The border scan (round six, `czL`)

Proved

J6Seg.PJ.Rn.czL

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

complexity-theoryinteger-multiplicationkappa-improvement

The border scan of row r.

The following holds (Lean statement, in the scope of module Solutions.J6SegILPreJD5):

theorem J6Seg.PJ.Rn.czL {u : Vr} {D : Cs} (hF : Fx Q kk m mw cB u) (hS : StS m L Mf D)
    (hB : BdS W m L Mf) (hLm : L.length = m) (hv : BV W u) {r : ℕ} (h1 : u 1 = r) (hr : r < m)
    (h4 : u 4 = Lg L r) (h3 : u 3 = 0) (hD51 : D 51 = []) (hmW : m + 1 ≤ 2 ^ W) :
    Rn (PJ.forL 51 28 2 czB) u D (fun u' => u' 3 = ((czP m mw cB L Mf r).length : ℤ) ∧
      (∀ i ∉ ([2, 3, 6, 7, 8, 29] : List (Fin HS)), u' i = u i) ∧ BV W u')
      (update D 41 ((czP m mw cB L Mf r).reverse ++ D 41)) (cFor W m (cCz W m))

Context. B-line IL, J pre-table (B5-J), jTab part D5: the inner loops of serP

Rn.czL (border scan), Rn.wbL (band-window count), Rn.cbL (band-window push), Rn.cnL (forward-window count), Rn.fsL (forward push): each as an Rn of a forL on stack 51, pushing its list (reversed) onto 41.

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_Defs71

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.PJ
open IntMul NLogN NLogN.KIL NLogN.RS Function J6Seg.PJ.E
variable {Q kk m mw cB W : ℕ} {L : List ℕ} {Mf : ℕ → ℕ → ℤ}

Formal statement

theorem J6Seg.PJ.Rn.czL {u : Vr} {D : Cs} (hF : Fx Q kk m mw cB u) (hS : StS m L Mf D)
    (hB : BdS W m L Mf) (hLm : L.length = m) (hv : BV W u) {r : ℕ} (h1 : u 1 = r) (hr : r < m)
    (h4 : u 4 = Lg L r) (h3 : u 3 = 0) (hD51 : D 51 = []) (hmW : m + 1 ≤ 2 ^ W) :
    Rn (PJ.forL 51 28 2 czB) u D (fun u' => u' 3 = ((czP m mw cB L Mf r).length : ℤ) ∧
      (∀ i ∉ ([2, 3, 6, 7, 8, 29] : List (Fin HS)), u' i = u i) ∧ BV W u')
      (update D 41 ((czP m mw cB L Mf r).reverse ++ D 41)) (cFor W m (cCz W m)) := 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.PJ.Rn.czL`, Solutions/J6SegILPreJD5.lean (lines 47–118) of the round-six development.

View graph