HvdH machine lemma CR.chirpP_spec (The chirp table program, part 4: sort, scan, gather (namespace `NLogN.)

Proved

NLogN.CR.chirpP_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.HCrChirp4):

theorem NLogN.CR.chirpP_spec (M : MulSpec) (q : Params) (he : 1 ≤ q.e) (hdh : q.dh < q.K) (tw : RM)
    (Ctw : ℕ) (htw : TwSpec M tw q.p (2 * q.r) Ctw) (s : IState HS) (hw : ChirpRegs q s)
    (LP : List Item) (hp : s.G hPp = .int q.p :: LP) (kb : ℕ) (L2 : List Item)
    (h2 : s.G 2 = .int kb :: L2) (hkb1 : 4 * q.r ≤ 2 ^ kb) (hkb2 : q.tdims.prod ≤ 2 ^ kb) :
    (chirpP tw).ok M s ∧ (chirpP tw).run s = s.upd 0 (gints (chirpL q) ++ s.G 0)

Context. The lemma belongs to the part The chirp table program, part 4: sort, scan, gather (namespace NLogN.CR).

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
set_option linter.unusedVariables false
open NLogN
open NLogN.CR
open IntMul RS IntMul.HvdH IntMul.HvdH.Ref

Formal statement

theorem NLogN.CR.chirpP_spec (M : MulSpec) (q : Params) (he : 1 ≤ q.e) (hdh : q.dh < q.K) (tw : RM)
    (Ctw : ℕ) (htw : TwSpec M tw q.p (2 * q.r) Ctw) (s : IState HS) (hw : ChirpRegs q s)
    (LP : List Item) (hp : s.G hPp = .int q.p :: LP) (kb : ℕ) (L2 : List Item)
    (h2 : s.G 2 = .int kb :: L2) (hkb1 : 4 * q.r ≤ 2 ^ kb) (hkb2 : q.tdims.prod ≤ 2 ^ kb) :
    (chirpP tw).ok M s ∧ (chirpP tw).run s = s.upd 0 (gints (chirpL q) ++ s.G 0) := 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.CR.chirpP_spec` of the Lean development NLogN/HCrChirp4.lean (lines 199–231), written for the prove2.me integer-multiplication campaign.

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