HvdH machine lemma ST.stepOK (The recursive step, part 12: `StepOK (stepP K) n₀ (lin K) (calls K)`)

Proved

NLogN.ST.stepOK

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.HSt12):

theorem NLogN.ST.stepOK (K n₀ : ℕ) (hK : 20 ≤ K) (h0 : n0K K ≤ n₀) (hnum : ∀ N, n₀ ≤ N → NumOK K N) :
    Top.StepOK (stepP K) n₀ (lin K) (calls K)

Context. The lemma belongs to the part The recursive step, part 12: StepOK (stepP K) n₀ (lin K) (calls K).

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
set_option linter.unusedVariables false
open NLogN
open NLogN.ST
open IntMul NLogN IntMul.HvdH IntMul.HvdH.Ref

Formal statement

theorem NLogN.ST.stepOK (K n₀ : ℕ) (hK : 20 ≤ K) (h0 : n0K K ≤ n₀) (hnum : ∀ N, n₀ ≤ N → NumOK K N) :
    Top.StepOK (stepP K) n₀ (lin K) (calls K) := 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.ST.stepOK` of the Lean development NLogN/HSt12.lean (lines 68–93), written for the prove2.me integer-multiplication campaign.

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