The forward rows (round six, `fLoop`)

Proved

J6Seg.PJ.Rn.fLoop

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

complexity-theoryinteger-multiplicationkappa-improvement

The forward rows.

The following holds (Lean statement, in the scope of module Solutions.J6SegILPreJD7):

theorem J6Seg.PJ.Rn.fLoop {v : Vr} {D : Cs} (hF : Fx Q kk m mw cB v) (hS : StS m L Mf D)
    (hB : BdS W m L Mf) (hLm : L.length = m) (hv : BV W v) {t : ℕ} (h23 : v 23 = t)
    (hD50 : D 50 = []) (hD51 : D 51 = [])
    (ht : t + ((List.range m).flatMap (fun r' => rowP Q m mw cB L Mf (m - 1 - r'))).length ≤ 2 ^ W)
    (hW8 : 8 * (m + 1) ^ 2 ≤ 2 ^ W) :
    Rn (PJ.forL 50 28 1 J6Seg.PJ.fBody) v D (fun v' =>
      v' 23 = ((t + ((List.range m).flatMap (fun r' => rowP Q m mw cB L Mf (m - 1 - r'))).length
        : ℕ) : ℤ) ∧
      (∀ i ∉ ([1, 2, 3, 4, 5, 6, 7, 8, 23, 29] : List (Fin HS)), v' i = v i) ∧ BV W v')
      (update D 41 (((List.range m).flatMap (fun r' => rowP Q m mw cB L Mf (m - 1 - r'))).reverse
        ++ D 41))
      (cFor W m (cfB W m))

Context. B-line IL, J pre-table (B5-J), jTab part D7: the serialization stage serP

Rn.serPs: serP pushes (serL …).reverse onto 41, sets 22 to the number of border rows and 23 to the number of pushed entries; registers outside 1–8, 22, 23, 29 and all stacks but 41 are unchanged.

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.fLoop {v : Vr} {D : Cs} (hF : Fx Q kk m mw cB v) (hS : StS m L Mf D)
    (hB : BdS W m L Mf) (hLm : L.length = m) (hv : BV W v) {t : ℕ} (h23 : v 23 = t)
    (hD50 : D 50 = []) (hD51 : D 51 = [])
    (ht : t + ((List.range m).flatMap (fun r' => rowP Q m mw cB L Mf (m - 1 - r'))).length ≤ 2 ^ W)
    (hW8 : 8 * (m + 1) ^ 2 ≤ 2 ^ W) :
    Rn (PJ.forL 50 28 1 J6Seg.PJ.fBody) v D (fun v' =>
      v' 23 = ((t + ((List.range m).flatMap (fun r' => rowP Q m mw cB L Mf (m - 1 - r'))).length
        : ℕ) : ℤ) ∧
      (∀ i ∉ ([1, 2, 3, 4, 5, 6, 7, 8, 23, 29] : List (Fin HS)), v' i = v i) ∧ BV W v')
      (update D 41 (((List.range m).flatMap (fun r' => rowP Q m mw cB L Mf (m - 1 - r'))).reverse
        ++ D 41))
      (cFor W m (cfB 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.fLoop`, Solutions/J6SegILPreJD7.lean (lines 107–164) of the round-six development.

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