Proof-relevant π-calculus Roly Perera James Cheney University of Glasgow University of Edinburgh Glasgow, UK Edinburgh, UK
[email protected] [email protected] Formalising the π-calculus is an illuminating test of the expressiveness of logical frameworks and mechanised metatheory systems, because of the presence of name binding, labelled transitions with name extrusion, bisimulation, and structural congruence. Formalisations have been undertaken in a variety of systems, primarily focusing on well-studied (and challenging) properties such as the theory of process bisimulation. We present a formalisation in Agda that instead explores the theory of concurrent transitions, residuation, and causal equivalence of traces, which has not previously been formalised for the π-calculus. Our formalisation employs de Bruijn indices and dependently- typed syntax, and aligns the “proved transitions” proposed by Boudol and Castellani in the context of CCS with the proof terms naturally present in Agda’s representation of the labelled transition relation. Our main contributions are proofs of the “diamond lemma” for residuation of concurrent transitions and a formal deVnition of equivalence of traces up to permutation of transitions. 1 Introduction The π-calculus [18, 19] is an expressive model of concurrent and mobile processes. It has been investigated extensively and many variations, extensions and reVnements have been proposed, including the asynchronous, polyadic, and applied π-calculus (among many others). The π-calculus has also attracted considerable attention from the logical frameworks and meta-languages community, and formalisations of its syntax and semantics have been performed using most of the extant mechanised metatheory techniques, including (among others) Coq [13, 12, 15], Nominal Isabelle [2], Abella [1] (building on Miller and Tiu [26]), CLF [6], and Agda [21].