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

Towards correct-by-construction product variants of a software product line: GFML, a formal language for feature modules

Software Product Line Engineering (SPLE) is a software engineering paradigm that focuses on reuse and variability. Although feature-oriented programming (FOP) can implement software product line efficiently, we still need a method to generate and prove correctness of all product variants more efficiently and automatically. In this context, we propose to manipulate feature modules which contain three kinds of artifacts: specification, code and correctness proof. We depict a methodology and a platform that help the user to automatically produce correct-by-construction product variants from the related feature modules. As a first step of this project, we begin by proposing a language, GFML, allowing the developer to write such feature modules. This language is designed so that the artifacts can be easily reused and composed. GFML files contain the different artifacts mentioned above.The idea is to compile them into FoCaLiZe, a language for specification, implementation and formal proof with some object-oriented flavor. In this paper, we define and illustrate this language. We also introduce a way to compose the feature modules on some examples.Comment: In Proceedings FMSPLE 2015, arXiv:1504.0301

FoCaLiZe: Inside an F-IDE

For years, Integrated Development Environments have demonstrated their usefulness in order to ease the development of software. High-level security or safety systems require proofs of compliance to standards, based on analyses such as code review and, increasingly nowadays, formal proofs of conformance to specifications. This implies mixing computational and logical aspects all along the development, which naturally raises the need for a notion of Formal IDE. This paper examines the FoCaLiZe environment and explores the implementation issues raised by the decision to provide a single language to express specification properties, source code and machine-checked proofs while allowing incremental development and code reusability. Such features create strong dependencies between functions, properties and proofs, and impose an particular compilation scheme, which is described here. The compilation results are runnable OCaml code and a checkable Coq term. All these points are illustrated through a running example.Comment: In Proceedings F-IDE 2014, arXiv:1404.578

Mécanismes Orientés-Objets pour l'Interopérabilité entre Systèmes de Preuve

Dedukti is a Logical Framework resulting from the combination ofdependent typing and rewriting. It can be used to encode many logicalsystems using shallow embeddings preserving their notion of reduction.These translations of logical systems in a common format are anecessary first step for exchanging proofs between systems. Thisobjective of interoperability of proof systems is the main motivationof this thesis.To achieve it, we take inspiration from the world of programminglanguages and more specifically from object-oriented languages becausethey feature advanced mechanisms for encapsulation, modularity, anddefault definitions. For this reason we start by a shallowtranslation of an object calculus to Dedukti. The most interestingpoint in this translation is the treatment of subtyping.Unfortunately, it seems very hard to incorporate logic in this objectcalculus. To proceed, object-oriented mechanisms should be restrictedto static ones which seem enough for interoperability. Such acombination of static object-oriented mechanisms and logic is alreadypresent in the FoCaLiZe environment so we propose a shallow embeddingof FoCaLiZe in Dedukti. The main difficulties arise from theintegration of FoCaLiZe automatic theorem prover Zenon and from thetranslation of FoCaLiZe functional implementation language featuringtwo constructs which have no simple counterparts in Dedukti: localpattern matching and recursion.We then demonstrate how this embedding of FoCaLiZe to Dedukti can beused in practice for achieving interoperability of proof systemsthrough FoCaLiZe, Zenon, and Dedukti. In order to avoid strengtheningto much the theory in which the final proof is expressed, we useDedukti as a meta-language for eliminating unnecessary axioms.Dedukti est un cadre logique résultant de la combinaison du typagedépendant et de la réécriture. Il permet d'encoder de nombreuxsystèmes logiques au moyen de plongements superficiels qui préserventla notion de réduction.Ces traductions de systèmes logiques dans un format commun sont unepremière étape nécessaire à l'échange de preuves entre cessystèmes. Cet objectif d'interopérabilité des systèmes de preuve estla motivation principale de cette thèse.Pour y parvenir, nous nous inspirons du monde des langages deprogrammation et plus particulièrement des langages orientés-objetparce qu'ils mettent en œuvre des mécanismes avancés d'encapsulation,de modularité et de définitions par défaut. Pour cette raison, nouscommençons par une traduction superficielle d'un calcul orienté-objeten Dedukti. L'aspect le plus intéressant de cette traduction est letraitement du sous-typage.Malheureusement, ce calcul orienté-objet ne semble pas adapté àl'incorporation de traits logiques. Afin de continuer, nous devonsrestreindre les mécanismes orientés-objet à des mécanismes statiques,plus faciles à combiner avec la logique et apparemment suffisant pournotre objectif d'interopérabilité. Une telle combinaison de mécanismesorientés-objet et de logique est présente dans l'environnementFoCaLiZe donc nous proposons un encodage superficiel de FoCaLiZe dansDedukti. Les difficultés principales proviennent de l'intégration deZenon, le prouveur automatique de théorèmes sur lequel FoCaLiZerepose, et de la traduction du langage d'implantation fonctionnel deFoCaLiZe qui présente deux constructions qui n'ont pas decorrespondance simple en Dedukti : le filtrage de motif local et larécursivité.Nous démontrons finalement comment notre encodage de FoCaLiZe dansDedukti peut servir en pratique à l'interopérabilité entre dessystèmes de preuve à l'aide de FoCaLiZe, Zenon et Dedukti. Pour éviterde trop renforcer la théorie dans laquelle la preuve finale estobtenue, nous proposons d'utiliser Dedukti en tant que méta-langagepour éliminer des axiomes superflus

ML Pattern-Matching, Recursion, and Rewriting: From FoCaLiZe to Dedukti

International audienceThe programming environment FoCaLiZe allows the user to specify, implement, and prove programs with the help of the theorem prover Zenon. In the actual version, those proofs are verified by Coq. In this paper we propose to extend the FoCaLiZe compiler by a backend to the Dedukti language in order to benefit from Zenon Modulo, an extension of Zenon for Deduction modulo. By doing so, FoCaLiZe can benefit from a technique for finding and verifying proofs more quickly. The paper focuses mainly on the process that overcomes the lack of local pattern-matching and recursive definitions in Dedukti

Compiling Programs and Proofs: FoCaLiZe Internals

Designing a tool to ease the development of high-level security or safety systems must consider to facilitate not only design and coding but also formal demonstrations of correctness and compliance to standards. This entails some requirements on the tool as these demonstrations ask to link together computational and logical aspects of the development. These requirements are briefly considered and a solution is proposed: functions, statements and proofs are handled in a unique language, offering inheritance and parametrized modules. The FoCaLiZe environment implements this language, which remains simple enough to be used in a usual engineering process. The code generation produces an executable functional code (in OCaml) and a checkable term of a logical Type Theory (verified by Coq), close enough to truly ease traceability. It ensures that OCaml and Coq produced codes are error-free and provides compact generated code. The main contribution of this paper is a detailed presentation of the compilation scheme, which is supported by an original treatment of the dependencies induced by the combination of computational and logical constructs. As the whole source code is translated to a logical term verified by Coq, we get a strong assurance in the correctness of the generated code, hence avoiding the need to prove correctness of the compiler itself

Combining Type Checking and Set Constraint Solving to Improve Automated Software Verification

In this paper we show how prescritive type checking and constraint solving can be combined to increase automation during software verification. We do so by defining a type system and implementing a typechecker for {log} (read `setlog'), a Constraint Logic Programming (CLP) language and satisfiability solver based on set theory. Hence, we proceed as follows: a) a type system for {log} is defined; b) the constraint solver is proved to be safe w.r.t. the type system; c) the implementation of a concrete typechecker is presented; d) the integration of type checking and set constraint solving to increase automation during software verification is discussed; and f) two industrial-strength case studies are presented where this combination is used with very good results