Prove2Me

Round-six lemma `body_le6`

Proved
IntMul.Kappa.R6.body_le6

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.body_le6 :
    (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) : ℝ) ≤
      10 ^ 30 * (2 * (32500 * (Cfm : ℝ) + 6) + 6 ^ 12) * ((Q.T : ℝ) * Q.p *
        (Real.log Q.p + 1) ^ 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.body_le6 :
    (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) : ℝ) ≤
      10 ^ 30 * (2 * (32500 * (Cfm : ℝ) + 6) + 6 ^ 12) * ((Q.T : ℝ) * Q.p *
        (Real.log Q.p + 1) ^ 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.body_le6`, Solutions/R6ZCost.lean (lines 73–147) of the round-six development.

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