HvdH machine lemma CR.evalP_cost (Chunking, `crtIn` and `evalDigits` as register programs (namespace `NL)

Proved

NLogN.CR.evalP_cost

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.HCrIn):

theorem NLogN.CR.evalP_cost (M : MulSpec) (s : IState HS) (b B : ℕ) (c : List ℤ) (RA LR LM : List Item)
    (hA : s.G CA = ints c ++ RA) (hR : s.G CR = .int (2 ^ b) :: LR)
    (hM : s.G CM = .int c.length :: LM) (hc : ∀ x ∈ c, 0 ≤ x ∧ x < 2 ^ B) (hb1 : 1 ≤ b)
    (hbB : b ≤ B) (hMB : zsz (c.length : ℤ) ≤ B) :
    evalP.cost M s ≤ 40000 * (c.length + 1) * (B + 3) + 20000 * (b * c.length + B + 4) + 103

Context. The lemma belongs to the part Chunking, crtIn and evalDigits as register programs (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 IntMul.HvdH IntMul.HvdH.Ref RS

Formal statement

theorem NLogN.CR.evalP_cost (M : MulSpec) (s : IState HS) (b B : ℕ) (c : List ℤ) (RA LR LM : List Item)
    (hA : s.G CA = ints c ++ RA) (hR : s.G CR = .int (2 ^ b) :: LR)
    (hM : s.G CM = .int c.length :: LM) (hc : ∀ x ∈ c, 0 ≤ x ∧ x < 2 ^ B) (hb1 : 1 ≤ b)
    (hbB : b ≤ B) (hMB : zsz (c.length : ℤ) ≤ B) :
    evalP.cost M s ≤ 40000 * (c.length + 1) * (B + 3) + 20000 * (b * c.length + B + 4) + 103 := 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.evalP_cost` of the Lean development NLogN/HCrIn.lean (lines 140–155), written for the prove2.me integer-multiplication campaign.

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