The number facts of the window on an axis (`win_num`)

Proved

J6Seg.Side.win_num

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

complexity-theoryinteger-multiplicationkappa-improvement

The number facts of the window on an axis.

The following holds (Lean statement, in the scope of module Solutions.J6SegJCost3):

theorem J6Seg.Side.win_num {d s t : ℕ} (hd1 : 1 ≤ d) (hs : 0 < s) (hst : s ≤ t) (h8 : 8 * d ^ 2 ≤ t)
    (hts : 8 * d ^ 2 * (t - s) ≤ 2 * t) (x : ℕ) :
    ((LJ s t (lamP d) (wF d)).toFinset.filter
      (fun j => x ≤ j ∧ j ≤ x + (lamP d + wF d))).card ≤ 182 * wF d

Context. J cost 3: the axis facts of the J pass, and the window bound on an axis

AxJ P n src D k: AxNum plus 0 < s ≤ t, 8d² ≤ t, 8d²(t-s) ≤ 2t, the SegReg of the line, t ≤ 2^ℓ, ℓ d ≤ 2p, w ≤ |LJ|; axJ_ev: eventually on every axis.
win_ax, bd_ax: on an axis, every window [x, x + λ + w] of LJ (index form), and the border [c, s), hold at most 182 w positions.

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
import Definitions.Def_IntMul_KappaFramework
import Definitions.Def_IntMul_KappaJainData

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg
open Finset
open IntMul NLogN IntMul.Kappa.R6 J6Seg

Formal statement

theorem J6Seg.Side.win_num {d s t : ℕ} (hd1 : 1 ≤ d) (hs : 0 < s) (hst : s ≤ t) (h8 : 8 * d ^ 2 ≤ t)
    (hts : 8 * d ^ 2 * (t - s) ≤ 2 * t) (x : ℕ) :
    ((LJ s t (lamP d) (wF d)).toFinset.filter
      (fun j => x ≤ j ∧ j ≤ x + (lamP d + wF d))).card ≤ 182 * wF d := 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_num`, Solutions/J6SegJCost3.lean (lines 83–126) of the round-six development.

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