Round-six lemma `p1U_runs`

Proved

NLogN.KFUU.p1U_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.KFUURev3):

theorem NLogN.KFUU.p1U_runs {pM : Fin P} (hM : MulOK I iprog pM MS) {pS : Fin HS} (hpS : LS pS)
    (u cz : ℕ) (bs : List Bool) (hcz : 1 ≤ cz) (w : ℕ) (xs0 : List ℤ)
    (hxs : xs0.length = vol (shP u bs) * cz) (hx : Bnd w xs0) (Z : ℕ)
    (hVZ : zsz ((vol (shP u bs) * cz : ℕ) : ℤ) ≤ Z) (hZ2 : 2 ≤ Z) (t : IState HS) (R : List Item)
    (h0 : t.G 0 = ints xs0 ++ R) {RS LZ LU LN : List Item}
    (hpS0 : t.G pS = ints (zl (shP u bs)) ++ RS)
    (hZ : t.G rCZ = .int (cz : ℤ) :: LZ) (hU : t.G rU2 = .int ((2 ^ u : ℕ) : ℤ) :: LU)
    (hN : t.G nR = .int (bs.length : ℤ) :: LN) :
    IRuns I iprog (p1U pM pS) t
      (t.upd 0 (ints (Dl (mzB u bs.length cz) (bgB bs) (List.range bs.length ++ [bs.length])
        (lay1 (bgB bs) bs.length 0) xs0 0) ++ R))
      ((bs.length + 1) * (KFU.cMv (vol (shP u bs) * cz) w + 50 * cU Z))

Context. KFUU §2c: phase 1 of the uniform reversal

p1U pM pS (sizes S = shP u bs on pS in order, n = |bs| on nR, 2^u on rU2, cz on rCZ) moves every extra bit behind the main atoms: Dl (lay1 n) ↦ Dl (lay1 0).

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
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_MachineRef

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFUU
open IntMul NLogN.KIL NLogN.KFC NLogN.KFT NLogN.KFU AModel AModelL12
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}

Formal statement

theorem NLogN.KFUU.p1U_runs {pM : Fin P} (hM : MulOK I iprog pM MS) {pS : Fin HS} (hpS : LS pS)
    (u cz : ℕ) (bs : List Bool) (hcz : 1 ≤ cz) (w : ℕ) (xs0 : List ℤ)
    (hxs : xs0.length = vol (shP u bs) * cz) (hx : Bnd w xs0) (Z : ℕ)
    (hVZ : zsz ((vol (shP u bs) * cz : ℕ) : ℤ) ≤ Z) (hZ2 : 2 ≤ Z) (t : IState HS) (R : List Item)
    (h0 : t.G 0 = ints xs0 ++ R) {RS LZ LU LN : List Item}
    (hpS0 : t.G pS = ints (zl (shP u bs)) ++ RS)
    (hZ : t.G rCZ = .int (cz : ℤ) :: LZ) (hU : t.G rU2 = .int ((2 ^ u : ℕ) : ℤ) :: LU)
    (hN : t.G nR = .int (bs.length : ℤ) :: LN) :
    IRuns I iprog (p1U pM pS) t
      (t.upd 0 (ints (Dl (mzB u bs.length cz) (bgB bs) (List.range bs.length ++ [bs.length])
        (lay1 (bgB bs) bs.length 0) xs0 0) ++ R))
      ((bs.length + 1) * (KFU.cMv (vol (shP u bs) * cz) w + 50 * cU Z)) := 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.p1U_runs`, Solutions/KFUURev3.lean (lines 147–221) of the round-six development.

View graph