We present a higher-order calculus ECC which can be seen as an extension of the calculus of constructions [CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics may be adequately formalized. It is shown that ECC is strongly normalizing and has other nice proof-theoretic properties. An !\GammaSet (realizability) model is described to show how the essential properties of the calculus can be captured set-theoretically.

The aim of this paper is to present the s-calculus which is a very simple -calculus with explicit substitutions and to prove its confluence on closed terms and the preservation of strong normalisation of -terms. We shall prove strong normalisation of the corresponding calculus of substitution by translating it into the oe-calculus [ACCL91], and therefore the relation between both calculi will be made explicit. The confluence of the s-calculus is obtained by the "interpretation method" ([Har89], [CHL92]). The proof of the preservation of normalisation follows the lines of an analogous result for the AE-calculus (cf. [BBLRD95]). The relation between s and AE is also studied.

We present a new system of intersection types for a composition-free calculus of explicit substitutions with a rule for garbage collection, and show that it characterizes those terms which are strongly normalizing. This system extends previous work on the natural generalization of the classical intersection types system, which characterized head normalization and weak normalization, but was not complete for strong normalization. An important role is played by the notion of available variable in a term, which is a generalization of the classical notion of free variable.

Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding ff-conversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renewed interest in systems of explicit substitutions using variable names. Formal systems of explicit substitutions using variable names is a new area however and we believe, it should not develop without being well-tied to existing work on explicit substitutions. The aim of this paper is to establish a bridge between explicit substitutions using de Bruijn indices and using variable names. In our aim to do so, we provide the t-calculus: a -calculus `a la de Bruijn which can be translated into a -calculus with explicit substitutions written with variables names. We present explicitly this translation and use it to obtain preservation of strong normalisation for t. Moreover, we show several prope...

Extending the λ-calculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the - calculus, because it allows postponement of work in two different but complementary ways. Moreover, gs (and also s) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that gs preserves strong normalization, is a conservative extension of g, and simulates fi-reduction of g and the classical -calculus. Furthermore, we study the simply typed versions of s and gs, and show that well-typed terms are strongly normalizing and that other properties,...

Abstract. We study the π-calculus, enriched with pairing and non-blocking input, and define a notion of type assignment that uses the type constructor →. We encode the circuits of the calculus X into this variant of π, and show that all reduction (cut-elimination) and assignable types are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen’s calculus LK, this implies that all proofs in LK have a representation in π.

The last fifteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the oe-calculus. In [KR95a], we extended the -calculus with explicit substitutions by turning de Bruijn's meta-operators into object-operators offering a style of explicit substitution that differs from that of oe. The resulting calculus, s, remains as close as possible to the -calculus from an intuitive point of view and, while preserving strong normalisation ([KR95a]), is extended in this paper to a confluent calculus on open terms: the s e -caculus. Since the establishment of the results of this paper 1 , another calculus, i, came into being in [MH95] which preserves strong normalisation and is itself confluent on open terms. However, we believe that s e still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical fi-reduction, whereas i is not. To prove confluence we introduce a ge...

A unication method based on the se -style of explicit substitution is proposed. This method together with appropriate translations, provide a Higher Order Unication (HOU) procedure for the pure -calculus. Our method is inuenced by the treatment introduced by Dowek, Hardin and Kirchner using the -style of explicit substitution. Correctness and completeness properties of the proposed se-unication method are shown and its advantages, inherited from the qualities of the se -calculus, are pointed out. Our method needs only one sort of objects: terms. And in contrast to the HOU approach based on the -calculus, it avoids the use of substitution objects. This makes our method closer to the syntax of the -calculus. Furthermore, detection of redices depends on the search for solutions of simple arithmetic constraints which makes our method more operational than the one based on the -style of explicit substitution. Keywords: Higher order unication, explicit substitution, lambda-calculi. 1

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...

Explicit substitutions were proposed by Abadi, Cardelli, Curien, Hardin and Lévy to internalise substitutions into λ-calculus and to propose a mechanism for computing on substitutions. λν is another view of the same concept which aims to explain the process of substitution and to decompose it in small steps. λν is simple and preserves strong normalisation. Apparently that important property cannot stay with another important one, namely confluence on open terms. The spirit of λν is closely related to another calculus of explicit substitutions proposed by de Bruijn and called Cλξφ. In this paper, we introduce λν, we present Cλξφ in the same framework as λν and we compare both calculi. Moreover, we prove properties of λν; namely λν correctly implements β reduction, λν is confluent on closed terms, i.e., on terms of classical λ-calculus and on all terms that are derived from those terms, and finally λν preserves strong normalization of β-reduction.