HvdH machine lemma AX.sTabR_all (One resampling axis, part 3: tables of the `S̃` axis, and the axis pro)

Proved

NLogN.AX.sTabR_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.HAx3):

theorem NLogN.AX.sTabR_all (p s t α Bq : ℕ) (st : IState HS) (ht : 0 < t) (hR : STabRegs p s t α st)
    (H : TabHyp p s t α Bq) :
    sTabR.ok st ∧ sTabR.run st = st.upd aW (ints (sTab p s t α).flatten ++ st.G aW) ∧
      sTabR.cost st ≤ tabCost (2 * HvdH.sWidth p α + 1) t (sWCost p Bq)

Context. The lemma belongs to the part One resampling axis, part 3: tables of the S̃ axis, and the axis program sapAxP: * sTabR = sTabP.ren σB, dTabR = dTabP.ren σB push the S weight table on aW = 39 and the step table on aD = 40 (registers: see HAx2).

popK removes them again.

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.AX
open IntMul NLogN IntMul.HvdH.Ref NLogN.GL NLogN.Rsp

Formal statement

theorem NLogN.AX.sTabR_all (p s t α Bq : ℕ) (st : IState HS) (ht : 0 < t) (hR : STabRegs p s t α st)
    (H : TabHyp p s t α Bq) :
    sTabR.ok st ∧ sTabR.run st = st.upd aW (ints (sTab p s t α).flatten ++ st.G aW) ∧
      sTabR.cost st ≤ tabCost (2 * HvdH.sWidth p α + 1) t (sWCost p Bq) := 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.AX.sTabR_all` of the Lean development NLogN/HAx3.lean (lines 90–103), written for the prove2.me integer-multiplication campaign.

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