HvdH machine lemma FW.prcOK_fwP (Fermat-ring convolution machine, part 10: the program `fwP` computing )

Proved

NLogN.FW.prcOK_fwP

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.HFW10):

theorem NLogN.FW.prcOK_fwP (hD : iprog (pc 1) = (difBody (pc 1)).ren ssH id)
    (hT : iprog (pc 2) = (ditBody (pc 2)).ren ssH id) : RM.PrcOK I iprog pc fwP

Context. The lemma belongs to the part Fermat-ring convolution machine, part 10: the program fwP computing Ref.fermatW.

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.FW
open IntMul IntMul.HvdH NLogN
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P} (pc : ℕ → Fin P)

Formal statement

theorem NLogN.FW.prcOK_fwP (hD : iprog (pc 1) = (difBody (pc 1)).ren ssH id)
    (hT : iprog (pc 2) = (ditBody (pc 2)).ren ssH id) : RM.PrcOK I iprog pc fwP := 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.FW.prcOK_fwP` of the Lean development NLogN/HFW10.lean (lines 438–444), written for the prove2.me integer-multiplication campaign.

View graph