The aim of this article is to compare two styles of Explicit Substitutions: the - and s-styles. We start by introducing a criterion of adequacy to simulate -reduction in calculi of explicit substitutions and we apply it to several calculi: , * , , s, t and u. The latter is presented here for the rst time and may be considered as an adequate variant of s. By doing so, we establish that calculi a la s are usually more adequate at simulating-reduction than calculi in the -style. In fact, we prove that t is more adequate than and that u is more adequate than , * and s. We also give counterexamples to show that all other comparisons are impossible according to our criterion. Our next step consists in presenting the ! and !e calculi, the two-sorted (term and substitution) versions of the s (cf. [KR95]) and se (cf. [KR97]) calculi, respectively. We establish an isomorphism between the s-calculus and the term restriction of the !-calculus, which extends to an isomorphism between se and the term restriction of !e. Since the ! and !e calculi are given in the style of the -calculus (cf. [ACCL91]) they are bridge calculi between s and and between se and and thus we are able to better understand one calculus in terms of the other. Finally, we present typed versions of all the calculi and check that the above mentioned isomorphism preserves types. As a consequence, the !-calculus is a calculus in the -style that has the following properties a..g: a) ! simulates one step -reduction, b) ! is con uent (on closed terms), c) ! preserves strong normalisation, d) !'s associated calculus of substitutions is SN, e) the simply typed ! calculus is SN, f) the !-calculus possesses an extension !e that is con uent on open terms (terms with eventual metavariables of sort term only), and g) the simply typed !e calculus is weakly normalising (on open term). As far as we know, the !-calculus is the rst calculus in the -style that has all those properties a..g. However, the open problem of the SN of the associated calculus of substitution of !e remains unsolved and like in the case of , and se, !e does not have PSN.

Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proof-search in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proof-search and strongly related to natural deduction. PTSC are equipped with a normalisation procedure, adapted from Herbelin’s and defined by local rewrite rules as in Cut-elimination, using explicit substitutions. It satisfies Subject Reduction and it is confluent. A PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising if and only if the latter is. We show how the conversion rules can be incorporated inside logical rules (as in syntax-directed rules for type checking), so that basic proofsearch tactics in type theory are merely the root-first application of our inference rules.

Abstract Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proof-search in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proof-search and strongly related to natural deduction. PTSC are equipped with a normalisation procedure, adapted from Herbelin’s and defined by local rewrite rules as in Cut-elimination, using explicit substitutions. It satisfies Subject Reduction and it is confluent. A PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising if and only if the latter is. We show how the conversion rules can be incorporated inside logical rules (as in syntax-directed rules for type checking), so that basic proofsearch tactics in type theory are merely the root-first application of our inference rules.

Abstract. The intuitionistic fragment of the call-by-name version of Curien and Herbelin’s λµ˜µ-calculus is isolated and proved strongly normalising by means of an embedding into the simply-typed λ-calculus. Our embedding is a continuation-and-garbage-passing style translation, the inspiring idea coming from Ikeda and Nakazawa’s translation of Parigot’s λµ-calculus. The embedding simulates reductions while usual continuation-passing-style transformations erase permutative reduction steps. For our intuitionistic sequent calculus, we even only need “units of garbage ” to be passed. We apply the same method to other calculi, namely successive extensions of the simply-typed λ-calculus leading to our intuitionistic system, and already for the simplest extension we consider (λ-calculus with generalised application), this yields the first proof of strong normalisation through a reduction-preserving embedding. 1