Johnson den lb e pos
JohnsonBound.johnson_den_lb_e_pos
Plain-language statement
Lower bound on the Johnson denominator when e > 0.
Exact Lean statement
lemma johnson_den_lb_e_pos {n d e : ℕ} {q frac : ℚ}
(hn_pos : (0 : ℚ) < n) (he0 : e ≠ 0)
(one_div_q_le : (1 : ℚ) / q ≤ frac - 1) (hfrac1_pos : (0 : ℚ) < frac - 1)
(hbase_nonneg : (0 : ℚ) ≤ (d / n : ℚ) - 2 * (e / n : ℚ) + (e / n : ℚ) ^ 2) :
(1 : ℚ) / (q * (n : ℚ) ^ 2) ≤
(d / n : ℚ) - 2 * (e / n : ℚ) + frac * (e / n : ℚ) ^ 2Formal artifact
Lean source
lemma johnson_den_lb_e_pos {n d e : ℕ} {q frac : ℚ} (hn_pos : (0 : ℚ) < n) (he0 : e ≠ 0) (one_div_q_le : (1 : ℚ) / q ≤ frac - 1) (hfrac1_pos : (0 : ℚ) < frac - 1) (hbase_nonneg : (0 : ℚ) ≤ (d / n : ℚ) - 2 * (e / n : ℚ) + (e / n : ℚ) ^ 2) : (1 : ℚ) / (q * (n : ℚ) ^ 2) ≤ (d / n : ℚ) - 2 * (e / n : ℚ) + frac * (e / n : ℚ) ^ 2 := by have h_e_div_n_ge : (e / n : ℚ) ^ 2 ≥ 1 / (n : ℚ) ^ 2 := by field_simp; exact_mod_cast Nat.one_le_pow _ _ (Nat.pos_of_ne_zero he0) by_cases hq0 : q = 0 · subst hq0 simp nlinarith [hbase_nonneg, hfrac1_pos, h_e_div_n_ge] · ring_nf at * nlinarith [mul_inv_cancel₀ hq0]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/JohnsonBound/Lemmas.lean:596-609
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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.