HvdH machine lemma AX.bapAxP_all (One resampling axis, part 8: the `B̃` axis program `bapAxP`)

Proved

NLogN.AX.bapAxP_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.HAx8):

theorem NLogN.AX.bapAxP_all (M : MulSpec) (p s t α m kb Bi Bo mm Φ Bacc Q Bq : ℕ) (rest : List ℕ)
    (hM : MulBnd M mm Φ) (H : BapHyp p s t α kb Bi Bo mm Bacc Q) (HT : TabHypB p s t α Bq)
    (hα : 1 ≤ α) (HQ : AxQ t s m kb Bi Bo Q) (hmr : m = rest.prod)
    (st : IState HS) (L : List (ℤ × ℤ)) (Rest : List Item) (hR : BapRegs p s t α m kb st)
    (hD : st.G gD = gints L ++ Rest) (hL : L.length = t * m) (hLB : GB Bi L) :
    bapAxP.run st = st.upd gD (gints (tensorGo [(s, bapL p s t α)] (t :: rest) L) ++ Rest) ∧
      bapAxP.ok M st ∧ bapAxP.cost M st ≤ bapAxCost p s t α m kb Bi mm Φ Bacc Q Bq ∧
      (tensorGo [(s, bapL p s t α)] (t :: rest) L).length = m * s ∧
      GB Bo (tensorGo [(s, bapL p s t α)] (t :: rest) L)

Context. The lemma belongs to the part One resampling axis, part 8: the B̃ axis program bapAxP: bapAxP on gD = gints L ++ R (with L of dims t :: rest, m = rest.prod) computes gints (tensorGo [(s, bapL p s t α)] (t :: rest) L) ++ R: it pushes the derived scalars e, s², 2α²-2, builds the seven tables of bapP (layout: HAx5), pops the derived scalars, runs GL.axisP with the slice program bapSl, and pops the seven tables again. Every stack except gD is restored.

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
set_option maxHeartbeats 16000000

Formal statement

theorem NLogN.AX.bapAxP_all (M : MulSpec) (p s t α m kb Bi Bo mm Φ Bacc Q Bq : ℕ) (rest : List ℕ)
    (hM : MulBnd M mm Φ) (H : BapHyp p s t α kb Bi Bo mm Bacc Q) (HT : TabHypB p s t α Bq)
    (hα : 1 ≤ α) (HQ : AxQ t s m kb Bi Bo Q) (hmr : m = rest.prod)
    (st : IState HS) (L : List (ℤ × ℤ)) (Rest : List Item) (hR : BapRegs p s t α m kb st)
    (hD : st.G gD = gints L ++ Rest) (hL : L.length = t * m) (hLB : GB Bi L) :
    bapAxP.run st = st.upd gD (gints (tensorGo [(s, bapL p s t α)] (t :: rest) L) ++ Rest) ∧
      bapAxP.ok M st ∧ bapAxP.cost M st ≤ bapAxCost p s t α m kb Bi mm Φ Bacc Q Bq ∧
      (tensorGo [(s, bapL p s t α)] (t :: rest) L).length = m * s ∧
      GB Bo (tensorGo [(s, bapL p s t α)] (t :: rest) L) := 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.bapAxP_all` of the Lean development NLogN/HAx8.lean (lines 115–418), written for the prove2.me integer-multiplication campaign.

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