Packing a digit list into one integer: result

Proved

NLogN.packOne_run

by xuanji · Oct 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

harvey-van-der-hoeven-tminteger-multiplicationturing-machines

Let stack $E_i$ hold the reversed list $L$ of nonnegative integers, register CR hold $2^b$ and CM hold $|L|$ , with $E_i$ and $X$ distinct from the scratch registers CA, CY, CR, CM, EC (and from each other). Then the program packOne E_i X consumes $L$ and pushes on $X$ the integer kpack (2^b) L $= \sum_k L_k 2^{bk}$ .

Formalization Note. kpack B L is the base- $B$ Horner evaluation of the digit list $L$ (least significant first). The machine layer: IState HS is a state with $64$ stacks of items (Item = integer or mark), an input position and an output; RM programs (straight-line register code RS, loops rep, conditionals, calls of the multiplication procedure through a MulSpec $M$ giving its domain and cost) have a functional semantics p.run s, a definedness predicate p.ok M s and a cost p.cost M s; Spec M p s s' c abbreviates p.ok M s ∧ p.run s = s' ∧ p.cost M s ≤ c. ints L / gints U push an integer list / a list of pairs as items (head = top); zsz z is the bit length of $|z|$ . MulBnd M m Φ says the multiplication procedure accepts all pairs of integers of bit length $\le m$ at cost $\le Φ$ . Register names (GU, CR, KB, XN, …) are fixed stack indices.

Preamble

-- Harvey–van der Hoeven multitape-TM development (NLogN), generated by skeleton subtraction.

import Definitions.Def_IntMul_HvdH_M_Defs4

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

set_option linter.unusedSimpArgs false
open NLogN
open IntMul IntMul.HvdH IntMul.HvdH.Ref

Formal statement

theorem NLogN.packOne_run (Ei X : Fin HS) (s : IState HS) (b : ℕ) (L : List ℤ) (Ri LR LM : List Item)
    (hEi : s.G Ei = ints L.reverse ++ Ri) (hR : s.G CR = .int (2 ^ b) :: LR)
    (hM : s.G CM = .int L.length :: LM) (hL : ∀ x ∈ L, 0 ≤ x)
    (h1 : Ei ≠ CA) (h2 : Ei ≠ CY) (h3 : Ei ≠ CR) (h4 : Ei ≠ CM) (h5 : Ei ≠ EC) (h6 : Ei ≠ X)
    (h7 : X ≠ CA) (h8 : X ≠ CY) (h9 : X ≠ CR) (h10 : X ≠ CM) (h11 : X ≠ EC) :
    (packOne Ei X).run s = (s.upd Ei Ri).upd X (.int (kpack (2 ^ b) L) :: s.G X) := 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.packOne_run` of the Lean development NLogN/HEnc.lean (lines 100–128), written for the prove2.me integer-multiplication campaign.

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