Rewriting Modulo \beta in the \lambda\Pi-Calculus Modulo

The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with dependent types where beta-conversion is extended with user-defined rewrite rules. It is an expressive logical framework and has been used to encode logics and type systems in a shallow way. Basic properties such as subject reduction or uniqueness of types do not hold in general in the lambda-Pi-calculus Modulo. However, they hold if the rewrite system generated by the rewrite rules together with beta-reduction is confluent. But this is too restrictive. To handle the case where non confluence comes from the interference between the beta-reduction and rewrite rules with lambda-abstraction on their left-hand side, we introduce a notion of rewriting modulo beta for the lambda-Pi-calculus Modulo. We prove that confluence of rewriting modulo beta is enough to ensure subject reduction and uniqueness of types. We achieve our goal by encoding the lambda-Pi-calculus Modulo into Higher-Order Rewrite System (HRS). As a consequence, we also make the confluence results for HRSs available for the lambda-Pi-calculus Modulo.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

A First-Order Representation of Pure Type Systems Using Superdeduction

International audienceSuperdeduction is a formalism closely related to deduction modulo which permits to enrich a deduction system (especially a first-order one such as natural deduction or sequent calculus) with new inference rules automatically computed from the presentation of a theory. We give a natural encoding from every functional Pure Type System (PTS) into superdeduction by defining an appropriate first-order theory. We prove that this translation is correct and conservative, showing a correspondence between valid typing judgments in the PTS and provable sequents in the corresponding superdeductive system. As a byproduct, we also introduce the superdeductive sequent calculus for intuitionistic logic, which was until now only defined for classical logic. We show its equivalence with the superdeductive natural deduction. This implies that superdeduction can be easily used as a logical framework. These results lead to a better understanding of the implementation and the automation of proof search for PTS, as well as to more cooperation between proof assistants