Round-six lemma `coreCost_le`

Proved

NLogN.KFL.coreCost_le

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.KFLMain):

theorem NLogN.KFL.coreCost_le {K ρ f : ℕ} (hK : 6 ≤ K) (hρK : ρ < K) (hf : 1 ≤ f) (g : Geo) (hg : g.Pos)
    (hQe : g.Q = 2 ^ (f * K)) (Ks Kcr : ℝ) :
    coreCost Ks Kcr K ρ f g ≤ 1 + 16 * Ks * ((g.V : ℝ) + 1) * ((f * K : ℕ) : ℝ) ^ (1 - (2 : ℝ)⁻¹ ^ 50) +
      2 * 10 ^ 15 * ((g.recG.V : ℝ) * ((zsz (g.V : ℤ) : ℝ) + 1) ^ 3 + g.V) +
      Kcr * ((g.recG.V : ℝ) * ((zsz (g.V : ℤ) : ℝ) + 1) ^ 3 + (g.V : ℝ) +
        min 1 (80 * ((f : ℝ) - 1) * 2⁻¹ ^ K) * (g.recG.V : ℝ) *
          ((zsz (g.V : ℤ) : ℝ) + 1) * ((g.B : ℝ) + (zsz (g.V : ℤ) : ℝ) + 1))

Context. KFL §18: Lemma 13 in IL

lem13_il (the corrected statement of KFH_WIP.lem13_iruns, see KFL_WIP): integers → bits (toBitsI), the slot canonicalisation by at most two Lemma-12 swaps (swSeq (preL px py)), the packed lines (linesI), the guard-set correction (corrI), the last selected bit (xorStep), the inverse swaps and bits → integers (fromBitsI). The composite is permRecs (rowAddAddr …) (bits_permRecs, pipeline_rixT, Fpp_conj); the cost is of the form (19) with τ = 1 - 2^{-50}.

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

Formal statement

theorem NLogN.KFL.coreCost_le {K ρ f : ℕ} (hK : 6 ≤ K) (hρK : ρ < K) (hf : 1 ≤ f) (g : Geo) (hg : g.Pos)
    (hQe : g.Q = 2 ^ (f * K)) (Ks Kcr : ℝ) :
    coreCost Ks Kcr K ρ f g ≤ 1 + 16 * Ks * ((g.V : ℝ) + 1) * ((f * K : ℕ) : ℝ) ^ (1 - (2 : ℝ)⁻¹ ^ 50) +
      2 * 10 ^ 15 * ((g.recG.V : ℝ) * ((zsz (g.V : ℤ) : ℝ) + 1) ^ 3 + g.V) +
      Kcr * ((g.recG.V : ℝ) * ((zsz (g.V : ℤ) : ℝ) + 1) ^ 3 + (g.V : ℝ) +
        min 1 (80 * ((f : ℝ) - 1) * 2⁻¹ ^ K) * (g.recG.V : ℝ) *
          ((zsz (g.V : ℤ) : ℝ) + 1) * ((g.B : ℝ) + (zsz (g.V : ℤ) : ℝ) + 1)) := 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.KFL.coreCost_le`, Solutions/KFLMain.lean (lines 389–434) of the round-six development.

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