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 λσ and the λse which have proved useful for solving practical problems like higher order unification, and the suspension calculus related to the implementation of the language λProlog. 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 λσ and λse as well as λσ and the suspension calculus are non comparable while λse is more adequate than the suspension calculus.