HvdH machine lemma AX.tabB_facts (One resampling axis, part 7: derived scalars and table removal for the)

Proved

NLogN.AX.tabB_facts

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.HAx7):

theorem NLogN.AX.tabB_facts (p s t α Bq : ℕ) (H : TabHypB p s t α Bq) (hs : 1 ≤ s) (hst : s ≤ t) :
    4 * (α ^ 2 + s + t + 2) ^ 6 < 2 ^ Bq ∧ 4 * (s + t + 2) ^ 2 < 2 ^ Bq ∧
    (∀ u < t, 4 * (u + t + 2) ^ 2 < 2 ^ Bq) ∧
    2 * HvdH.eWidth p α + 2 < 2 ^ Bq ∧ s ^ 2 < 2 ^ Bq ∧ 2 * α ^ 2 + 2 < 2 ^ Bq ∧
    s * (2 * HvdH.eWidth p α + 1) < 2 ^ Bq ∧ 2 * t * s + s < 2 ^ Bq ∧ t < 2 ^ Bq

Context. The lemma belongs to the part One resampling axis, part 7: derived scalars and table removal for the B̃ axis.

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.tabB_facts (p s t α Bq : ℕ) (H : TabHypB p s t α Bq) (hs : 1 ≤ s) (hst : s ≤ t) :
    4 * (α ^ 2 + s + t + 2) ^ 6 < 2 ^ Bq ∧ 4 * (s + t + 2) ^ 2 < 2 ^ Bq ∧
    (∀ u < t, 4 * (u + t + 2) ^ 2 < 2 ^ Bq) ∧
    2 * HvdH.eWidth p α + 2 < 2 ^ Bq ∧ s ^ 2 < 2 ^ Bq ∧ 2 * α ^ 2 + 2 < 2 ^ Bq ∧
    s * (2 * HvdH.eWidth p α + 1) < 2 ^ Bq ∧ 2 * t * s + s < 2 ^ Bq ∧ t < 2 ^ 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.tabB_facts` of the Lean development NLogN/HAx7.lean (lines 165–211), written for the prove2.me integer-multiplication campaign.

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