  Pitts and Stark's $\nu$-calculus is a paradigmatic total language for
studying the problem of contextual equivalence in higher-order languages with
name generation. Models for the $\nu$-calculus that validate basic equivalences
concerning names may be constructed using functor categories or nominal sets,
with a dynamic allocation monad used to model computations that may allocate
fresh names. If recursion is added to the language and one attempts to adapt
the models from (nominal) sets to (nominal) domains, however, the direct-style
construction of the allocation monad no longer works. This issue has previously
been addressed by using a monad that combines dynamic allocation with
continuations, at some cost to abstraction.
  This paper presents a direct-style model of a $\nu$-calculus-like language
with recursion using the novel framework of proof-relevant logical relations,
in which logical relations also contain objects (or proofs) demonstrating the
equivalence of (the semantic counterparts of) programs. Apart from providing a
fresh solution to an old problem, this work provides an accessible setting in
which to introduce the use of proof-relevant logical relations, free of the
additional complexities associated with their use for more sophisticated
languages.
