Round-six lemma `ph2`

Proved

J6Seg.PJ.Rn.ph2

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

complexity-theoryinteger-multiplicationkappa-improvement

R, kF, λF, w, λ.

The following holds (Lean statement, in the scope of module Solutions.J6SegILPreJ7):

theorem J6Seg.PJ.Rn.ph2 {s t p α d W : ℕ} {c : Cs} (hcL : c cLC = []) {vb : Vr} (hb : BV W vb)
    (h1 : vb 1 = p) (h2 : vb 2 = d) (hp : p < 2 ^ W) (hd : 2 * d + 8 < 2 ^ W)
    (hl : 2 * lamF d + 8 < 2 ^ W) (hw : wF d < 2 ^ W) :
    Rn J6Seg.PJ.ph2 vb c (fun v => v = update (update (update (update (update (update vb 21 ((p / 7 : ℕ) : ℤ))
      5 (kF d)) 6 (lamF d)) 26 (wF d)) 24 1) 24 (lamP d) ∧ BV W v) c (cPh2 W d)

Context. B-line IL, J pre-table (B5-J), part 7: the parameter phase

parP Z kA kC wA wC: from the axis descriptor [k, pre, post, s, t, p, α, d] on Z, the registers (ParV)

1 p   2 d   3 pre   4 post   5 kF   6 λF   7 wd = wA·Q+wC   9 2^Q   10 2^(Q+1)
11 kJ = kA·Q+kC   12 2^kJ   20 Q   21 R = p/7   22 s   23 t   24 λ   25 α   26 w = 7 kF

(Q = segPF p d s, λ = lamP d), every other stack unchanged (Rn.parP).

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_Defs65

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.PJ
open IntMul NLogN NLogN.KIL NLogN.RS J6Seg.PJ.E Function

Formal statement

theorem J6Seg.PJ.Rn.ph2 {s t p α d W : ℕ} {c : Cs} (hcL : c cLC = []) {vb : Vr} (hb : BV W vb)
    (h1 : vb 1 = p) (h2 : vb 2 = d) (hp : p < 2 ^ W) (hd : 2 * d + 8 < 2 ^ W)
    (hl : 2 * lamF d + 8 < 2 ^ W) (hw : wF d < 2 ^ W) :
    Rn J6Seg.PJ.ph2 vb c (fun v => v = update (update (update (update (update (update vb 21 ((p / 7 : ℕ) : ℤ))
      5 (kF d)) 6 (lamF d)) 26 (wF d)) 24 1) 24 (lamP d) ∧ BV W v) c (cPh2 W 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.PJ.Rn.ph2`, Solutions/J6SegILPreJ7.lean (lines 180–220) of the round-six development.

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