Round-six lemma `cost_final6`

Proved

IntMul.Kappa.R6.cost_final6

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.R6ZCost):

theorem IntMul.Kappa.R6.cost_final6 (hP : Q.P.Ok) (C18 : ℝ) (hC18 : 0 ≤ C18) (A18 c18 : ℕ)
    (hc : (c18 : ℝ) ≤ C18 * cU6 Q.P Q.n A18) :
    ((c18 + NLogN.KFZ.cBody (BN6 Q) (BN6 Q) (Q.p / 6) (dsG Q.p Q.h (D + 1) (σ6 Q) Q.K).length Q.p Q.h
      D (σ6 Q) (Q.sB D) (Q.tB D) + 2 : ℕ) : ℝ) ≤
      (C18 + Kc * ((Q.P.x : ℝ) + 4) ^ 2) * cU6 Q.P Q.n (A18 + 2)

Context. Round six, chirp: the cost of the chirp procedure against cU6

Ncost6_ge_one, lgp_le: Ncost6 ≥ 1, log p + 1 ≤ (x + 4)(log lg n + 1).
body_le6: the body without the Prop 18 call is O(T p (log p + 1)^2).
cost_final6: the Prop 18 call within C₁₈ · cU6 A₁₈ plus the body and the call is at most (C₁₈ + Kc (x+4)^2) · cU6 (A₁₈ + 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 IntMul NLogN NLogN.KIL NLogN.KFB NLogN.KFM IntMul.OpenAI.Glue NLogN.KFZ
attribute [local irreducible] fmA Cfm
variable {Q : Q6} (hQ : QOk Q) {D : ℕ} (hD : Q.d = D + 1)
include hQ hD

Formal statement

theorem IntMul.Kappa.R6.cost_final6 (hP : Q.P.Ok) (C18 : ℝ) (hC18 : 0 ≤ C18) (A18 c18 : ℕ)
    (hc : (c18 : ℝ) ≤ C18 * cU6 Q.P Q.n A18) :
    ((c18 + NLogN.KFZ.cBody (BN6 Q) (BN6 Q) (Q.p / 6) (dsG Q.p Q.h (D + 1) (σ6 Q) Q.K).length Q.p Q.h
      D (σ6 Q) (Q.sB D) (Q.tB D) + 2 : ℕ) : ℝ) ≤
      (C18 + Kc * ((Q.P.x : ℝ) + 4) ^ 2) * cU6 Q.P Q.n (A18 + 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.cost_final6`, Solutions/R6ZCost.lean (lines 149–211) of the round-six development.

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