Round-six lemma `atan_body_cost`

Proved

NLogN.atan_body_cost

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 NLogN.HNumAtan):

theorem NLogN.atan_body_cost (x N K : ℕ) (σ : ℤ) (hσ : |σ| = 1) (s0 t : IState HS) (k : ℕ)
    (h : atanInv' x N K σ s0 k t) (hk : k < K) (hx : 0 < x) :
    (atanBody x σ).cost t + asgCost 45 (decE 45) ((atanBody x σ).run t) ≤
      30 * 10 ^ 7 * (atanB x N K + 2) ^ 2

Context. atanInv / atanhInv as a register program

Inputs N on register 0, K on register 1; the result is pushed on register 2. Scratch registers 40 … 45.

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_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel
import Definitions.Def_IntMul_HvdH_Numeric

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN
open IntMul NLogN.RS

Formal statement

theorem NLogN.atan_body_cost (x N K : ℕ) (σ : ℤ) (hσ : |σ| = 1) (s0 t : IState HS) (k : ℕ)
    (h : atanInv' x N K σ s0 k t) (hk : k < K) (hx : 0 < x) :
    (atanBody x σ).cost t + asgCost 45 (decE 45) ((atanBody x σ).run t) ≤
      30 * 10 ^ 7 * (atanB x N K + 2) ^ 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 `NLogN.atan_body_cost`, Solutions/HNumAtan.lean (lines 194–285) of the round-six development.

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