HvdH machine lemma expNegFix_bnds (`expNegFix`, `expNeg`, `expPos` as register programs)

Proved

NLogN.expNegFix_bnds

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.HNumExp2):

theorem NLogN.expNegFix_bnds (n M : ℕ) (X : ℤ) :
    zsz (M : ℤ) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (n : ℤ) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz ((n : ℤ) + 1) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz ((M : ℤ) + -((n : ℤ) + 1)) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz X ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (log2Bits M) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (-X / log2Bits M) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (-X / log2Bits M * log2Bits M) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (X + -X / log2Bits M * log2Bits M) ≤ 3 * M + 14 ∧
    zsz ((((M : ℤ) + -((n : ℤ) + 1)).toNat : ℤ) + ((-X / log2Bits M).toNat : ℤ)) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (((((M : ℤ) + -((n : ℤ) + 1)).toNat : ℤ) + ((-X / log2Bits M).toNat : ℤ)) + 1) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    ((((M : ℤ) + -((n : ℤ) + 1)).toNat : ℤ) + ((-X / log2Bits M).toNat : ℤ)).toNat + 3 ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (expBits M (X + -X / log2Bits M * log2Bits M) 0).1 ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (expBits M (X + -X / log2Bits M * log2Bits M) 0).2 ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz ((expBits M (X + -X / log2Bits M * log2Bits M) 0).1 + 2 ^ ((((M : ℤ) + -((n : ℤ) + 1)).toNat : ℤ) + ((-X / log2Bits M).toNat : ℤ)).toNat) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat + 1

Context. The lemma belongs to the part expNegFix, expNeg, expPos as register programs: expNegFixP: inputs n on 10, M on 11, X on 12; result pushed on 13. Uses (with pushes popped) registers 3 … 9 and the scratch registers of log2P (0, 1, 2, 40 … 47) and expP (30 … 39).

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 IntMul RS IntMul.HvdH
set_option maxHeartbeats 1000000

Formal statement

theorem NLogN.expNegFix_bnds (n M : ℕ) (X : ℤ) :
    zsz (M : ℤ) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (n : ℤ) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz ((n : ℤ) + 1) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz ((M : ℤ) + -((n : ℤ) + 1)) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz X ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (log2Bits M) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (-X / log2Bits M) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (-X / log2Bits M * log2Bits M) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (X + -X / log2Bits M * log2Bits M) ≤ 3 * M + 14 ∧
    zsz ((((M : ℤ) + -((n : ℤ) + 1)).toNat : ℤ) + ((-X / log2Bits M).toNat : ℤ)) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (((((M : ℤ) + -((n : ℤ) + 1)).toNat : ℤ) + ((-X / log2Bits M).toNat : ℤ)) + 1) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    ((((M : ℤ) + -((n : ℤ) + 1)).toNat : ℤ) + ((-X / log2Bits M).toNat : ℤ)).toNat + 3 ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (expBits M (X + -X / log2Bits M * log2Bits M) 0).1 ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz (expBits M (X + -X / log2Bits M * log2Bits M) 0).2 ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat ∧
    zsz ((expBits M (X + -X / log2Bits M * log2Bits M) 0).1 + 2 ^ ((((M : ℤ) + -((n : ℤ) + 1)).toNat : ℤ) + ((-X / log2Bits M).toNat : ℤ)).toNat) ≤ expNegB n M (zsz X) (-X / log2Bits M).toNat + 1 := 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.expNegFix_bnds` of the Lean development NLogN/HNumExp2.lean (lines 107–187), written for the prove2.me integer-multiplication campaign.

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