HvdH machine lemma expPosP_spec (`expPos` as a register program)

Proved

NLogN.expPosP_spec

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.HNumExpPos):

theorem NLogN.expPosP_spec (n k j σ : ℕ) (s : IState HS) (L16 L17 L18 L27 : List Item)
    (h16 : s.G 16 = .int n :: L16) (h17 : s.G 17 = .int k :: L17)
    (h18 : s.G 18 = .int j :: L18) (h27 : s.G 27 = .int σ :: L27) :
    expPosP.ok s ∧ expPosP.run s = s.upd 19 (.int (expPos n k j σ) :: s.G 19)

Context. The lemma belongs to the part expPos as a register program: expPosP: inputs n on 16, k on 17, j on 18, σ on 27; expPos n k j σ pushed on 19. It uses (with pushes popped) registers 3 … 15 and the scratch registers of piP/log2P (0, 1, 2, 40 … 47) and expP (30 … 39).

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

Formal statement

theorem NLogN.expPosP_spec (n k j σ : ℕ) (s : IState HS) (L16 L17 L18 L27 : List Item)
    (h16 : s.G 16 = .int n :: L16) (h17 : s.G 17 = .int k :: L17)
    (h18 : s.G 18 = .int j :: L18) (h27 : s.G 27 = .int σ :: L27) :
    expPosP.ok s ∧ expPosP.run s = s.upd 19 (.int (expPos n k j σ) :: s.G 19) := 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.expPosP_spec` of the Lean development NLogN/HNumExpPos.lean (lines 86–91), written for the prove2.me integer-multiplication campaign.

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