Johnson unrefined by M
JohnsonBound.johnson_unrefined_by_M'
Plain-language statement
Johnson bound scaled by |F| / (|F| - 1).
Exact Lean statement
lemma johnson_unrefined_by_M' [Zero F]
(h_n : 0 < n) (h_B : 2 ≤ B.card) (h_card : 2 ≤ card F) :
B.card * (card F / (card F - 1)) *
((1 - e B 0 / n) ^ 2 + e B 0 ^ 2 /
((card F - 1) * n ^ 2) - 1 + d B / n) ≤
(card F / (card F - 1)) * d B / nFormal artifact
Lean source
lemma johnson_unrefined_by_M' [Zero F] (h_n : 0 < n) (h_B : 2 ≤ B.card) (h_card : 2 ≤ card F) : B.card * (card F / (card F - 1)) * ((1 - e B 0 / n) ^ 2 + e B 0 ^ 2 / ((card F - 1) * n ^ 2) - 1 + d B / n) ≤ (card F / (card F - 1)) * d B / n := by rw [mul_comm (B.card : ℚ), mul_assoc, ← mul_div] exact mul_le_mul_of_nonneg_left (johnson_unrefined_by_M h_n h_B h_card) (le_of_lt (div_pos (by exact_mod_cast lt_of_lt_of_le (by decide : 0 < 2) h_card) (by linarith [show (2 : ℚ) ≤ (card F : ℚ) from by exact_mod_cast h_card])))- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/JohnsonBound/Lemmas.lean:384-393
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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.