Round-six lemma `stageU_sp`

Proved

NLogN.KIL.stageU_sp

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.KIL4Tbl):

theorem NLogN.KIL.stageU_sp (t : IState HS) {Q Bb : ℕ} (T : List ℤ) (L : List Item)
    (h14 : t.G 14 = ints T ++ L) (a c : ℕ) (LA LC LM LAM LN : List Item)
    (hSA : t.G sSA = .int a :: LA) (hSC : t.G sSC = .int c :: LC)
    (hSM : t.G sSM = .int T.length :: LM) (hSAM : t.G sSAM = .int ((a * T.length : ℕ) : ℤ) :: LAM)
    (hNQ : t.G sNQ = .int (-(Q : ℤ)) :: LN) (hc : c < Q) (hT : InR Q T)
    (hQ : zsz ((2 * Q : ℕ) : ℤ) ≤ Bb) (hM : zsz (T.length : ℤ) ≤ Bb) (ha : zsz (a : ℤ) ≤ Bb)
    (haM : zsz ((a * T.length : ℕ) : ℤ) ≤ Bb) :
    Sp stageU t (((((t.upd 14 (ints (stg Q a c T) ++ L)).upd sSA LA).upd sSC LC).upd sSM LM).upd
      sSAM LAM) (cStageU a T.length Bb)

Context. KIL4: the offset table generator with run-time stage data

KIL2's genP Q ac has one stage per field of the prefix A of the target, every number (Q, the radix a, the control c, the suffix volume M) a literal. The uniform generator genPU n reads the stages from four stage stacks (34 … 37: a, c, M, a·M, the last stage on top) and -Q from the parameter stack 23; its code depends only on the number n of stages.

gStageT, rsim_gStage: the stage as a template; the literal stage simulates it.
stagesU_sp, genPU_sp: the generator pushes tbl Q ac on stack 14, pops the stage stacks.
tbl_zeros, tbl_app_congr: the table only depends on the controls; runs of control-0 fields collapse to one field of their volume (the "collapsed" stage lists of the uniform shifts have at most 2m + 3 stages whatever the shape).

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_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KIL
open IntMul NLogN.RS AModel

Formal statement

theorem NLogN.KIL.stageU_sp (t : IState HS) {Q Bb : ℕ} (T : List ℤ) (L : List Item)
    (h14 : t.G 14 = ints T ++ L) (a c : ℕ) (LA LC LM LAM LN : List Item)
    (hSA : t.G sSA = .int a :: LA) (hSC : t.G sSC = .int c :: LC)
    (hSM : t.G sSM = .int T.length :: LM) (hSAM : t.G sSAM = .int ((a * T.length : ℕ) : ℤ) :: LAM)
    (hNQ : t.G sNQ = .int (-(Q : ℤ)) :: LN) (hc : c < Q) (hT : InR Q T)
    (hQ : zsz ((2 * Q : ℕ) : ℤ) ≤ Bb) (hM : zsz (T.length : ℤ) ≤ Bb) (ha : zsz (a : ℤ) ≤ Bb)
    (haM : zsz ((a * T.length : ℕ) : ℤ) ≤ Bb) :
    Sp stageU t (((((t.upd 14 (ints (stg Q a c T) ++ L)).upd sSA LA).upd sSC LC).upd sSM LM).upd
      sSAM LAM) (cStageU a T.length Bb) := 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.KIL.stageU_sp`, Solutions/KIL4Tbl.lean (lines 117–160) of the round-six development.

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