Round-six lemma `globU_runs`

Proved

NLogN.KFUU.R6Crt.globU_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.R6CrtP2):

theorem NLogN.KFUU.R6Crt.globU_runs {pM : Fin P} (hMul : MulOK I iprog pM MS) (Q u n c2 V Z : ℕ) (hQd : Q = 2 ^ u)
    (hu : 1 ≤ u) (hn : 2 ≤ n) (hV : Q ^ n * (c2 * 2) ≤ V) (hc2 : 1 ≤ c2) (hVZ : zsz ((V : ℕ) : ℤ) ≤ Z)
    (t : IState HS) {L2 LM LB LH LN LU : List Item}
    (h2 : t.G rU2 = .int ((Q : ℕ) : ℤ) :: L2) (hM : t.G rM = .int ((Q ^ n : ℕ) : ℤ) :: LM)
    (hB : t.G rB = .int ((c2 : ℕ) : ℤ) :: LB) (hH : t.G rH = .int ((n / 2 : ℕ) : ℤ) :: LH)
    (hN : t.G nR = .int (n : ℤ) :: LN) (hU : t.G rU = .int (u : ℤ) :: LU) :
    IRuns I iprog (globU pM) t (gSt t Q u n (c2 * 2)) (cGlob n Z)

Context. CRT reversal §7: the global registers (globU, gdropU) and the prologue (proU)

proU: the prologue of KFUU's p2U (volume, Q^n, rB = vol ctB · cr, rH = ⌊n/2⌋).
globU: gQm = Q^m, gP = Q^{n-m}, gMd = Q^{n-2m}, gD = Q^{m-1}, gB = Q^{m-1}·vol T, gI = n - m, gUm = u·m, and n copies of Q on oSP.
gdropU: pops them again.

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_R6_Defs14

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFUU NLogN.KFUU.R6Crt
open IntMul NLogN.KIL NLogN.KFB NLogN.KFC NLogN.KFT NLogN.KFU AModel AModelL12 NLogN.KFUU
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}

Formal statement

theorem NLogN.KFUU.R6Crt.globU_runs {pM : Fin P} (hMul : MulOK I iprog pM MS) (Q u n c2 V Z : ℕ) (hQd : Q = 2 ^ u)
    (hu : 1 ≤ u) (hn : 2 ≤ n) (hV : Q ^ n * (c2 * 2) ≤ V) (hc2 : 1 ≤ c2) (hVZ : zsz ((V : ℕ) : ℤ) ≤ Z)
    (t : IState HS) {L2 LM LB LH LN LU : List Item}
    (h2 : t.G rU2 = .int ((Q : ℕ) : ℤ) :: L2) (hM : t.G rM = .int ((Q ^ n : ℕ) : ℤ) :: LM)
    (hB : t.G rB = .int ((c2 : ℕ) : ℤ) :: LB) (hH : t.G rH = .int ((n / 2 : ℕ) : ℤ) :: LH)
    (hN : t.G nR = .int (n : ℤ) :: LN) (hU : t.G rU = .int (u : ℤ) :: LU) :
    IRuns I iprog (globU pM) t (gSt t Q u n (c2 * 2)) (cGlob n 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.R6Crt.globU_runs`, Solutions/R6CrtP2.lean (lines 119–269) of the round-six development.

View graph