Comparing calculi of explicit substitutions with eta-reduction

The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. Three styles of explicit substitutions are treated in this paper: the ?s and the ?s_e which have proved useful for solving practical problems like higher order unification, and the suspension calculus related to the implementation of the language ?-Prolog. We enlarge the suspension calculus with an adequate eta-reduction which we show to preserve termination and confluence of the associated substitution calculus and to correspond to the eta-reductions of the other two calculi. Additionally, we prove that ?s and ?s_e as well as ?s and the suspension calculus are non comparable while ?s_e is more adequate than the suspension calculus. ©2002 Published by Elsevier Science B.V.