Nominal Lambda Calculus: An Internal Language for FM-Cartesian Closed Categories

Reasoning about atoms (names) is difficult. The last decade has seen the development of numerous novel techniques. For equational reasoning, Clouston and Pitts introduced Nominal Equational Logic (NEL), which provides judgements of equality and freshness of atoms. Just as Equational Logic (EL) can be enriched with function types to yield the lambda-calculus (LC), we introduce NLC by enriching NEL with (atom-dependent) function types and abstraction types. We establish meta-theoretic properties of NLC; define NFM-cartesian closed categories, hence a categorical semantics for NLC; and prove soundness & completeness by way of NLC-classifying categories. A corollary of these results is that NLC is an internal language for NFM-cccs. A key feature of NLC is that it provides a novel way of encoding freshness via dependent types, and a new vehicle for studying the interaction of freshness and higher order types.