Formalization of metatheory of the Lambda Calculus in constructive type theory using the Barendregt variable convention

Abstract

Abstarct We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.

More Church-Rosser Proofs in BELUGA

“…In the following years, the Church-Rosser theorem became a benchmark to showcase how to mechanize the variable binding problem. Some partial data-points, with the understanding that these are not exhaustive not disjoint: if interested in mirroring the informal practice of working mathematicians, see the paper by Vestergaard and Brotherston [27] and the recent work by Copello et al [6]. If you want to reason about α-conversion explicitly via quotients, see Ford and Mason's [9].…”