => (x.1.coeff k).valMinAbs.natAbs) hk) 2 _ = Φ.φ.natDegree * (Rq.lInftyNorm Φ x) ^ 2 := by rw [Finset.sum_const, Finset.card_range, smul_eq_mul]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/NormBounds/Basic.lean:300-309
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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.
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Plain-language statement
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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.