Round-six lemma `finB_runs`

Proved

J6Seg.Slab.finB_runs

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

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

The following holds (Lean statement, in the scope of module Solutions.J6SegILCtl4):

theorem J6Seg.Slab.finB_runs (s : IState HS) (G : Fin HS → List ℤ) (X Sv LC T Cd A D W : List ℤ) (n cur : ℕ)
    {L2 : List ℤ} (hN : G bN = (n : ℤ) :: L2) (hle : cur ≤ n) {Z : ℕ} (hn : zsz (n : ℤ) ≤ Z)
    (hA : G lA = []) (hC : G lC = []) (hD : G lD = []) :
    IRuns I iprog finB (encK s (cS G X Sv ((cur : ℤ) :: LC) T Cd A D W))
      (encK s (cS G X Sv LC ((flatC (List.replicate (n - cur) J6Seg.dropC)).reverse ++ T)
        Cd A D W))
      (6 * cU Z + 4000 * (n - cur + 1))

Context. B-line IL, B1 follow-up: building the control list, §4 the main loop and the final drops

mainLoop_runs: rptX bN' entP over the whole table: the reversed control prefix (ctlAux src argOf (range n') 0).1 on bT, the position pos n' on bCur, the table moved (reversed) to bSv; cost n' (21 cU Z + 4000) + 4000 n + 3000 (n' + 1) (the drops telescope).
finB_runs: the final n - pos n' drops, bCur popped: bT holds (flatC (ctlOf …)).reverse.
lenB_runs: n + n' pushed on rNC.

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.Slab
set_option maxRecDepth 20000
open IntMul NLogN NLogN.KIL Function
open NLogN.KFUU (lA lC lD cU)
open NLogN.KFT (cf3 rptX)
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}

Formal statement

theorem J6Seg.Slab.finB_runs (s : IState HS) (G : Fin HS → List ℤ) (X Sv LC T Cd A D W : List ℤ) (n cur : ℕ)
    {L2 : List ℤ} (hN : G bN = (n : ℤ) :: L2) (hle : cur ≤ n) {Z : ℕ} (hn : zsz (n : ℤ) ≤ Z)
    (hA : G lA = []) (hC : G lC = []) (hD : G lD = []) :
    IRuns I iprog finB (encK s (cS G X Sv ((cur : ℤ) :: LC) T Cd A D W))
      (encK s (cS G X Sv LC ((flatC (List.replicate (n - cur) J6Seg.dropC)).reverse ++ T)
        Cd A D W))
      (6 * cU Z + 4000 * (n - cur + 1)) := 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.Slab.finB_runs`, Solutions/J6SegILCtl4.lean (lines 87–145) of the round-six development.

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