CrtNeed6 holds for an admissible round-six parameter record

Proved

IntMul.Kappa.R6.crtNeed6

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

For a round-six parameter record $P$ with $P.\mathrm{Ok}$ , CrtNeed6 P holds: given IL procedures for the bit-network swap (SwapNeedτ) and the compact-controls row addition (RowUCP 576), the CRT maps $\Phi$ and $\Phi^{-1}$ (with the E′ reversal of the CRT axis) are implemented on the IL machine within cost $C_0\cdot\mathrm{cU6}(P,n,A)$ for all large $n$ .

In Lean:

theorem IntMul.Kappa.R6.crtNeed6 (P : Prm) (hP : P.Ok) : CrtNeed6 P

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 IntMul IntMul.Kappa IntMul.Kappa.R6
open IntMul NLogN NLogN.KIL NLogN.KFB NLogN.KFM AModel NLogN.KFUU NLogN.KFUU.R6Crt NLogN.KFU

Formal statement

theorem IntMul.Kappa.R6.crtNeed6 (P : Prm) (hP : P.Ok) : CrtNeed6 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.crtNeed6`, Solutions/R6CrtNeed.lean (lines 262–459) of the round-six development.

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