One axis (round six, `side_at`)

Proved

J6Seg.Side.side_at

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

complexity-theoryinteger-multiplicationkappa-improvement

One axis.

The following holds (Lean statement, in the scope of module Solutions.J6SegSideCost5):

theorem J6Seg.Side.side_at (P : Prm) (ap : AxPlan6) {n : ℕ} {src : ℕ → ℕ} {D k : ℕ}
    (h : AxNum P n src D k) (hD : dP6 P n = D + 1) :
    SideCostAt P ap n src D k (Kc P ap * cU6 P n 2 / (D + 1))

Context. Side costs 5: sideCost6_all

Against cU6 P n 2 = T p · Ncost6 · (log lg n + 1)^2 with Ncost6 ≥ d + ℓ, d = D + 1: every side cost is ≤ Kc · cU6 P n 2 / (D + 1) with Kc = 10^12 (W+15)^3 (W+x+14)^2 (Cfm+1), W = 3 wA + wC.

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_Defs68

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.Side
open IntMul NLogN IntMul.Kappa.R6 J6Seg

Formal statement

theorem J6Seg.Side.side_at (P : Prm) (ap : AxPlan6) {n : ℕ} {src : ℕ → ℕ} {D k : ℕ}
    (h : AxNum P n src D k) (hD : dP6 P n = D + 1) :
    SideCostAt P ap n src D k (Kc P ap * cU6 P n 2 / (D + 1)) := 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 `J6Seg.Side.side_at`, Solutions/J6SegSideCost5.lean (lines 128–227) of the round-six development.

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