Zmod Digit nat Abs le
ArkLib.Lattices.Ajtai.zmodDigit_natAbs_le
Plain-language statement
Core digit bound. Each base-b digit of zmodDigitDecomposition, viewed as a centered residue, has absolute value at most b - 1 , provided b - 1 ≤ q/2, so the digit (a natural number < b) does not wrap to a negative centered representative.
Exact Lean statement
theorem zmodDigit_natAbs_le {b digits : ℕ} (hb : 1 < b) (hq : q ≤ b ^ digits)
(hbq : b - 1 ≤ q / 2) (c : ZMod q) (e : Fin digits) :
((zmodDigitDecomposition b digits hb hq).digit c e).valMinAbs.natAbs ≤ b - 1Formal artifact
Lean source
theorem zmodDigit_natAbs_le {b digits : ℕ} (hb : 1 < b) (hq : q ≤ b ^ digits) (hbq : b - 1 ≤ q / 2) (c : ZMod q) (e : Fin digits) : ((zmodDigitDecomposition b digits hb hq).digit c e).valMinAbs.natAbs ≤ b - 1 := by simp only [zmodDigitDecomposition] set d := (Nat.digits b c.val).getD (e : ℕ) 0 with hd have hdb : d < b := by rcases lt_or_ge (e : ℕ) (Nat.digits b c.val).length with hlt | hge · rw [hd, List.getD_eq_getElem _ _ hlt] exact Nat.digits_lt_base hb (List.getElem_mem _) · rw [hd, List.getD_eq_default _ _ hge]; omega rw [ZMod.valMinAbs_natCast_of_le_half (by omega : d ≤ q / 2)] simp only [Int.natAbs_natCast] omega- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/Hachi/Gadget/Norms.lean:68-80
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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.
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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.