Almost johnson lhs div B card
JohnsonBound.almost_johnson_lhs_div_B_card
Plain-language statement
LHS of the almost-Johnson bound divided by |B| in terms of e and d.
Exact Lean statement
lemma almost_johnson_lhs_div_B_card [Zero F] (h_n : 0 < n) (h_B : 2 ≤ B.card) :
(k B * (k B - 1) + (B.card - k B) * ((B.card - k B) / (card F - 1) - 1)) / B.card =
(1 - e B 0 / n) ^ 2 * B.card + B.card * (e B 0) ^ 2 / ((card F - 1) * n ^ 2) - 1Formal artifact
Lean source
lemma almost_johnson_lhs_div_B_card [Zero F] (h_n : 0 < n) (h_B : 2 ≤ B.card) : (k B * (k B - 1) + (B.card - k B) * ((B.card - k B) / (card F - 1) - 1)) / B.card = (1 - e B 0 / n) ^ 2 * B.card + B.card * (e B 0) ^ 2 / ((card F - 1) * n ^ 2) - 1 := by set E := (n - e B 0) / n generalize eqrhs : (_ + _ - 1 : ℚ) = rhs have eqE : E = k B / B.card := by grind only [= k_and_e'] suffices (B.card * E - 1) * E + ((B.card - B.card * E) / (card F - 1) - 1) * (1 - E) = rhs by rw [eqE, mul_div_cancel₀ _ (by simp only [ne_eq, Rat.natCast_eq_zero_iff]; omega)] at this rw [← this]; field_simp rw [← eqrhs] have : E = 1 - (e B 0) / n := by simp only [E] field_simp [show (n : ℚ) ≠ 0 from by exact_mod_cast Nat.pos_iff_ne_zero.mp h_n] grind only- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/JohnsonBound/Lemmas.lean:340-354
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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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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.