A Beluga Formalization of the Harmony Lemma in the $\pi$-Calculus

TL;DR

This paper provides a formal proof of the Harmony Lemma in a fragment of the $$-calculus using Beluga, addressing a key challenge in mechanizing concurrent system semantics.

Contribution

It offers the first formal, mechanized proof of the Harmony Lemma for a fragment of the $$-calculus in Beluga, with new encoding techniques for telescopes and induction.

Findings

Formalization in Beluga successfully proves the Harmony Lemma.

Introduces encoding techniques for telescopes and lexicographic induction.

Addresses a key challenge in mechanizing concurrent semantics.

Abstract

The "Harmony Lemma", as formulated by Sangiorgi & Walker, establishes the equivalence between the labelled transition semantics and the reduction semantics in the $\pi$-calculus. Despite being a widely known and accepted result for the standard $\pi$-calculus, this assertion has never been rigorously proven, formally or informally. Hence, its validity may not be immediately apparent when considering extensions of the $\pi$-calculus. Contributing to the second challenge of the Concurrent Calculi Formalization Benchmark -- a set of challenges tackling the main issues related to the mechanization of concurrent systems -- we present a formalization of this result for the fragment of the $\pi$-calculus examined in the Benchmark. Our formalization is implemented in Beluga and draws inspiration from the HOAS formalization of the LTS semantics popularized by Honsell et al. In passing, we introduce a couple of useful encoding techniques for handling telescopes and lexicographic induction.