this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call N-systems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via left-introduction and right-introduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them L-systems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in L-systems. The advantage of N-systems is that they seem closer to actual reasoning, while L-systems on the other hand seem to have an easier proof theory. L-systems are often extended with a "cut" rule as part of showing that for a given L-system and N-system, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider

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.

Nowadays, type theory has many applications and is used in many different disciplines. Within computer science, logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. A general framework called Pure Type Systems (PTSs for short) has been introduced independently by Terlouw and Berardi in 1988 and 1989, in order to provide a unified formalism in which many type systems can be represented. In particular, PTSs allow the representation of the simple theory of types, the polymophic theory of types, the dependent theory of types and various other well-known type systems such as the Edinburgh Logical Frameworks LF and the Automath system. Pure Type Systems are usually presented using variable names. In this article, we present a formulation of PTSs with de Bruijn indices. De Bruijn indices [6] avoid the problems caused by variable names during the implementation of type systems. We show that PTSs with variable names and PTSs with de Bruijn indices are isomorphic. This isomorphism enables us to answer questions about PTSs with de Bruijn indices including confluence, termination (strong normalisation) and safety (subject reduction).

We introduce a criterion of efficiency to simulate fi-reduction in calculi of explicit substitutions and we apply it to several calculi: oe, oe * , AE, s, t and u. The latter is presented here for the first time and may be considered as an efficient variant of s. The results of this paper imply that calculi `a la s are usually more efficient at simulating fi-reduction than calculi in the oe-style. In fact, we prove that t is more efficient than AE and that u is more efficient than AE, oe * and s. We also give counterexamples to show that all other comparisons are impossible. 1 Introduction The classical -calculus (cf. [2]) deals with substitution in an implicit way. This means that the computations to perform substitution are usually described with operators which do not belong to the language of the -calculus. There has however been an interest in formalising substitution explicitly in order to provide a theoretical framework for the implementation of functional programming langua...

) Eduardo Bonelli 1;2 , Delia Kesner 2 , Alejandro R'ios 1 1 Departamento de Computaci'on - Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabell'on I, Ciudad Universitaria (1428), Buenos Aires, Argentina. febonelli,riosg@dc.uba.ar 2 LRI (UMR 8623) - Bat 490, Universit'e de Paris-Sud, 91405 Orsay Cedex, France. kesner@lri.fr Abstract. We propose a formalism for higher-order rewriting in de Bruijn notation. This notation not only is used for terms (as usually done in the literature) but also for metaterms, which are the syntactical objects used to express general higher-order rewrite systems. We give formal translations from higher-order rewriting with names to higher-order rewriting with de Bruijn indices, and vice-versa. These translations can be viewed as an interface in programming languages based on higher-order rewrite systems, and they are also used to show some properties, namely, that both formalisms are operationally equivalent, and th...

A higher order unification (HOU) method based on the ...-style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...-style of explicit substitution. Correctness and completeness properties of the proposed approach are shown and advantages of the method, inherited from the qualities of the ... calculus, are pointed out.

In recent years a large number of `explicit substitution calculi' have been proposed with various combinations of properties. One property that has attracted special attention is `PSN:' whether the set of fi-strongly normalising terms is still strongly normalising with explicit substitution. Several calculi with this property have been found: we discuss AE, Ø, s, t, and x; in this note we add two new variants: AE1 and AE0. We show that these calculi all have essentially the same reductions, or put differently: the renaming overhead is negligible with respect to normalisation. Furthermore x -- the only one of the lot with implicit binding using usual -calculus variables -- is a least common denominator in the sense that all the others are (strict) conservative extensions of it. A consequence of this is that all the PSN results proven for these calculi are equivalent (and follow from PSN for x).

Kamareddine and Nederpelt [9], resp. Kamareddine and Ríos [11] gave two calculi of explicit of substitutions highly inuenced by de Bruijn's notation of the -calculus. These calculi added to the explosive pool of work on explicit substitution in the past 15 years. As far as we know, calculi of explicit substitutions: a) are unable to handle local substitutions, and b) have answered (positively or negatively) the question of the termination of the underlying calculus of substitutions. The exception to a) is the calculus of [9] where substitution is handled both locally and globally. However, the calculus of [9] does not satisfy properties like conuence and termination. The exception to b) is the s e -calculus [11] for which termination of the s e -calculus, the underlying calculus of substitutions, remains unsolved. This paper has two aims: (i) To provide a calculus a la de Bruijn which deals with local substitution and whose underlying calculus of substitutions is terminating and conuent.