SUBSEXPL: A tool for Simulating and Comparing Explicit Substitutions Calculi A Tutorial

In this tutorial we present the system SUBSEXPL that is used for simulating and comparing explicit substitutions calculi. This framework was developed in Ocaml, a language of the ML family, and it allows the manipulation of expressions of the λ-calculus and of several styles of explicit substitutions calculi. Applications of this framework include: the visualisation of the contractions of the λ-calculus, and of guided one-step reductions and normalisation via each of the associated substitution calculi. Many useful facilities are available: reductions can be easily recorded and stored into files, latex output and useful examples for dealing with, among other things, arithmetic operations and computational operators such as conditionals and repetitions in the λ-calculus. The current implementation of SUBSEXPL includes treatment of three different calculi of explicit substitutions: the λσ, the λse, the suspension calculus and a refinement of the suspension calculus called combining suspension calculus which allows for combination of steps of β-contraction; other explicit substitutions calculi can be easily incorporated into the system. An implementation of the η-reduction is provided for each of these explicit substitutions calculi. This system has been of great help for systematically comparing explicit substitutions calculi, as well as for understanding properties of explicit substitutions such as the Preservation of Strong Normalisation. In addition, it has been used for teaching basic properties of the λ-calculus such as: computational adequacy, the importance of de Bruijn’s notation and of making explicit substitutions in real implementations based on the λ-calculus.