Rensets and renaming-based recursion for syntax with bindings

I introduce renaming-enriched sets (rensets for short), which are algebraic structures axiomatizing fundamental properties of renaming (also known as variable-for-variable substitution) on syntax with bindings. Rensets compare favorably in some respects with the well-known foundation based on nominal sets. In particular, renaming is a more fundamental operator than the nominal swapping operator and enjoys a simpler, equationally expressed relationship with the variable-freshness predicate. Together with some natural axioms matching properties of the syntactic constructors, rensets yield a truly minimalistic characterization of λ-calculus terms as an abstract datatype – one involving an infinite set of unconditional equations, referring only to the most fundamental term operators: the constructors and renaming. This characterization yields a recursion principle, which (similarly to the case of nominal sets) can be improved by incorporating Barendregt’s variable convention. When interpreting syntax in semantic domains, my renaming-based recursor is easier to deploy than the nominal recursor. My results have been validated with the proof assistant Isabelle/HOL.