Nominal Reasoning Techniques in Coq

We explore an axiomatized nominal approach to variable binding in Coq, using an untyped lambda-calculus as our test case. In our nominal approach, alpha-equality of lambda terms coincides with Coq’s builtin equality. Our axiomatization includes a nominal induction principle and functions for calculating free variables and substitution. These axioms are collected in a module signature and proved sound using locally nameless terms as the underlying representation. Our experience so far suggests that it is feasible to work from such axiomatized theories in Coq and that the nominal style of variable binding corresponds closely with paper proofs. We are currently working on proving the soundness of a primitive recursion combinator and developing a method of generating these axioms and their proof of soundness from a grammar describing the syntax of terms and binding.