Abstract. We characterise the strongly normalising terms of a composition-free calculus of explicit substitutions (with or without garbage collection) by means of an intersection type assignment system. The main novelty is a cut-rule which allows to forget the context of the minor premise when the context of the main premise does not have an assumption for the cut variable.

this paper we carry out a semantical investigation of perpetual strategies in

We show how to characterise compositionally a number of evaluation properties of λ-terms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalisation, normalisation, head normalisation, and weak head normalisation. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterisation results are new, to our knowledge, or else they streamline, strengthen, or generalise earlier results in the literature. The completeness parts of the characterisations are proved uniformly for all the properties, using a set-theoretical semantics of intersection types over suitable kinds of stable sets. This technique generalises Krivine's and Mitchell's methods for strong normalisation to other evaluation properties.

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.