HvdH machine lemma CR.stB_cost (Parameter computation, part 5: costs of stages B–D and of `parP` (name)

Proved

NLogN.CR.stB_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.HCrPar5):

theorem NLogN.CR.stB_cost (K b E e : ℕ) (t : IState HS) (LB LE LR : List Item)
    (hB : t.G NLogN.CR.rB = .int b :: LB) (hE : t.G NLogN.CR.rE = .int E :: LE) (hR : t.G hRr = .int ((2 ^ e : ℕ) : ℤ) :: LR)
    (hK : 1 ≤ K) (B : ℕ) (hV : 2 ^ (b + 1) + 4 * 2 ^ e * PV K b E + b * b + 2 * b + 1 < 2 ^ B) :
    (NLogN.CR.stB K).cost t ≤ (3 * b + 40) * 10 ^ 9 * (B + 2) ^ 2

Context. The lemma belongs to the part Parameter computation, part 5: costs of stages B–D and of parP (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
set_option maxHeartbeats 4000000

Formal statement

theorem NLogN.CR.stB_cost (K b E e : ℕ) (t : IState HS) (LB LE LR : List Item)
    (hB : t.G NLogN.CR.rB = .int b :: LB) (hE : t.G NLogN.CR.rE = .int E :: LE) (hR : t.G hRr = .int ((2 ^ e : ℕ) : ℤ) :: LR)
    (hK : 1 ≤ K) (B : ℕ) (hV : 2 ^ (b + 1) + 4 * 2 ^ e * PV K b E + b * b + 2 * b + 1 < 2 ^ B) :
    (NLogN.CR.stB K).cost t ≤ (3 * b + 40) * 10 ^ 9 * (B + 2) ^ 2 := 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.stB_cost` of the Lean development NLogN/HCrPar5.lean (lines 15–170), written for the prove2.me integer-multiplication campaign.

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