Terminal semantics for codata types in intensional Martin-L\"of type theory

In this work, we study the notions of relative comonad and comodule over a relative comonad, and use these notions to give a terminal coalgebra semantics for the coinductive type families of streams and of infinite triangular matrices, respectively, in intensional Martin-L\"of type theory. Our results are mechanized in the proof assistant Coq.Comment: 14 pages, ancillary files contain formalized proof in the proof assistant Coq; v2: 20 pages, title and abstract changed, give a terminal semantics for streams as well as for matrices, Coq proof files updated accordingl

Non-wellfounded trees in Homotopy Type Theory

We prove a conjecture about the constructibility of coinductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable. Indeed, in this work, we construct coinductive types in a subsystem of Homotopy Type Theory; this subsystem is given by Intensional Martin-L\"of type theory with natural numbers and Voevodsky's Univalence Axiom. Our results are mechanized in the computer proof assistant Agda.Comment: 14 pages, to be published in proceedings of TLCA 2015; ancillary files contain Agda files with formalized proof

Towards a verified transformation from AADL to the formal component-based language FIACRE

International audienceDuring the last decade, aadl  is an emerging architecture description languages addressing the modeling of embedded systems. Several research projects have shown that aadl  concepts are well suited to the design of embedded systems. Moreover, aadl  has a precise execution model which has proved to be one key feature for effective early analysis. In this paper, we are concerned with the foundational aspects of the verification support for aadl. More precisely, we propose a verification toolchain for aadl  models through its transformation to the Fiacre language which is the pivot verification language of the TOPCASED project: high level models can be transformed to Fiacre  models and then model-checked. Then, we investigate how to prove the correctness of the transformation from AADL into Fiacre and present related elementary ingredients: the semantics of aadl  and Fiacre  subsets expressed in a common framework, namely timed transition systems. We also briefly discuss experimental validation of the work

Une théorie mécanisée des arbres réguliers en théorie des types dépendants

We propose two characterizations of regular trees. The first one is semantic and is based on coinductive types. The second one is syntactic and represents regular trees by means of cyclic terms. We prove that both of these characterizations are isomorphic. Then, we study the problem of defining tree morphisms preserving the regularity property. We show, by using the formalism of tree transducers, the existence of syntactic criterion ensuring that this property is preserved. Finally, we consider applications of the theory of regular trees such as the definition of the parallel composition operator of a process algebra or, the decidability problems on regular trees through a mechanization of a model-checker for a coalgebraic mu-calculus. All the results were mechanized and proved correct in the Coq proof assistant.Nous proposons deux caractérisations des arbres réguliers. La première est sémantique et s'appuie sur les types co-inductifs. La seconde est syntaxique et repose sur une représentation des arbres réguliers par des termes cycliques. Nous prouvons que ces deux caractérisations sont isomorphes.Ensuite, nous étudions le problème de la définition de morphisme d'arbres préservant la propriété de régularité. Nous montrons en utilisant le formalisme des transducteurs d'arbres, l'existence d'un critère syntaxique garantissant la préservation de cette propriété. Enfin, nous considérons des applications de la théorie des arbres réguliers comme la définition de l'opérateur de composition parallèle d'une algèbre de processus ou encore, les problèmes de décidabilité sur les arbres réguliers via une mécanisation d'un vérificateur de modèles pour un mu-calcul coalgébrique. Tous les résultats ont été mécanisés et prouvés corrects dans l'assistant de preuve Coq