HvdH machine lemma GL.seq_all (`fwdL` and `exactConvL` programs)

Proved

NLogN.GL.seq_all

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.HGl8):

theorem NLogN.GL.seq_all {M : MulSpec} (a b : RM) (s t u : IState HS) (Ca Cb : ℕ)
    (ha : a.run s = t ∧ a.ok M s ∧ a.cost M s ≤ Ca)
    (hb : b.run t = u ∧ b.ok M t ∧ b.cost M t ≤ Cb) :
    (RM.seq a b).run s = u ∧ (RM.seq a b).ok M s ∧ (RM.seq a b).cost M s ≤ Ca + Cb

Context. The lemma belongs to the part fwdL and exactConvL programs: * fwdP axsS axsB fw: gD : gints U ++ R ↦ gD : gints (fwdL q U) ++ R (S̃ along all axes via tgP axsS, FtL via ftlP fw, B̃ along all axes via tgP axsB).

exactP axsS axsB fw: gD : ints u ++ R, gV : ints v ++ RV ↦ gD : ints (exactConvL q u v) ++ R, gV : RV.

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 NLogN.GL
open IntMul NLogN IntMul.HvdH IntMul.HvdH.Ref

Formal statement

theorem NLogN.GL.seq_all {M : MulSpec} (a b : RM) (s t u : IState HS) (Ca Cb : ℕ)
    (ha : a.run s = t ∧ a.ok M s ∧ a.cost M s ≤ Ca)
    (hb : b.run t = u ∧ b.ok M t ∧ b.cost M t ≤ Cb) :
    (RM.seq a b).run s = u ∧ (RM.seq a b).ok M s ∧ (RM.seq a b).cost M s ≤ Ca + Cb := 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.GL.seq_all` of the Lean development NLogN/HGl8.lean (lines 18–27), written for the prove2.me integer-multiplication campaign.

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