HvdH machine lemma octExpP_ok (`octExp` and `twiddle` as register programs)

Proved

NLogN.octExpP_ok

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.HNumTw):

theorem NLogN.octExpP_ok (n k a : ℕ) (s : IState HS) (L48 L49 L50 : List Item)
    (h48 : s.G 48 = .int n :: L48) (h49 : s.G 49 = .int k :: L49)
    (h50 : s.G 50 = .int a :: L50) : octExpP.ok s

Context. The lemma belongs to the part octExp and twiddle as register programs: * octExpP: inputs n on 48, k on 49, a on 50; pushes (octExp n k a).1 on 51 and (octExp n k a).2 on 52. Uses (pushed and popped) registers 3 … 9 and the scratch registers of piP (0, 1, 2, 40 … 47) and expP (30 … 39).

twiddleP: inputs n on 28, k on 29, j on 53; pushes (twiddle n k j).1 on 51 and (twiddle n k j).2 on 52. Uses in addition registers 10 … 14 and 48 … 50.

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 IntMul.HvdH
set_option maxHeartbeats 2000000

Formal statement

theorem NLogN.octExpP_ok (n k a : ℕ) (s : IState HS) (L48 L49 L50 : List Item)
    (h48 : s.G 48 = .int n :: L48) (h49 : s.G 49 = .int k :: L49)
    (h50 : s.G 50 = .int a :: L50) : octExpP.ok s := 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.octExpP_ok` of the Lean development NLogN/HNumTw.lean (lines 69–90), written for the prove2.me integer-multiplication campaign.

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