Checking Zenon Modulo Proofs in Dedukti

Dedukti has been proposed as a universal proof checker. It is a logical framework based on the lambda Pi calculus modulo that is used as a backend to verify proofs coming from theorem provers, especially those implementing some form of rewriting. We present a shallow embedding into Dedukti of proofs produced by Zenon Modulo, an extension of the tableau-based first-order theorem prover Zenon to deduction modulo and typing. Zenon Modulo is applied to the verification of programs in both academic and industrial projects. The purpose of our embedding is to increase the confidence in automatically generated proofs by separating untrusted proof search from trusted proof verification.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

Soundly Proving B Method Formulae Using Typed Sequent Calculus

International audienceThe B Method is a formal method mainly used in the railway industry to specify and develop safety-critical software. To guarantee the consistency of a B project, one decisive challenge is to show correct a large amount of proof obligations, which are mathematical formulae expressed in a classical set theory extended with a specific type system. To improve automated theorem proving in the B Method, we propose to use a first-order sequent calculus extended with a polymorphic type system, which is in particular the output proof-format of the tableau-based automated theorem prover Zenon. After stating some modifications of the B syntax and defining a sound elimination of comprehension sets, we propose a translation of B formulae into a polymorphic first-order logic format. Then, we introduce the typed sequent calculus used by Zenon, and show that Zenon proofs can be translated to proofs of the initial B formulae in the B proof system

Automated Deduction in the B Set Theory using Typed Proof Search and Deduction Modulo

International audienceWe introduce an encoding of the set theory of the B method using polymorphic types and deduction modulo, which is used for the automated verication of proof obligations in the framework of theBWare project. Deduction modulo is an extension of predicate calculus with rewriting both on terms and propositions. It is well suited for proof search in theories because it turns many axioms into rewrite rules. We also present the associated automated theorem prover Zenon Modulo, an extension of Zenon to polymorphic types and deduction modulo, along with its backend to the Dedukti universal proof checker, which also relies on types and deduction modulo, and which allows us to verify the proofs produced by Zenon Modulo. Finally, we assess our approach over the proof obligation benchmark of BWare

Proof Certification in Zenon Modulo: When Achilles Uses Deduction Modulo to Outrun the Tortoise with Shorter Steps

International audienceWe present the certifying part of the Zenon Modulo automated theorem prover, which is an extension of the Zenon tableau-based first order automated theorem prover to deduction modulo. The theory of deduction modulo is an extension of predicate calculus, which allows us to rewrite terms as well as propositions, and which is well suited for proof search in axiomatic theories, as it turns axioms into rewrite rules. In addition, deduction modulo allows Zenon Modulo to compress proofs by making computations implicit in proofs. To certify these proofs, we use Dedukti, an external proof checker for the λΠ-calculus modulo, which can deal natively with proofs in deduction modulo. To assess our approach, we rely on some experimental results obtained on the benchmarks provided by the TPTP library

Dedukti: a Logical Framework based on the λ\lambdaΠ\Pi-Calculus Modulo Theory

Dedukti is a Logical Framework based on the $\lambda$$\Pi$-Calculus Modulo Theory. We show that many theories can be expressed in Dedukti: constructive and classical predicate logic, Simple type theory, programming languages, Pure type systems, the Calculus of inductive constructions with universes, etc. and that permits to used it to check large libraries of proofs developed in other proof systems: Zenon, iProver, FoCaLiZe, HOL Light, and Matita

Using Deduction Modulo in Set Theory

International audienceWe present some improvements of Zenon Modulo and the application of this tool to sets of problems coming from set theory. Zenon Modulo is an extension of the tableau-based first order automated theorem prover Zenon to deduction modulo. Deduction modulo is an ex-tension of predicate calculus, which allows us to rewrite terms as well as propositions, and which is well-suited for proof-search in axiomatic theories, as it turns axioms into rewrite rules. The improvements dis-cussed here consist in a better heuristic to automatically build rewrite systems given a set of axioms, and some optimizations in the rewrit-ing process used during the proof search. We also present some updated results obtained on benchmarks provided by the TPTP library for set theory categories. Finally, we discuss some recent work about the appli-cation of our tool to the B method set theory, in particular the way we treat equality and the comprehension scheme

Déduction automatique et certification de preuve pour la méthode B

The B Method is a formal method heavily used in the railway industry to specify and develop safety-critical software. It allows the development of correct-by-construction programs, thanks to a refinement process from an abstract specification to a deterministic implementation of the program. The soundness of the refinement steps depends on the validity of logical formulas called proof obligations, expressed in a specific typed set theory. Typical industrial projects using the B Method generate thousands of proof obligations, thereby relying on automated tools to discharge as many as possible proof obligations. A specific tool, called Atelier B, designed to implement the B Method and provided with a theorem prover, helps users verify the validity of proof obligations, automatically or interactively. Improving the automated verification of proof obligations is a crucial task for the speed and ease of development. The solution developed in our work is to use Zenon, a first-orderlogic automated theorem prover based on the tableaux method. The particular feature of Zenon is to generate proof certificates, i.e. proof objects that can be verified by external tools. The B Method is based on first-order logic and a specific typed set theory. To improve automated theorem proving in this theory, we extend the proof-search algorithm of Zenon to polymorphism and deduction modulo theory, leading to a new tool called Zenon Modulo which is the main contribution of our work. The extension to polymorphism allows us to deal with problems combining several sorts, like booleans and integers, and generic axioms, like B set theory axioms, without relying on encodings. Deduction modulo theory is an extension of first-order logic with rewriting both on terms and propositions. It is well suited for proof search in axiomatic theories, as it turns axioms into rewrite rules. This way, we turn proof search among axioms into computations, avoiding unnecessary combinatorial explosion, and reducing the size of proofs by recording only their meaningful steps. To certify Zenon Modulo proofs, we choose to rely on Dedukti, a proof-checker used as a universal backend to verify proofs coming from different theorem provers,and based on deduction modulo theory. This work is part of a larger project called BWare, which gathers academic entities and industrial companies around automated theorem proving for the B Method. These industrial partners provide to BWare a large benchmark of proof obligations coming from real industrial projects using the B Method and allowing us to test our tool Zenon Modulo. The experimental results obtained on this benchmark are particularly conclusive since Zenon Modulo proves more proof obligations than state-of-the-art first-order provers. In addition, all the proof certificates produced by Zenon Modulo on this benchmark are well checked by Dedukti, increasing our confidence in the soundness of our work.La Méthode B est une méthode formelle de spécification et de développement de logiciels critiques largement utilisée dans l'industrie ferroviaire. Elle permet le développement de programmes dit corrects par construction, grâce à une procédure de raffinements successifs d'une spécification abstraite jusqu'à une implantation déterministe du programme. La correction des étapes de raffinement est garantie par la vérification de la correction de formules mathématiques appelées obligations de preuve et exprimées dans la théorie des ensembles de la Méthode B. Les projets industriels utilisant la Méthode B génèrent généralement des milliers d'obligation de preuve. La faisabilité et la rapidité du développement dépendent donc fortement d'outils automatiques pour prouver ces formules mathématiques. Un outil logiciel, appelé Atelier B, spécialement développé pour aider au développement de projet avec la Méthode B, aide les utilisateurs a se décharger des obligations de preuve, automatiquement ou interactivement. Améliorer la vérification automatique des obligations de preuve est donc une tache importante. La solution que nous proposons est d'utiliser Zenon, un outils de déduction automatique pour la logique du premier ordre et qui implémente la méthode des tableaux. La particularité de Zenon est de générer des certificats de preuve, des preuves écrites dans un certain format et qui permettent leur vérification automatique par un outil tiers. La théorie des ensembles de la Méthode B est une théorie des ensembles en logique du premier ordre qui fait appel à des schémas d'axiomes polymorphes. Pour améliorer la preuve automatique avec celle-ci, nous avons étendu l'algorithme de recherche de preuve de Zenon au polymorphisme et à la déduction modulo théorie. Ce nouvel outil, qui constitue le cœur de notre contribution, est appelé Zenon Modulo. L'extension de Zenon au polymorphisme nous a permis de traiter, efficacement et sans encodage, les problèmes utilisant en même temps plusieurs types, par exemple les booléens et les entiers, et des axiomes génériques, tels ceux de la théorie des ensembles de B. La déduction modulo théorie est une extension de la logique du premier ordre à la réécriture des termes et des propositions. Cette méthode est parfaitement adaptée à la recherche de preuve dans les théories axiomatiques puisqu'elle permet de transformer des axiomes en règles de réécriture. Par ce moyen, nous passons d'une recherche de preuve dans des axiomes à du calcul, réduisant ainsi l'explosion combinatoire de la recherche de preuve en présence d'axiomes et compressant la taille des preuves en ne gardant que les étapes intéressantes. La certification des preuves de Zenon Modulo, une autre originalité de nos travaux, est faite à l'aide de Dedukti, un vérificateur universel de preuve qui permet de certifier les preuves provenant de nombreux outils différents, et basé sur la déduction modulo théorie. Ce travail fait parti d'un projet plus large appelé BWare, qui réunit des organismes de recherche académique et des industriels autour de la démonstration automatique d'obligations de preuve dans l'Atelier B. Les partenaires industriels ont fournit à BWare un ensemble d'obligation de preuve venant de vrais projets industriels utilisant la Méthode B, nous permettant ainsi de tester notre outil Zenon Modulo.Les résultats expérimentaux obtenus sur cet ensemble de référence sont particulièrement convaincant puisque Zenon Modulo prouve plus d'obligation de preuve que les outils de déduction automatique de référence au premier ordre. De plus, tous les certificats de preuve produits par Zenon Modulo ont été validés par Dedukti, nous permettant ainsi d'être très confiant dans la correction de notre travail

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Implementing Polymorphism in Zenon

International audienceExtending first-order logic with ML-style polymorphism allows to define generic axioms dealing with several sorts. Until recently, most automated theorem provers relied on preprocess encodings into mono/many-sorted logic to reason within such theories. In this paper, we discuss the implementation of polymorphism into the first-order tableau-based automated theorem prover Zenon. This implementation led us to modify some basic parts of the code, from the representation of expressions to the proof-search algorithm

Proof Certification in Zenon Modulo: When Achilles Uses Deduction Modulo to Outrun the Tortoise with Shorter Steps

International audienceWe present the certifying part of the Zenon Modulo automated theorem prover, which is an extension of the Zenon tableau-based first order automated theorem prover to deduction modulo. The theory of deduction modulo is an extension of predicate calculus, which allows us to rewrite terms as well as propositions, and which is well suited for proof search in axiomatic theories, as it turns axioms into rewrite rules. In addition, deduction modulo allows Zenon Modulo to compress proofs by making computations implicit in proofs. To certify these proofs, we use Dedukti, an external proof checker for the λΠ-calculus modulo, which can deal natively with proofs in deduction modulo. To assess our approach, we rely on some experimental results obtained on the benchmarks provided by the TPTP library