 
Summary: SUBSEXPL: A tool for Simulating and Comparing Explicit
Substitutions Calculi
A Tutorial
Fl’avio L. C. de Moura # + Mauricio AyalaRinc’on + Fairouz Kamareddine #
April 4, 2006
Abstract
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 #calculus and of several styles of explicit substitutions calculi.
Applications of this framework include: the visualisation of the contractions of the #calculus, and of
guided onestep 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 #calculus. The current implementation of SUBSEXPL includes
treatment of three di#erent calculi of explicit substitutions: the ##, the #se , the suspension calculus and
a refinement of the suspension calculus called combining suspension calculus which allows for combination
of steps of #contraction; other explicit substitutions calculi can be easily incorporated into the system.
An implementation of the #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
