Round-six lemma `setupNeed6`

Proved

IntMul.Kappa.R6.setupNeed6

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.R6SNeed):

theorem IntMul.Kappa.R6.setupNeed6 (P : Prm) (hP : P.Ok) : SetupNeed6 P

Context. Round-six setup, part 9: SetupNeed6

The procedure is startP (frozen: n and b = lg n), the compiled register program setRS, and oP (frozen: 2^{2n} on rO). Its cost is O(n) + cSet · kU; the second term is O(d r² p²) ≤ T p for large n (since T ≥ r³/4 once d ≥ 3, and r = 2^ℓ beats every fixed power of b).

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_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_HvdH_MachineRef
import Definitions.Def_IntMul_KappaFramework
import Definitions.Def_IntMul_KappaJainData
import Definitions.Def_IntMul_Kappa_R6_Defs16

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open IntMul NLogN NLogN.KIL NLogN.KFM NLogN.RS Finset Function

Formal statement

theorem IntMul.Kappa.R6.setupNeed6 (P : Prm) (hP : P.Ok) : SetupNeed6 P := 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 `IntMul.Kappa.R6.setupNeed6`, Solutions/R6SNeed.lean (lines 182–279) of the round-six development.

View graph