Reviewing the Classical and the de Bruijn Notation for [lambda]-calculus and Pure Type Systems

This article is a brief review of the type-free -calculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type-free -calculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the -calculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn’s notation for the -calculus is introduced and some of its advantages are outlined.