A Formal Proof of the Strong Normalization Theorem for System T in Agda

Abstract

We present a framework for the formal meta-theory of lambda calculi in first-order syntax, with two sorts of names, one to represent both free and bound variables, and the other for constants, and by using Stoughton's multiple substitutions. On top of the framework we formalize Girard's proof of the Strong Normalization Theorem for both the simply-typed lambda calculus and System T. As to the latter, we also present a simplification of the original proof. The whole development has been machine-checked using the Agda system. * This work is partially supported by Agencia Nacional de Investigación e Innovación (ANII), Uruguay. 1 Both the previous framework and the one presented here use named variables, it bears repeating. In a contrary sense, there are nameless approaches, e.g., or locally nameless syntax [6], which use numbers to identify the variables.2 Or without using Barendregt's variable convention.