One backward record (round six, `bBodyS`)

Proved

J6Seg.PJ.Rn.bBodyS

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

complexity-theoryinteger-multiplicationkappa-improvement

One backward record.

The following holds (Lean statement, in the scope of module Solutions.J6SegILPreJD6):

theorem J6Seg.PJ.Rn.bBodyS {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 n22 t : ℕ} (h1 : u 1 = r)
    (hr : r < m) (h22 : u 22 = n22) (h23 : u 23 = t) (hD51 : D 51 = []) (hmW : m + 1 ≤ 2 ^ W)
    (hn22 : n22 + 1 ≤ 2 ^ W) (ht : t + (recP kk m mw cB L Mf r).length ≤ 2 ^ W) :
    Rn J6Seg.PJ.bBody u D (fun u' => u' 22 = (n22 : ℤ) + flg cB L r ∧
      u' 23 = ((t + (recP kk m mw cB L Mf r).length : ℕ) : ℤ) ∧
      (∀ i ∉ ([2, 3, 4, 5, 6, 7, 8, 22, 23, 29] : List (Fin HS)), u' i = u i) ∧ BV W u')
      (update D 41 ((recP kk m mw cB L Mf r).reverse ++ D 41)) (cbB W m)

Context. B-line IL, J pre-table (B5-J), jTab part D6: one backward record, one forward row

Rn.bBodyS: bBody at row r pushes (recP r).reverse onto 41, adds flg r to 22 and (recP r).length to 23; Rn.fBodyS: fBody at r' pushes (rowP (m - 1 - r')).reverse and adds its length to 23.

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.bBodyS {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 n22 t : ℕ} (h1 : u 1 = r)
    (hr : r < m) (h22 : u 22 = n22) (h23 : u 23 = t) (hD51 : D 51 = []) (hmW : m + 1 ≤ 2 ^ W)
    (hn22 : n22 + 1 ≤ 2 ^ W) (ht : t + (recP kk m mw cB L Mf r).length ≤ 2 ^ W) :
    Rn J6Seg.PJ.bBody u D (fun u' => u' 22 = (n22 : ℤ) + flg cB L r ∧
      u' 23 = ((t + (recP kk m mw cB L Mf r).length : ℕ) : ℤ) ∧
      (∀ i ∉ ([2, 3, 4, 5, 6, 7, 8, 22, 23, 29] : List (Fin HS)), u' i = u i) ∧ BV W u')
      (update D 41 ((recP kk m mw cB L Mf r).reverse ++ D 41)) (cbB 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.bBodyS`, Solutions/J6SegILPreJD6.lean (lines 45–108) of the round-six development.

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