Window bound (round six, `win_le`)

Proved

J6Seg.Side.win_le

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

complexity-theoryinteger-multiplicationkappa-improvement

Window bound.

The following holds (Lean statement, in the scope of module Solutions.J6SegJCost1):

theorem J6Seg.Side.win_le (hs : 0 < s) (hst : s ≤ t) (hl : 1 ≤ lam) (w x X : ℕ) :
    s * ((LJ s t lam w).toFinset.filter (fun j => x ≤ j ∧ j ≤ x + X)).card ≤
      2 * w * (s * ((X + 2 * w) / lam + 3) + (t - s) * (X + 2 * w + 1) + s)

Context. J cost 1: windows of the near-boundary list

LJ_cover: every j ∈ LJ is within w of a cut < s or of s;
bad_Ico: s · #{l ∈ [a, a+Y) : ¬ stepOK l} < (t - s) Y + s;
mult_Ico: #{c ∈ [a, a+Y) : λ ∣ c} ≤ Y/λ + 2;
win_le: #{j ∈ LJ : x ≤ j ≤ x + X} ≤ 2w ((X+2w)/λ + 4 + (t-s)(X+2w+1)/s).

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_Defs61

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg
open Finset
open J6Seg
variable {s t lam : ℕ}

Formal statement

theorem J6Seg.Side.win_le (hs : 0 < s) (hst : s ≤ t) (hl : 1 ≤ lam) (w x X : ℕ) :
    s * ((LJ s t lam w).toFinset.filter (fun j => x ≤ j ∧ j ≤ x + X)).card ≤
      2 * w * (s * ((X + 2 * w) / lam + 3) + (t - s) * (X + 2 * w + 1) + s) := 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.Side.win_le`, Solutions/J6SegJCost1.lean (lines 135–176) of the round-six development.

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