Round-six lemma `withSplitF_runs`

Proved

NLogN.KFD.withSplitF_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.KFDLevel):

theorem NLogN.KFD.withSplitF_runs (p : IStmt HS Pp) (W P L' h Bb : ℕ) (t : IState HS) (xs ys : List ℤ) (R : List Item)
    (hP1 : 1 ≤ P) (hL1 : 1 ≤ L') (hh : 1 ≤ h)
    (zW : zsz (W : ℤ) ≤ Bb) (zv : zsz ((P * L' * h : ℕ) : ℤ) ≤ Bb) (hB1 : 1 ≤ Bb)
    {L1 L2 L3 : List Item} (hPp : t.G dP = .int (P : ℤ) :: L1)
    (hLp : t.G rLp = .int (L' : ℤ) :: L2)
    (hVv : t.G rV = .int ((P * L' * h : ℕ) : ℤ) :: L3) (cp : ℕ)
    (hp : ∀ u : IState HS, u.G 0 = ints xs ++ R → u.G 1 = .int ((1 : ℕ) : ℤ) :: t.G 1 →
      u.G 2 = .int (W : ℤ) :: t.G 2 → u.G 3 = .int ((P * L' : ℕ) : ℤ) :: t.G 3 →
      u.G 4 = .int (h : ℤ) :: t.G 4 →
      IRuns I iprog p u (u.upd 0 (ints ys ++ R)) cp) :
    IRuns I iprog (withPushI (splitL W) p) (t.upd 0 (ints xs ++ R)) (t.upd 0 (ints ys ++ R))
      (cSplit W Bb + cp)

Context. KFD: the fine body of one uniform level

bodyF C tpl j0 is KIL5's bodyU with every per-element piece replaced by its fine form: fcode (subU …), the row split / merge by KFC's bwdF / fwdF, the fine network netCF with fine frame changes edgeCF (fine level steps fcode (stepU …)), the fine routing fcode (rebuildCU …). bodyF_run: the same denotation levelR as bodyU_run, with the fine cost bodyCF.

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.KFD
open IntMul NLogN.RS
open NLogN.KIL AModel AModelL12
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}

Formal statement

theorem NLogN.KFD.withSplitF_runs (p : IStmt HS Pp) (W P L' h Bb : ℕ) (t : IState HS) (xs ys : List ℤ) (R : List Item)
    (hP1 : 1 ≤ P) (hL1 : 1 ≤ L') (hh : 1 ≤ h)
    (zW : zsz (W : ℤ) ≤ Bb) (zv : zsz ((P * L' * h : ℕ) : ℤ) ≤ Bb) (hB1 : 1 ≤ Bb)
    {L1 L2 L3 : List Item} (hPp : t.G dP = .int (P : ℤ) :: L1)
    (hLp : t.G rLp = .int (L' : ℤ) :: L2)
    (hVv : t.G rV = .int ((P * L' * h : ℕ) : ℤ) :: L3) (cp : ℕ)
    (hp : ∀ u : IState HS, u.G 0 = ints xs ++ R → u.G 1 = .int ((1 : ℕ) : ℤ) :: t.G 1 →
      u.G 2 = .int (W : ℤ) :: t.G 2 → u.G 3 = .int ((P * L' : ℕ) : ℤ) :: t.G 3 →
      u.G 4 = .int (h : ℤ) :: t.G 4 →
      IRuns I iprog p u (u.upd 0 (ints ys ++ R)) cp) :
    IRuns I iprog (withPushI (splitL W) p) (t.upd 0 (ints xs ++ R)) (t.upd 0 (ints ys ++ R))
      (cSplit W Bb + cp) := 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.KFD.withSplitF_runs`, Solutions/KFDLevel.lean (lines 56–138) of the round-six development.

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