Round-six lemma `fineE_runs`

Proved

NLogN.KFUU.R6Fine.fineE_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.R6FineRun):

theorem NLogN.KFUU.R6Fine.fineE_runs (core : IStmt HS P) (w f a Z : ℕ) (t : IState HS) (Ld : List Item)
    (hd : t.G NLogN.KFUU.dP = .int (w : ℤ) :: .int (f : ℤ) :: .int (a : ℤ) :: Ld)
    (zw : zsz (w : ℤ) ≤ Z) (zf : zsz (f : ℤ) ≤ Z) (za : zsz (a : ℤ) ≤ Z) (X0 : List Item) (c : ℕ)
    (hcore : 1 ≤ a → IRuns I iprog core
      ((((t.upd NLogN.KFUU.dP Ld).upd NLogN.KFUU.rW (.int (w : ℤ) :: t.G NLogN.KFUU.rW)).upd rU (.int (f : ℤ) :: t.G rU)).upd
        rCZ (.int (a : ℤ) :: t.G rCZ))
      (((((t.upd NLogN.KFUU.dP Ld).upd NLogN.KFUU.rW (.int (w : ℤ) :: t.G NLogN.KFUU.rW)).upd rU (.int (f : ℤ) :: t.G rU)).upd
        rCZ (.int (a : ℤ) :: t.G rCZ)).upd 0 X0) c)
    (hz : a = 0 → X0 = t.G 0) :
    IRuns I iprog (fineE core) t (t.upd 0 X0) (c + 6 * NLogN.KFUU.cU Z + 26)

Context. Fine-bit exposure §2: the procedures fineF / fineB in IL

bodyF: qX /= 2, one mvFU move, qZ *= 2; bodyB: qZ /= 2, one mvBU move, qX *= 2. loopF_runs / loopB_runs: f iterations (rptX nR) take stage 0 to stage f.
coreF / coreB: with y on dP and a, f in rCZ, rU, set qB = y, qX, qZ, nR = f, tq = 2, run the loop, clear. coreF_runs / coreB_runs.
fineE core: pop w, f, a from dP (into rW, rU, rCZ), run core if a > 0, push them back. fineE_runs.

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_Defs70

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFUU NLogN.KFUU.R6Fine
set_option maxRecDepth 20000
open IntMul NLogN.KIL NLogN.KFB NLogN.KFC NLogN.KFT NLogN.KFU AModel AModelL12 NLogN.KFUU IntMul.Kappa.R6
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}

Formal statement

theorem NLogN.KFUU.R6Fine.fineE_runs (core : IStmt HS P) (w f a Z : ℕ) (t : IState HS) (Ld : List Item)
    (hd : t.G NLogN.KFUU.dP = .int (w : ℤ) :: .int (f : ℤ) :: .int (a : ℤ) :: Ld)
    (zw : zsz (w : ℤ) ≤ Z) (zf : zsz (f : ℤ) ≤ Z) (za : zsz (a : ℤ) ≤ Z) (X0 : List Item) (c : ℕ)
    (hcore : 1 ≤ a → IRuns I iprog core
      ((((t.upd NLogN.KFUU.dP Ld).upd NLogN.KFUU.rW (.int (w : ℤ) :: t.G NLogN.KFUU.rW)).upd rU (.int (f : ℤ) :: t.G rU)).upd
        rCZ (.int (a : ℤ) :: t.G rCZ))
      (((((t.upd NLogN.KFUU.dP Ld).upd NLogN.KFUU.rW (.int (w : ℤ) :: t.G NLogN.KFUU.rW)).upd rU (.int (f : ℤ) :: t.G rU)).upd
        rCZ (.int (a : ℤ) :: t.G rCZ)).upd 0 X0) c)
    (hz : a = 0 → X0 = t.G 0) :
    IRuns I iprog (fineE core) t (t.upd 0 X0) (c + 6 * NLogN.KFUU.cU Z + 26) := 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 `NLogN.KFUU.R6Fine.fineE_runs`, Solutions/R6FineRun.lean (lines 549–600) of the round-six development.

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