Categorical combinators [12, 21, 43] and more recently oe-calculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the -calculus. We reintroduce here the ingredients of these calculi in a self-contained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper w.r.t. [12, 21, 1, 23] are the following: 1. We present a confluent weak calculus of substitutions, where no variable clashes can be feared. 2. We solve a conjecture raised in [1]: oe-calculus is not confluent (it is confluent on ground terms only). This unfortunate result is "repaired" by presenting a confluent version of oe-calculus, named the Env-calculus in [23], called here the confluent oe-calculus.

Explicit substitutions calculi are formal systems that implement fi-reduction by means of an internal substitution operator. In that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. The oe -calculus of explicit substitutions, proposed by Abadi, Cardelli, Curien andL evy, is a first-order rewriting system that implements substitution and renaming mechanism of -calculus. However, oe does not preserve strong normalisation of -calculus and it is not a confluent system. Typed variants of oe without composition are strongly normalising but not confluent, while variants with composition are confluent but do not preserve strong normalisation. Neither of them enjoys both properties. In this paper we propose the i -calculus. This is, as far as we know, the first confluent calculus of explicit substitutions that preserves strong normalisation. 1. Explicit substitutions The -calculus is a higher-order theor...

: 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 are formal systems that implement fi-reduction by means of an internal substitution operator. Thus, in that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. This feature is useful, for instance, to represent incomplete proofs in type based proof systems. The oe -calculus of explicit substitutions proposed by Abadi, Cardelli, Curien and L'evy gives an elegant way to deal with management of variable names and substitutions of -calculus. However, oe does not preserve strong normalisation of -calculus and it is not a confluent system. Typed variants of oe without composition are strongly normalising but not confluent, while variants with composition are confluent but do not preserve strong normalisation. Neither of them enjoys both properties. In this paper we propose the i - calculus an we present the full proofs of its main properties. This is, as far as we know, the...

We present a dependent-type system for a #-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.

: In this paper we consider explicit substitutions calculi that allow open terms. In particular, we propose a variant of the oe -calculus, that we call OE . For this calculus and its simply-typed version, we study its meta-theoretical properties. The OE -calculus enjoys the same general characteristics as oe , i.e. a simple and finitary first-order presentation, confluent on terms with meta-variables, with a composition operator and with simultaneous substitutions. However, OE does not have the non-left-linear surjective pairing rule of oe which raises technical problems in some frameworks. (R'esum'e : tsvp) Cesar.Munoz@inria.fr Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33 1) 39 63 55 11 -- T'el'ecopie : (33 1) 39 63 53 30 Propri'et'es m'eta-th'eoriques de OE : Une variante lin'eaire `a gauche de oe R'esum'e : Dans cet article, on s'int'eresse aux calculs avec substitutions explicites q...

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.

We present a theory of dependent types with explicit substitutions. We follow a meta-theoretical approach where open expressions ---expressions with meta-variables--- are first-class objects. The system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.

CÉSAR MUÑOZ∗ Abstract. We present a dependent-type system for a λ-calculus with explicit substitutions. In this system, meta-variables, as well as substitutions, are first-class objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.