Plain-language statement
Algebraic identity expressing the Johnson LHS as a difference of squares.
Exact Lean statement
lemma johnson_denom [Zero F] (h_card : 2 ≤ card F) :
(card F / (card F - 1)) *
((1 - e B 0 / n) ^ 2 + (e B 0) ^ 2 /
((card F - 1) * n ^ 2) - 1 + d B / n) =
(1 - (card F / (card F - 1)) *
(e B 0 / n)) ^ 2 - (1 - (card F / (card F - 1)) * (d B / n))Formal artifact
Lean source
lemma johnson_denom [Zero F] (h_card : 2 ≤ card F) : (card F / (card F - 1)) * ((1 - e B 0 / n) ^ 2 + (e B 0) ^ 2 / ((card F - 1) * n ^ 2) - 1 + d B / n) = (1 - (card F / (card F - 1)) * (e B 0 / n)) ^ 2 - (1 - (card F / (card F - 1)) * (d B / n)) := by set c := card F; set c1 := (c : ℚ) - 1 have n₂ : c1 ≠ 0 := by simp [c1, c, sub_eq_zero]; grind only suffices c / c1 * (d B / n - 2 * e B 0 / n + c / c1 * e B 0 ^ 2 / n ^ 2) = (1 - c / c1 * (e B 0 / n)) ^ 2 - (1 - c / c1 * (d B / n)) by rw [← this]; have : c / c1 = 1 + 1 / c1 := by grind only grind only [= e.eq_1] grind only- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/JohnsonBound/Lemmas.lean:396-408
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Affine gaps lifted to interleaved codes
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.