Round-six lemma `jcost_arith`

Proved

J6Seg.Side.jcost_arith

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.J6SegJCost4):

theorem J6Seg.Side.jcost_arith {d pre post T m G w u E Ckn Lj Tot N K : ℕ}
    (hK : K ≤ pre * post * Ckn + 100000 * E * (pre * post * (G + 1) + pre * (Lj + Tot + 2) + N + 2))
    (hC : Ckn ≤ 3100 * (w + 1) * (m + 1) * u + 2)
    (hL : Lj + Tot + 2 ≤ 3000 * (w + 1) * (m + 1))
    (hG : G + 1 ≤ 2 * (m + 1))
    (h1 : d * (pre * post * (m + 1)) ≤ 100 * T) (h2 : d * (pre * post) ≤ T) (h3 : d * N ≤ 198 * T)
    (h4 : d ≤ T) (hpost : 1 ≤ post) (hE : 1 ≤ E) :
    d * K ≤ T * (400000 * (w + 1) * u + 40000000000 * E * (w + 1))

Context. J cost 4: the J pass as a natural-number polynomial

kerC_le1: kerC Ckn B m post Lj 1 Tot pre N ≤ pre·post·Ckn + 10^5 (B+2)(pre·post·(m+1) + pre (Lj + Tot + 2) + N + 2);
jcost_arith: the generic arithmetic;
jcost_ax: on an axis, for every u and B, d · kerC (jC … u + 2) B (2^fJ) post |TJ6| 1 |TJ| pre NJ ≤ T (4·10^5 (w+1) u + 4·10^10 (B+2)(w+1));
jBody_ax: |jBody| ≤ 1460 w m + 24 m + G + 3w + 13.

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_RealExp
import Definitions.Def_IntMul_HvdH_MachineRef
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_SApprox
import Definitions.Def_IntMul_HvdH_DiagApprox
import Definitions.Def_IntMul_HvdH_EApprox
import Definitions.Def_IntMul_HvdH_JApprox
import Definitions.Def_IntMul_HvdH_BApprox
import Definitions.Def_IntMul_HvdH_FPApprox
import Definitions.Def_IntMul_KappaFramework
import Definitions.Def_IntMul_KappaJainData

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open Finset
open IntMul NLogN IntMul.Kappa.R6

Formal statement

theorem J6Seg.Side.jcost_arith {d pre post T m G w u E Ckn Lj Tot N K : ℕ}
    (hK : K ≤ pre * post * Ckn + 100000 * E * (pre * post * (G + 1) + pre * (Lj + Tot + 2) + N + 2))
    (hC : Ckn ≤ 3100 * (w + 1) * (m + 1) * u + 2)
    (hL : Lj + Tot + 2 ≤ 3000 * (w + 1) * (m + 1))
    (hG : G + 1 ≤ 2 * (m + 1))
    (h1 : d * (pre * post * (m + 1)) ≤ 100 * T) (h2 : d * (pre * post) ≤ T) (h3 : d * N ≤ 198 * T)
    (h4 : d ≤ T) (hpost : 1 ≤ post) (hE : 1 ≤ E) :
    d * K ≤ T * (400000 * (w + 1) * u + 40000000000 * E * (w + 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 `J6Seg.Side.jcost_arith`, Solutions/J6SegJCost4.lean (lines 31–94) of the round-six development.

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