Pitts and Starks $ u$-calculus is a paradigmatic total language for studying the problem of contextual equivalence in higher-order languages with name generation. Models for the $ u$-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 $ u$-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.
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