We present typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen sequent calculus for minimal logic. Our calculus is inspired by the Curry-Howard Isomorphism, in the sense that types, both for patterns and terms, correspond to propositions, terms correspond to proofs, and term reduction corresponds to sequent proof normalization performed by cut elimination. The calculus enjoys subject reduction, confluence, preservation of strong normalization w.r.t a system with meta-level substitutions, and strong normalization for well-typed terms, and, as a consequence, can be seen as an implementation calculus for functional formalisms using meta-level operations for pattern matching and substitutions.

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 show how higher-order rewriting may be encoded into

In this paper we consider -calculi of explicit substitutions that admit open expressions, i.e. expressions with meta-variables. In particular, we propose a variant of the oe-calculus that we call L . For this calculus and its simply-typed version, we study its meta-theoretical properties. The L-calculus enjoys the same general characteristics as oe, i.e. a simple and finitary first-order presentation, confluent on expressions with meta-variables of terms and weakly normalizing on typed expressions. Moreover, L does not have the non-left-linear surjective pairing rule of oe which raises technical problems in some frameworks.