Search

International audience; Equations is a plugin for the Coq proof assistant which provides a notation for defining programs by dependent pattern-matching and structural or well-founded recursion. It additionally derives useful proof principles for demonstrating properties about them. We present a general design and implementation that provides a robust and expressive function definition package as a definitional extension to the Coq kernel. At the core of the system is a new simplifier for dependent equalities that can be reused to define enhanced versions of dependent elimination tactics. We introduce verified optimizations of the simplifier that allow generating smaller and simpler Equations definitions and proof terms for these tactics in general.

International audience; We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isa-belle/HOL's recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL's new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion.

In this paper we prove the correctness of a compiler for a call-by-name language using step-indexed logical relations and biorthogonality. The source language is an extension of the simply typed lambda-calculus with recursion, and the target language is an extension of the Krivine abstract machine. We formalized the proof in the Coq proof assistant.

International audience; Our concern is the modular development of a certified static analyzer in Coq: we extend a certified abstract domain of convex polyhedra with a linearization procedure approximating polynomial expressions. In order to help such a development, we propose a proof framework, embedded in Coq, that implements a refinement calculus. It allows to hide for proofs several low-level aspects of the computations on abstract domains. Moreover, refinement proofs are naturally simplified thanks to computations of weakest preconditions.

In order to avoid well-known paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type{0} : Type{1} : · · · . Such type systems are called cumulative if for any type A we have that A : Type{i} implies A : Type{i+1}. The Predicative Calculus of Inductive Constructions (pCIC) which forms the basis of the Coq proof assistant, is one such system. In this paper we present the Predicative Calculus of Cumulative Inductive Constructions (pCuIC) which extends the cumulativity relation to inductive types. We discuss cumulative inductive types as present in Coq 8.7 and their application to formalization and definitional translations. ispartof: LIPIcs : Leibniz International Proceedings in Informatics ispartof: International Conference on Formal Structures for Computation and Deduction (FSCD) location:Oxford, UK date:9 Jul - 12 Jul 2018 status: accepted

Computational Logic is the use of computers to establish facts in a logical formalism. Originating in 19th-century attempts to understand the nature of mathematical reasoning, the subject now comprises a wide variety of formalisms, techniques and technologies. One strand of work follows the "LCF approach" pioneered by Robin Milner FRS, where proofs can be constructed interactively or with the help of users' code (which does not compromise correctness). A refinement of LCF, called Isabelle, retains these advantages while providing flexibility in the choice of logical formalism and much stronger automation. The main application of these techniques has been to prove the correctness of hardware and software systems, but increasingly researchers have been applying them to mathematics itself., Comment: accepted by Proceedings of the Royal Society Series A, subject to minor revisions

International audience; Relational properties describe multiple runs of one or more programs. They characterize many useful notions of security, program refinement, and equivalence for programs with diverse computational effects, and they have received much attention in the recent literature. Rather than developing separate tools for special classes of effects and relational properties, we advocate using a general purpose proof assistant as a unifying framework for the relational verification of effectful programs. The essence of our approach is to model effectful computations using monads and to prove relational properties on their monadic representations, making the most of existing support for reasoning about pure programs.We apply this method in F* and evaluate it by encoding a variety of relational program analyses, including information flow control, program equivalence and refinement at higher order, correctness of program optimizations and game-based cryptographic security. By relying on SMT-based automation, unary weakest preconditions, user-defined effects, and monadic reification, we show that, compared to unary properties, verifying relational properties requires little additional effort from the F* programmer.

International audience; We developed a formal framework for CDCL (conflict-driven clause learning) using the Isabelle/HOL proof assistant. Through a chain of refinements, an abstract CDCL calculus is connected first to a more concrete calculus, then to a SAT solver expressed in a functional programming language, and finally to a SAT solver in an imperative language, with total correctness guarantees. The framework offers a convenient way to prove metatheorems and experiment with variants, including the DPLL (Davis-Putnam-Logemann-Loveland) calculus. The imperative program relies on the two-watched-literal data structure and other optimizations found in modern solvers. We used Isabelle's Refinement Framework to automate the most tedious refinement steps. The most noteworthy aspects of our work are the inclusion of rules for forget, restart, and incremental solving and the application of stepwise refinement.

We present a formalization of the first half of Bachmair and Ganzinger'schapter on resolution theorem proving in Isabelle/HOL, culminating witha refutationally complete first-order prover based on ordered resolution withliteral selection. We develop general infrastructure and methodology that canform the basis of completeness proofs for related calculi, includingsuperposition. Our work clarifies several of the fine points in the chapter'stext, emphasizing the value of formal proofs in the field of automatedreasoning.

This paper presents a mechanically verified implementation of an algorithm for deciding the equivalence of Kleene algebra terms within the Coq proof assistant. The algorithm decides equivalence of two given regular expressions through an iterated process of testing the equivalence of their partial derivatives and does not require the construction of the corresponding automata. Recent theoretical and experimental research provides evidence that this method is, on average, more efficient than the classical methods based on automata. We present some performance tests, comparisons with similar approaches, and also introduce a generalization of the algorithm to decide the equivalence of terms of Kleene algebra with tests. The motivation for the work presented in this paper is that of using the libraries developed as trusted frameworks for carrying out certified program verification.

Current tools for automated deduction are often powerful and complex. Due to their complexity there is a risk that they contain bugs and thus deliver wrong results. To ensure reliability of these tools, one possibility is to develop certifiers which check the results of tools with the help of a trusted proof assistant. We present a framework which illustrates the essential steps to develop stand-alone certifiers which efficiently check generated proofs outside the employed proof assistant. Our framework has already been used to develop certifiers for various properties, including termination, confluence, completion, and tree automata related properties.

У овој тези представљена је интерактивна формализација модела разних геометрија и алгебарских метода аутоматског доказивања геометријских те- орема... In this thesis is presented interactive formalization of various models of geometry and algebraic methods for automated proving geometry theorems....

We present a collection of formalized results about finite nested multisets, developed using the Isabelle/HOL proof assistant. The nested multiset order is a generalization of the multiset order that can be used to prove termination of processes. Hereditary multisets, a variant of nested multisets, offer a convenient representation of ordinals below epsilon-0. In Isabelle/HOL, both nested and hereditary multisets can be comfortably defined as inductive datatypes. Our formal library also provides, somewhat nonstandardly, multisets with negative multiplicities and syntactic ordinals with negative coefficients. We present applications of the library to formalizations of Goodstein's theorem and the decidability of unary PCF (programming computable functions)., Leibniz International Proceedings in Informatics (LIPIcs), 84, ISSN:1868-8969, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017), ISBN:978-3-95977-047-7

Formal trust in an abstract property, be it a mathematical result or a quality of the behavior of a computer program or a piece of hardware, is founded on the existence of a proof of its correctness. Many different kinds of proofs are written by mathematicians or generated by theorem provers, with the common problem of ascertaining whether those claimed proofs are themselves correct. The recently proposed Foundational Proof Certificate (FPC) framework harnesses advances in proof theory to define the semantics of proof formats, which can be verified by an independent and trusted proof checking kernel written in a logic programming language. This thesis extends initial results in certification of first-order proofs in several directions. It covers various essential logical axes grouped in meaningful combinations as they occur in practice: first,classical logic without fixed points and proofs generated by automated theorem provers; later, intuitionistic logic with fixed points and equality as logical connectives and proofs generated by proof assistants. The role of proof certificates is no longer limited to representing complete proofs to enable independent checking, but is extended to model proof transformations where details can be added to or subtracted from a certificate. These transformations yield operationally simpler certificates, around which increasingly trustworthy and performant proof checkers are constructed. Another new role of proof certificates is writing high-level proof outlines, which can be used to represent standard proof patterns as written by mathematicians, as well as automated techniques like property-based testing. We apply these developments to fully certify results produced by two families of standard automated theorem provers: resolution- and satisfiability-based. Another application is the design of programmable proof description languages for a proof assistant., La confiance formelle en une propriété abstraite provient de l'existence d'une preuve de sa correction, qu'il s'agisse d'un théorème mathématique ou d'une qualité du comportement d'un logiciel ou processeur. Il existe de nombreuses définitions différentes de ce qu'est une preuve, selon par exemple qu'elle est écrite soit par des humains soit par des machines, mais ces définitions sont toutes concernées par le problème d'établir qu'un document représente en fait une preuve correcte. Le cadre des Certificats de Preuve Fondamentaux (Foundational Proof Certificates, FPC) est une approche proposée récemment pour étudier ce problème, fondée sur des progrès de la théorie de la démonstration pour définir la sémantique des formats de preuve. Les preuves ainsi définies peuvent être vérifiées indépendamment par un noyau vérificateur de confiance codé dans un langage de programmation logique. Cette thèse étend des résultats initiaux sur la certification de preuves du premier ordre en explorant plusieurs dimensions logiques essentielles, organisées en combinaisons correspondant à leur usage en pratique: d'abord, la logique classique sans points fixes, dont les preuves sont générées par des démonstrateurs automatiques de théorème; ensuite, la logique intuitionniste avec points fixes et égalité,dont les preuves sont générées par des assistants de preuve. Les certificats de preuve ne se limitent pas comme précédemment à servir de représentation des preuves complètes pour les vérifier indépendamment. Leur rôle s'étend pour englober des transformations de preuve qui peuvent enrichir ou compacter leur représentation. Ces transformations peuvent rendre des certificats plus simples opérationnellement, ce qui motive la construction d'une suite de vérificateurs de preuve de plus en plus fiables et performants. Une autre nouvelle fonction des certificats de preuve est l'écriture d'aperçus de preuve de haut niveau, qui expriment des schémas de preuve tels qu'ils sont employés dans la pratique des mathématiciens, ou dans des techniques automatiques comme le property-based testing. Ces développements s'appliquent à la certification intégrale de résultats générés par deux familles majeures de démonstrateurs automatiques de théorème, utilisant techniques de résolution et satisfaisabilité, ainsi qu'à la création de langages programmables de description de preuve pour un assistant de preuve.

This thesis presents a formalization of fundamental database theories and algorithms. This furthers the maturing state of the art in formal specification development in the database field, with contributions stemming from two foundational approches to database models: relational and logic based.As such, a first contribution is a Coq library for the relational model. This contains a mechanization of integrity constraints and of their inference procedures. We model two of the most common dependencies, namely functional and multivalued, together with their corresponding axiomatizations. We prove soundness of their inference algorithms and, for the case of functional ones, also completeness. These types of dependencies are instances of equality and, respectively, tuple generating dependencies, which fall under the yet wider class of general dependencies. We model these and their inference procedure,i.e, the chase, for which we establish soundness.A second contribution consists of a Coq/Ssreflect library for logic programming in the Datalog setting. As part of this work, we give (one of the) first mechanizations of the standard Datalog language and of its extension with negation. The library includes a formalization of their model theoretical semantics and of their fixpoint semantics, implemented through bottom-up and, respectively, through stratified evaluation procedures. This is complete with the corresponding soundness, termination and completeness proofs. In this context, we also construct a preliminary framework for dealing with stratified programs. This work paves the way towards the certification of data-centric applications., Cette thèse présente une formalisation en Coq des langages et des algorithmes fondamentaux portant sur les bases de données. Ainsi, ce fourni des spécifications formelles issues des deux approches différentes pour la définition des modèles de données: une basée sur l’algèbre et l'autre basée sur la logique.A ce titre, une première contribution de cette thèse est le développement d'une bibliothèque Coq pour le modèle relationnel. Cette bibliothèque contient les modélisations de l’algèbre relationnelle et des requêtes conjonctives. Il contient aussi une mécanisation des contraintes d'intégrité et de leurs procédures d'inférence. Nous modélisons deux types de contraintes: les dépendances, qui sont parmi les plus courantes: les dépendances fonctionnelles et les dépendances multivaluées, ainsi que leurs axiomatisations correspondantes. Nous prouvons formellement la correction de leurs algorithmes d'inférence et, pour le cas de dépendances fonctionnelles, aussi la complétude.Ces types de dépendances sont des instances de contraintes plus générales : les dépendances génératrices d'égalité (equality generating dependencies, EGD) et, respectivement, les dépendances génératrices de tuples (tuple generating dependencies, TGD), qui appartiennent a une classe encore plus large des dépendances générales (general dependencies). Nous modélisons ces dernières et leur procédure d'inférence, i.e, "the chase", pour lequel nous établissons la correction. Enfin, on prouve formellement les théorèmes principaux des bases de données, c'est-à-dire, les équivalences algébriques, la théorème de l' homomorphisme et la minimisation des requêtes conjonctives.Une deuxième contribution consiste dans le développement d'une bibliothèque Coq/ssreflect pour la programmation logique, restreinte au cas du Datalog. Dans le cadre de ce travail, nous donnons la première mécanisations d'un moteur Datalog standard et de son extension avec la négation. La bibliothèque comprend une formalisation de leur sémantique en theorie des modelés ainsi que de leur sémantique par point fixe, implémentée par une procédure d'évaluation stratifiée. La bibliothèque est complétée par les preuves de correction, de terminaison et de complétude correspondantes. Cette plateforme ouvre la voie a la certification d' applications centrées données.

As computer programs become increasingly complex, techniques for ensuring trustworthiness of information manipulated by them become critical. In this work, we use the Coq proof assistant to formalise a $$\lambda $$ λ -calculus with trust types, originally formulated by Ørbæk and Palsberg. We give formal proofs of type soundness, erasure and simulation theorems and also prove decidability of the typing problem. As a result of our formalisation a certified type checker is derived.

User interfaces for interactive proof assistants have always lagged behind those for mainstream programming languages. Whereas integrated development environments—IDEs—have support for features like project management, version control, dependency analysis and incremental project compilation, “IDE”s for proof assistants typically only operate on files in isolation, relying on external tools to integrate those files into larger projects. In this paper we present Coqoon, an IDE for Coq developments integrated into Eclipse. Coqoon manages proofs as projects rather than isolated source files, and compiles these projects using the Eclipse common build system. Coqoon takes advantage of the latest features of Coq, including asynchronous and parallel processing of proofs, and—when used together with a third-party OCaml extension for Eclipse—can even be used to work on large developments containing Coq plugins.

CAPEs Proof assistants are software-based tools that are used in the mechanization of proof construction and validation in mathematics and computer science, and also in certified program development. Different such tools are being increasingly used in order to accelerate and simplify proof checking, and the Coq proof assistant is one of the most known and used. Language and automata theory is a well-established area of mathematics, relevant to computer science foundations and information technology. In particular, context-free language theory is of fundamental importance in the analysis, design and implementation of computer programming languages. This work describes a formalization effort, using the Coq proof assistant, of fundamental results of the classical theory of context-free grammars and languages. These include closure properties (union, concatenation and Kleene star), grammar simplification (elimination of useless symbols, inaccessible symbols, empty rules and unit rules), the existence of a Chomsky Normal Form for context-free grammars and the Pumping Lemma for context-free languages. To achieve this, several steps had to be fulfilled, including (i) understanding of the characteristics, importance and benefits of mathematical formalization, specially in computer science, (ii) familiarization with the underlying mathematical theories used in proof assistants, (iii) familiarization with the Coq proof assistant, (iv) review of the strategies used in the informal proofs of the main results of the context-free language theory and finally (iv) selection and adequation of the representation and proof strategies adopted in order the achieve the desired objectives. The result is an important set of libraries covering the main results of context-free language theory, with more than 500 lemmas and theorems fully proved and checked. This is probably the most comprehensive formalization of the classical context-free language theory in the Coq proof assistant done to the present date, and includes the remarkable result that is the formalization of the Pumping Lemma for context-free languages. The perspectives for the further development of this work are diverse and can be grouped in three different areas: inclusion of new devices and results, code extraction and general enhancements of its libraries. Assistentes de prova são ferramentas de software que são usadas na mecanização da construção e da validação de provas na matemática e na ciência da computação, e também no desenvolvimento de programas certificados. Diferentes ferramentas estão sendo usadas de forma cada vez mais frequente para acelerar e simplificar a verificação de provas, e o assistente de provas Coq é uma das mais conhecidas e utilizadas. A teoria de linguagens e de autômatos é uma área bem estabelecida da matemática, com relevância para os fundamentos da ciência da computação e a tecnologia da informação. Em particular, a teoria das linguagens livres de contexto é de fundamental importância na análise, no projeto e na implementação de linguagens de programação de computadores. Este trabalho descreve um esforço de formalização, usando o assistente de provas Coq, de resultados fundamentais da teoria clássica das gramáticas e linguagens livres de contexto. Estes incluem propriedades de fechamento (união, concatenação e estrela de Kleene), simplificação gramatical (eliminação de símbolos inúteis, de símbolos inacessíveis, de regras vazias e de regras unitárias), a existência da Forma Normal de Chomsky para gramáticas livres de contexto e o Lema do Bombeamento para linguagens livres de contexto. Para alcançar estes resultados, diversas etapas precisaram ser cumpridas, incluindo (i) o entendimento das características, da importância e dos benefícios da formalização matemática, especialmente na ciência da computação, (ii) a familiarização com as teorias matemáticas fundamentais utilizadas pelos assistentes de provas, (iii) a familiarização com o assistente de provas Coq, (iv) a revisão das estratégias usadas nas provas informais dos principais resultados da teoria das linguagens livres de contexto e, finalmente, (v) a seleção e adequação das estratégias de representação e prova adotadas para permitir o alcance dos resultados pretendidos. O resultado é um importante conjunto de bibliotecas cobrindo os principais resultados da teoria das linguagens livres de contexto, com mais de 500 lemas e teoremas totalmente provados e verificados. Esta é provavelmente a formalização mais abrangente da teoria clássica das linguagens livres de contexto jamais feita no assistente de provas Coq, e inclui o importante resultado que é a formalização do Lema do Bombeamento para linguagens livres de contexto. As perspectivas para novos desenvolvimentos a partir deste trabalho são diversas e podem ser agrupadas em três áreas diferentes: inclusão de novos dispositivos e resultados, extração de código e aprimoramentos gerais das suas bibliotecas.

In this paper we prove the correctness of a compiler for a call-by-name language using step-indexed logical relations and biorthogonality. The source language is an extension of the simply typed lambda-calculus with recursion, and the target language is an extension of the Krivine abstract machine. We formalized the proof in the Coq proof assistant. https://www.sciencedirect.com/science/article/pii/S157106611630041X publishedVersion publishedVersion Fil: Rodríguez, Leonardo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Fil: Pagano, Miguel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Fil: Fridlender, Daniel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Ciencias de la Computación

Nowadays, we are using an increasing number of computer applications. Errors in critical applications (medicine, transport, . . .) may carry serious health or financial issues. Avoiding errors in programs is a challenge and may be achieved by deductive verification. Deductive verification applies to program written in a high-level languages, which are transformed into machine language by compilers. These compilers must be correct to ensure the nonpropagation of errors to machine code. Since 2005, multicore processors have spread in all electronic devices. So, these architectures need adapted compilers and proofs of correctness. Our work is the modular extension of a verified compiler for parallel languages targeting multicore architectures. Specifications of these languages (and their operational semantics) needed at all levels of the compiler and proofs of correctness of this compiler are parameterized by modules specifying elements of parallelism such as a relaxed memory model and notions of synchronization and scheduling between threads. This work is the first step in the conception of a certified compiler for high-level parallel languages such as algorithmic skeletons.; Les applications informatiques sont de plus en plus présentes dans nos vies. Pour les applications critiques (médecine, transport, . . .), les conséquences d’une erreur informatique ont un coût inacceptable, que ce soit sur le plan humain ou financier. Une des méthodes pour éviter la présence d’erreurs dans les programmes est la vérification déductive. Celle-ci s’applique à des programmes écrits dans des langages de haut-niveau transformés, par des compilateurs, en programmes écrits en langage machine. Les compilateurs doivent être corrects pour ne pas propager d’erreurs au langage machine. Depuis 2005, les processeurs multi-coeurs se sont répandus dans l’ensemble des systèmes informatiques. Ces architectures nécessitent des compilateurs et des preuves de correction adaptées. Notre contribution est l’extension modulaire d’un compilateur vérifié pour un langage parallèle ciblant des architectures parallèles multi-coeurs. Les spécifications des langages (et leurs sémantiques opérationnelles) présents aux divers niveaux du compilateur ainsi que les preuves de la correction du compilateur sont paramétrées par des modules spécifiant des éléments de parallélisme tels qu’un modèle mémoire faible et des notions de synchronisation et d’ordonnancement entre processus légers. Ce travail ouvre la voie à la conception d’un compilateur certifié pour des langages parallèles de haut-niveau tels que les langages à squelettes algorithmiques.