Round-six lemma `rowCodeU_runs`

Proved

NLogN.KFX.rowCodeU_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.KFX6Run):

theorem NLogN.KFX.rowCodeU_runs {q : ℕ} {Kc : ℝ} (H13 : L13U q Kc) {I : List Sym}
    {k K e T S0 B r H : ℕ} {i j : Fin mC} (hij : (i : ℕ) ≠ j) (hK : 6 ≤ K)
    (hp : PF k K e T S0 B r) {u : IState HS} (hA : Agree (rowσ k K e T S0 B r H i j) u)
    (zs : List GI) (R0 : List Item) (hg0 : u.G 0 = ints (gflat zs) ++ R0)
    (hl : zs.length = B * 2 ^ T * S0 * r) (hGB : GB H zs) :
    ∃ c, IRuns I l12prog (rowCodeU q) u
        (u.upd 0 (ints (gflat (permRecs (rect (N0R k K e T B i j) (N1R k K e i j) (N2R k K e i j)
          (N3R k K e S0 i j) (mC ^ k * K)) (rowAddAddr K (e % K) (mC ^ k) (pxR i j) (pyR i j)) r zs))
          ++ R0)) c ∧
      (c : ℝ) ≤ KFR.bnd Kc K (mC ^ k) (B * 2 ^ T * S0) r H +
        (cHd (zsz ((VbR T S0 B r H : ℕ) : ℤ)) : ℝ)

Context. KFX §6: the bit-length loop and the run of the uniform row addition

loopZ computes A = zsz V on stack 48 by halving V (pushed on stack 14) until it is 0. rowCodeU q = withPushI hdrL (zB (l13U q)) pushes the master registers computed from the parameter registers (hdrL), runs loopZ, then l13U q.

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.KFX
open IntMul NLogN.RS NLogN.KIL NLogN.KFD NLogN.KFE NLogN.KFB NLogN.KFL NLogN.KFW NLogN.KFH AModel Butterfly

Formal statement

theorem NLogN.KFX.rowCodeU_runs {q : ℕ} {Kc : ℝ} (H13 : L13U q Kc) {I : List Sym}
    {k K e T S0 B r H : ℕ} {i j : Fin mC} (hij : (i : ℕ) ≠ j) (hK : 6 ≤ K)
    (hp : PF k K e T S0 B r) {u : IState HS} (hA : Agree (rowσ k K e T S0 B r H i j) u)
    (zs : List GI) (R0 : List Item) (hg0 : u.G 0 = ints (gflat zs) ++ R0)
    (hl : zs.length = B * 2 ^ T * S0 * r) (hGB : GB H zs) :
    ∃ c, IRuns I l12prog (rowCodeU q) u
        (u.upd 0 (ints (gflat (permRecs (rect (N0R k K e T B i j) (N1R k K e i j) (N2R k K e i j)
          (N3R k K e S0 i j) (mC ^ k * K)) (rowAddAddr K (e % K) (mC ^ k) (pxR i j) (pyR i j)) r zs))
          ++ R0)) c ∧
      (c : ℝ) ≤ KFR.bnd Kc K (mC ^ k) (B * 2 ^ T * S0) r H +
        (cHd (zsz ((VbR T S0 B r H : ℕ) : ℤ)) : ℝ) := 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.KFX.rowCodeU_runs`, Solutions/KFX6Run.lean (lines 287–430) of the round-six development.

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