. This paper proposes a notion of reduction for the proof nets of Linear Logic modulo an equivalence relation on the contraction links, that essentially amounts to consider the contraction as an associative commutative binary operator that can float freely in and out of proof net boxes. The need for such a system comes, on one side, from the desire to make proof nets an even more parallel syntax for Linear Logic, and on the other side from the application of proof nets to l-calculus with or without explicit substitutions, which needs a notion of reduction more flexible than those present in the literature. The main result of the paper is that this relaxed notion of rewriting is still strongly normalizing. Keywords: Proof Nets. Linear Logic. Strong Normalization. 1 Introduction In his seminal paper [6], Girard proposed proof nets as a parallel syntax for Linear Logic, where uninteresting permutations in the order of application of logical rules are de-sequentialised and collapsed. Neve...

We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proof-nets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simply-typed terms, step by step simulation of β-reduction and full composition.

. We show that reducing any simply-typed oe-term (resp. oe*) by applying the rules in oe (resp. oe *) eagerly always terminates, by a translation to the simply-typed -calculus. This holds even with term and substitution meta-variables. In fact, every reduction terminates provided that (fi)-redexes are only contracted under so-called safe contexts; and in oe, resp. oe *-normal forms, all contexts around terms of sort T are safe. The result is then extended to second-order type systems. 1 Introduction The simply-typed oe-calculus does not terminate strongly [8, 9], but it terminates in the weak sense: every typed term has a normal form. We present a proof of this; in fact, we prove the widely believed claim that every reduction where oe steps are applied eagerly is finite, as a consequence of a more general theorem stating that every reduction where (fi)-contraction only occurs under so-called safe contexts terminates. For the sake of generality, we prove this result even in the pres...

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.