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. It has also been shown that eta reduction is useful for adapting substitution calculi for practical problems like higher order uni cation. This paper concentrates on rewrite rules for eta reduction in three dierent styles of explicit substitution calculi: , se and the suspension calculus. Both and se when extended with eta reduction, have proved useful for solving higher order uni cation. We enlarge the suspension calculus with an adequate eta-reduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the eta-reductions of the other two calculi. We prove that and se as well as and the suspension calculus are non comparable while se is more adequate than the suspension calculus in simulating one step of beta-contraction.

We present a new term indexing approach which shall support efficient automated theorem proving in classical higher order logic. Key features of our indexing method are a shared representation of terms, the use of partial syntax trees to speedup logical computations and indexing of subterm occurrences. For the implementation of explicit substitutions, additional support is offered by indexing of bound variable occurrences. A preliminary evaluation of our approach shows some encouraging first results. 1

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