Johnson unrefined
JohnsonBound.johnson_unrefined
Plain-language statement
Unrefined Johnson bound in terms of e, d, and |B|.
Exact Lean statement
lemma johnson_unrefined [Zero F]
(h_n : 0 < n) (h_B : 2 ≤ B.card) (h_card : 2 ≤ card F) :
(1 - e B 0 / n) ^ 2 * B.card + B.card * (e B 0) ^ 2 /
((card F - 1) * n ^ 2) - 1 ≤
(B.card - 1) * (1 - d B / n)Formal artifact
Lean source
lemma johnson_unrefined [Zero F] (h_n : 0 < n) (h_B : 2 ≤ B.card) (h_card : 2 ≤ card F) : (1 - e B 0 / n) ^ 2 * B.card + B.card * (e B 0) ^ 2 / ((card F - 1) * n ^ 2) - 1 ≤ (B.card - 1) * (1 - d B / n) := by have h_rewrite : (k B * (k B - 1) + (B.card - k B) * ((B.card - k B) / (card F - 1) - 1)) / B.card ≤ (B.card - 1) * (1 - d B / n) := by have this := almost_johnson_choose_2_elimed h_n h_B h_card rw [div_le_iff₀ (by positivity)] convert this using 1 · rfl · field_simp [h_n.ne'] convert h_rewrite using 1 convert almost_johnson_lhs_div_B_card h_n h_B |> Eq.symm using 1- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/JohnsonBound/Lemmas.lean:357-371
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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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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.