A Formalization of the Simply Typed Lambda Calculus in Coq

In this paper we present a formalization of the simply typed lambda calculus with constants and with typing à la Church. It has been accomplished using the theorem prover Coq. The formalization includes, among other results, definitions of typed terms over arbitrary many-sorted signature, a substitution operating on typing judgements, an equivalence relation generalizing the concept of α-convertibility to free variables and a proof of strong normalization of β-reduction. The formalization has been used in a bigger project of certification of the higher-order recursive path ordering where well-foundedness of the union of the higher-order recursive path ordering and β-reduction was the main result.