  Formalising the pi-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
pi-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 definition of equivalence of traces up to permutation of transitions.
