Round-six lemma `cBlk2_le`

Proved

J6Seg.KR.cBlk2_le

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

The following holds (Lean statement, in the scope of module Solutions.J6SegAxBud1):

theorem J6Seg.KR.cBlk2_le {n pk b φD φT B BH J Z Φ : ℕ} (hj : Nat.clog 2 (2 * n) ≤ J)
    (hZ : zsz ((b * (2 * n) : ℕ) : ℤ) ≤ Z) (hΦ : φT ≤ (n + 1) * Φ) :
    cBlk2 n pk b φD φT B BH + 30 ≤ (n + 1) * Yb pk b φD Φ B BH J Z

Context. B-line axis budget 1: closed-form bounds on the chunk kernel cKer2

cKG_le, cBlk2_le: one block (with j ≤ J, zsz (2nb) ≤ Z, φT ≤ Φ): cBlk2 n … ≤ (n+1) · Yb with Yb = 10^13 (BH+B+b+pk+10)(J+1)(Z+5) + 10 φD + Φ';
cLoop2_le, cKer2_le: cKer2 … ds ≤ 10^9 (λ+1)(BH+B+2) + 2(λ+1)·Yb + 10^4 for block lists with 1 ≤ n ≤ λ, Σ n ≤ λ.

Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_R6_Defs70

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.KR
open IntMul NLogN NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFV

Formal statement

theorem J6Seg.KR.cBlk2_le {n pk b φD φT B BH J Z Φ : ℕ} (hj : Nat.clog 2 (2 * n) ≤ J)
    (hZ : zsz ((b * (2 * n) : ℕ) : ℤ) ≤ Z) (hΦ : φT ≤ (n + 1) * Φ) :
    cBlk2 n pk b φD φT B BH + 30 ≤ (n + 1) * Yb pk b φD Φ B BH J Z := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `J6Seg.KR.cBlk2_le`, Solutions/J6SegAxBud1.lean (lines 41–106) of the round-six development.

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