Round-six lemma `ZU6_le`

Proved

IntMul.Kappa.R6.ZU6_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.R6MTotal):

theorem IntMul.Kappa.R6.ZU6_le {Q : Q6} (hQ : QOk Q) (hP : Q.P.Ok) (A : ℕ) :
    (ZU6 Q : ℝ) ≤ Kz6 Q.P * cU6 Q.P Q.n (A + 2)

Context. Round-six machine: the total cost of one context

cU6_mono, ZU6_le: the unit ZU6 is within Kz6 · cU6 A (A ≥ 2).
total_le6: the setup (within C_S · cU6 A_S) plus kMul6 + kFin6 + 2 of a context is at most Ctot6 · cU6 A' with A' = A_Φ + A_L + A_C + A_S + 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_R6_Defs17

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open Finset IntMul NLogN NLogN.KIL NLogN.KFB NLogN.KFT NLogN.KFU NLogN.RS IntMul.OpenAI.Glue IntMul.OpenAI.RCost NLogN.KFM

Formal statement

theorem IntMul.Kappa.R6.ZU6_le {Q : Q6} (hQ : QOk Q) (hP : Q.P.Ok) (A : ℕ) :
    (ZU6 Q : ℝ) ≤ Kz6 Q.P * cU6 Q.P Q.n (A + 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 `IntMul.Kappa.R6.ZU6_le`, Solutions/R6MTotal.lean (lines 67–145) of the round-six development.

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