Round-six lemma `atanhInv_bound`

Proved

IntMul.HvdH.Num.atanhInv_bound

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.KFMBasePi):

theorem IntMul.HvdH.Num.atanhInv_bound (x N K : ℕ) (hx : 2 ≤ x) :
    |(atanhInv x N K : ℝ) - 2 ^ N * ((Real.log (1 + 1 / x) - Real.log (1 - 1 / x)) / 2)| ≤ K + 2 * 2 ^ N / x ^ (2 * K + 1)

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 Definitions.Def_IntMul_HvdH_Numeric

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.HvdH
open Finset
open IntMul.HvdH

Formal statement

theorem IntMul.HvdH.Num.atanhInv_bound (x N K : ℕ) (hx : 2 ≤ x) :
    |(atanhInv x N K : ℝ) - 2 ^ N * ((Real.log (1 + 1 / x) - Real.log (1 - 1 / x)) / 2)| ≤ K + 2 * 2 ^ N / x ^ (2 * K + 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 `IntMul.HvdH.Num.atanhInv_bound`, Solutions/KFMBasePi.lean (lines 107–157) of the round-six development.

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