HvdH machine lemma Rsp.psKeys_length (List form of `bapL`: `ptL`, `cselL`, `japL`, `dapL`, `psL`)

Proved

NLogN.Rsp.psKeys_length

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.HRspBM):

theorem NLogN.Rsp.psKeys_length (s t : ℕ) : (psKeys s t).length = s

Context. The lemma belongs to the part List form of bapL: ptL, cselL, japL, dapL, psL.

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.Rsp
open IntMul NLogN

Formal statement

theorem NLogN.Rsp.psKeys_length (s t : ℕ) : (psKeys s t).length = s := 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.Rsp.psKeys_length` of the Lean development NLogN/HRspBM.lean (lines 220–220), written for the prove2.me integer-multiplication campaign.

View graph