Certified algorithms for numerical semigroups in Rocq

TL;DR

This paper introduces a formalized framework for numerical semigroups using Rocq, providing certified algorithms for key invariants, marking the first such formalization in a proof assistant.

Contribution

It presents the first formalization of numerical semigroups in a proof assistant and offers certified algorithms for computing fundamental invariants.

Findings

Certified algorithms for key invariants of numerical semigroups.

First formalization of numerical semigroups in a proof assistant.

Provides proofs of correctness for invariants computation.

Abstract

A numerical semigroup is a co-finite submonoid of the monoid of non-negative integers under addition. Many properties of numerical semigroups rely on some fundamental invariants, such as, among others, the set of gaps (and its cardinality), the Ap\'ery set or the Frobenius number. Algorithms for calculating invariants are currently based on computational tools, such as GAP, which lack proofs (either formal or informal) of their correctness. In this paper we introduce a Rocq formalization of numerical semigroups. Given the semigroup generators, we provide certified algorithms for computing some of the fundamental invariants: the set of gaps, of small elements, the Ap\'ery set, the multiplicity, the conductor and the Frobenius number. To the best of our knowledge this is the first formalization of numerical semigroups in any proof assistant.