Gentle Formalisation of Stoughton’s Lambda Calculus Substitution

In [5], Allen Stoughton proposed a notion of substitution for the Lambda calculus formulated in its original syntax with only one sort of symbols (names) for variables –without identifying α-convertible terms. According to such formulation, the action of substitution on terms is defined by simple structural recursion and an interesting theory arises concerning e.g. α conversion. In this paper we present a formalisation of Stoughton’s development in Constructive Type Theory using the language Agda, up to the Substitution Lemma for α conversion. The code obtained is quite concise and we are able to formulate some improvements over the original presentation. For instance, our definition of α conversion is just syntax directed but we are nevertheless able to prove in an easy way that it gives rise to an equivalence relation, whereas in [5] the latter was included as part of the definition. As a result of this work we are inclined to assert that Stoughton’s is the right way to formulate the Lambda calculus in its original, conventional syntax and that it is a formulation amenable to fully formal treatment.