Abstract not found

: In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the Curry-Howard isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambda-terms. In order to see the process of proof construction as an incremental process of term construction, it is necessary to extend the lambda-calculus with new operators. First, we consider typed meta-variables to represent the parts of a proof that are under construction, and second, we make explicit the substitution mechanism in order to deal with capture of variables that are bound in terms containing meta-variables. Unfortunately, the theory of explicit substitution calculi with typed meta-variables is more complex than that of lambda-calculus. And worse, in general they do not share the same properties, notably with respect to confluence and strong normalization. A contribution of this thesis is to show that the pr...

Explicit substitutions calculi provide a method to compute over -expressions by using first-order term rewriting systems. Many calculi have been proposed that differ by their signature, their number of rules and their theoretical properties. Our aim is to compare these systems from an operational point of view. For a selected class of such calculi, we have developed prototypes written in ELAN. In this paper, we briefly present our first experimental results. 1

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.