Round-six lemma `xorU_runs`

Proved

NLogN.KFX.xorU_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.KFX2Rot):

theorem NLogN.KFX.xorU_runs {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    {d Q j : ℕ} (hd : 0 < d) (hQ : 0 < Q) {Bb : ℕ} (hB2 : 2 ≤ Bb)
    (zd : zsz (d : ℤ) ≤ Bb) (zQ : zsz (Q : ℤ) ≤ Bb) (zr : zsz ((2 : ℤ) ^ j) ≤ Bb)
    (P B w : ℕ) (zP : zsz (P : ℤ) ≤ Bb)
    (t : IState HS) (xs : List ℤ) (R : List Item) (hlen : xs.length = P * 2 * B) (hbw : Bnd w xs)
    (h0 : t.G 0 = ints xs ++ R) {eP eB ed eQ ej : Ex HS}
    (hsz : ((xorPl eP eB ed eQ ej).map fun p => p.2.size + 1).sum ≤ 100)
    (n1 : NoLow eP) (n2 : NoLow eB) (n3 : NoLow ed) (n4 : NoLow eQ) (n5 : NoLow ej)
    (E1 : ESp eP t Bb (P : ℤ)) (E2 : ESp eB t Bb (B : ℤ)) (E3 : ESp ed t Bb (d : ℤ))
    (E4 : ESp eQ t Bb (Q : ℤ)) (E5 : ESp ej t Bb ((2 : ℤ) ^ j)) :
    IRuns I iprog (xorU eP eB ed eQ ej) t
      (t.upd 0 (ints (pull [P, 2, B] [P, 2, B] (KIL.rotAddr 2 (xorA d Q j)) 0 xs) ++ R))
      (cXorU P B w Bb)

Context. KFX §2: register-driven table rotations (the lines (22) and the last bit)

pushSt_G_mem, pushOK_of: pushed parameter lists built from KFL.ESp facts.
tabRot_runs: a generic withPushI pl (fcode U ; rotF) where U pushes the rotation table.
tabGenU sgn b: KFL.tabGen with the literals P, Q, d0, d1, 2^ρ, f-1, 2^K read from the registers 1, 2, 3, 5, 6, 7, 8 (tabGenU_sub: the literal instance is RS.sub).
rotU_geo: one line rixT g (rot2 Q (amt K ρ f l)) with parameters from expressions.
xorGenU, xorU_geo: the last-bit step rixT g (XLc K ρ f).

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 AModel Butterfly

Formal statement

theorem NLogN.KFX.xorU_runs {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    {d Q j : ℕ} (hd : 0 < d) (hQ : 0 < Q) {Bb : ℕ} (hB2 : 2 ≤ Bb)
    (zd : zsz (d : ℤ) ≤ Bb) (zQ : zsz (Q : ℤ) ≤ Bb) (zr : zsz ((2 : ℤ) ^ j) ≤ Bb)
    (P B w : ℕ) (zP : zsz (P : ℤ) ≤ Bb)
    (t : IState HS) (xs : List ℤ) (R : List Item) (hlen : xs.length = P * 2 * B) (hbw : Bnd w xs)
    (h0 : t.G 0 = ints xs ++ R) {eP eB ed eQ ej : Ex HS}
    (hsz : ((xorPl eP eB ed eQ ej).map fun p => p.2.size + 1).sum ≤ 100)
    (n1 : NoLow eP) (n2 : NoLow eB) (n3 : NoLow ed) (n4 : NoLow eQ) (n5 : NoLow ej)
    (E1 : ESp eP t Bb (P : ℤ)) (E2 : ESp eB t Bb (B : ℤ)) (E3 : ESp ed t Bb (d : ℤ))
    (E4 : ESp eQ t Bb (Q : ℤ)) (E5 : ESp ej t Bb ((2 : ℤ) ^ j)) :
    IRuns I iprog (xorU eP eB ed eQ ej) t
      (t.upd 0 (ints (pull [P, 2, B] [P, 2, B] (KIL.rotAddr 2 (xorA d Q j)) 0 xs) ++ R))
      (cXorU P B w 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.KFX.xorU_runs`, Solutions/KFX2Rot.lean (lines 386–456) of the round-six development.

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