HvdH machine lemma atanP_cost (`atanInv` / `atanhInv` as a register program)

Proved

NLogN.atanP_cost

by xuanji · Oct 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

harvey-van-der-hoeven-tminteger-multiplicationturing-machines

The following holds (Lean statement, in the scope of module NLogN.HNumAtan):

theorem NLogN.atanP_cost (x N K : ℕ) (σ : ℤ) (hx : 0 < x) (hσ : |σ| = 1) (s : IState HS)
    (L0 L1 : List Item) (h0 : s.G 0 = .int N :: L0) (h1 : s.G 1 = .int K :: L1) :
    (atanP x σ).cost s ≤ (K + 2) * 10 ^ 9 * (atanB x N K + 2) ^ 2

Context. The lemma belongs to the part 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. This is a lemma of the Lean development NLogN, a multitape-Turing-machine implementation of the Harvey–van der Hoeven $O(n\log n)$ integer multiplication algorithm, used in the proof of NLogN.FIN.theorem_1_1_nlogn : IntMul.MulTimeBound fun n => n * Real.log n. All names refer to the definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8 (namespace NLogN), built on the platform model Def_IntMul_MultitapeModel and the published reference Def_IntMul_HvdH_MachineRef. The machine layer: IState HS is a state of 64 item stacks; RM programs have a semantics p.run s, a definedness predicate p.ok M s and a cost p.cost M s relative to a multiplication spec M : MulSpec; zsz z is the bit length of $|z|$ .

Preamble

-- Harvey–van der Hoeven multitape-TM development (NLogN), generated by skeleton subtraction.

import Definitions.Def_IntMul_HvdH_M_Defs8

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

set_option linter.unusedSimpArgs false
open NLogN
open IntMul RS
set_option maxHeartbeats 1000000

Formal statement

theorem NLogN.atanP_cost (x N K : ℕ) (σ : ℤ) (hx : 0 < x) (hσ : |σ| = 1) (s : IState HS)
    (L0 L1 : List Item) (h0 : s.G 0 = .int N :: L0) (h1 : s.G 1 = .int K :: L1) :
    (atanP x σ).cost s ≤ (K + 2) * 10 ^ 9 * (atanB x N K + 2) ^ 2 := by sorry

Source

D. Harvey, J. van der Hoeven, Integer multiplication in time O(n log n), Ann. of Math. 193 (2021), 563–617 (Theorem 1.1: multitape Turing machine multiplication in O(n log n)). Lemma `NLogN.atanP_cost` of the Lean development NLogN/HNumAtan.lean (lines 288–391), written for the prove2.me integer-multiplication campaign.

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