Developing (Meta)Theory of λ-calculus in the Theory of Contexts

We present a case study on the formal development of a non trivial (meta)theory in the Theory of Contexts using the Coq proof assistant. The methodology underlying the Theory of Contexts for reasoning on systems presented in HOAS is based on an axiomatic syntactic standpoint. We feel that one of the main advantages of this approach, is that it requires a very low logical overhead. The object logic we focus on is the lazy, call-by-name λ-calculus (λcbn), both untyped and simply typed. We will see that the formal, fully detailed development of the theory of λcbn in the Theory of Contexts introduces a small, sustainable overhead with respect to the proofs “on the paper”. Moreover, this will allow for comparison with similar case studies developed in other approaches to the metatheoretical reasoning in higher-order abstract syntax.