Round-six lemma `fmB_le_log`

Proved

NLogN.KFZ.fmB_le_log

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.KFZFmb):

theorem NLogN.KFZ.fmB_le_log (p m : ℕ) (hp : 1 ≤ p) (hm : m ≤ 64 * p) :
    (KFM.fmB m : ℝ) ≤ 32500 * (Cfm : ℝ) * p * (Real.log p + 1) ^ 2

Context. KFZ Fmb: fmB is O(m log² m)

fmB_le_log: for 1 ≤ p and m ≤ 64 p, KFM.fmB m ≤ 32500 (8 fmA + 10^7 + 10^5) · p · (log p + 1)^2.

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 Definitions.Def_IntMul_MultitapeModel
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFZ
open IntMul NLogN.KFM
attribute [local irreducible] fmA Cfm

Formal statement

theorem NLogN.KFZ.fmB_le_log (p m : ℕ) (hp : 1 ≤ p) (hm : m ≤ 64 * p) :
    (KFM.fmB m : ℝ) ≤ 32500 * (Cfm : ℝ) * p * (Real.log p + 1) ^ 2 := 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.KFZ.fmB_le_log`, Solutions/KFZFmb.lean (lines 54–94) of the round-six development.

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