Hedberg's theorem in the minimalist foundation

Le collezioni nella teoria dei tipi intensionale MF (Minimalist Foundation) dotate di uguaglianza decidibile soddisfano l'unicità delle prove di identità.ope

Constructions, inductive types and strong normalization

This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and type-checking, based on the equality-as-judgement presentation. We present a set-theoretic notion of model, CC-structures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to non-algebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a non-trivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the..

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

On the Formalisation of the Metatheory of the Lambda Calculus and Languages with Binders

Este trabajo trata sobre el razonamiento formal veri cado por computadora involucrando lenguajes con operadores de ligadura. Comenzamos presentando el Cálculo Lambda, para el cual utilizamos la sintaxis histórica, esto es, sintaxis de primer orden con sólo un tipo de nombres para las variables ligadas y libres. Primeramente trabajamos con términos concretos, utilizando la operación de sustitución múltiple de nida por Stoughton como la operación fundamental sobre la cual se de nen las conversiones alfa y beta. Utilizando esta sintaxis desarrollamos los principales resultados metateóricos del cálculo: los lemas de sustitución, el teorema de Church-Rosser y el teorema de preservación de tipo (Subject Reduction) para el sistema de asignación de tipos simples. En una segunda formalización reproducimos los mismos resultados, esta vez basando la conversion alfa sobre una operación más sencilla, que es la de permutación de nombres. Utilizando este mecanismo, derivamos principios de inducción y recursión que permiten trabajar identificando términos alfa equivalentes, de modo tal de reproducir la llamada convención de variables de Barendregt. De este modo, podemos imitar las demostraciones al estilo lápiz y papel dentro del riguroso entorno formal de un asistente de demostración. Como una generalización de este último enfoque, concluimos utilizando técnicas de programación genérica para definir una base para razonar sobre estructuras genéricas con operadores de ligadura. Definimos un universo de tipos de datos regulares con información de variables y operadores de ligadura, y sobre éstos definimos operadores genéricos de formación, eliminación e inducción. También introducimos una relación de alfa equivalencia basada en la operación de permutación y derivamos un principio de iteración/inducción que captura la convención de variables anteriormente mencionada. A modo de ejemplo, mostramos cómo definir el Cálculo Lambda y el sistema F en nuestro universo, ilustrando no sólo la reutilización de las pruebas genéricas, sino también cuán sencillo es el desarrollo de nuevas pruebas en estos casos. Todas las formalizaciones de esta tesis fueron realizadas en Teoría Constructiva de Tipos y verificadas utilizando el asistente de pruebas AgdaThis work is about formal, machine-checked reasoning on languages with name binders. We start by considering the ʎ-calculus using the historical ( rst order) syntax with only one sort of names for both bound and free variables. We rst work on the concrete terms taking Stoughton's multiple substitution operation as the fundamental operation upon which the ά and ß-conversion are de ned. Using this syntax we reach well-known meta-theoretical results, namely the Substitution lemmas, the Church-Rosser theorem and the Subject Reduction theorem for the system of assignment of simple types. In a second formalisation we reproduce the same results, this time using an approach in which -conversion is de ned using the simpler operation of name permutation. Using this we derive induction and recursion principles that allow us to work by identifying terms up to -conversion and to reproduce the so-called Barendregt's variable convention [4]. Thus, we are able to mimic pencil and paper proofs inside the rigorous formal setting of a proof assistant. As a generalisation of the latter, we conclude by using generic programming techniques to de ne a framework for reasoning over generic structures with binders. We de ne a universe of regular datatypes with variables and binders information, and over these we de ne generic formation, elimination, and induction operations. We also introduce an ά equivalence relation based on the swapping operation, and are able to derive an -iteration/induction principle that captures Barendregt's variable convention. As an example, we show how to de ne the ʎ calculus and System F in our universe, and thereby we are able to illustrate not only the reuse of the generic proofs but also how simple the development of new proofs becomes in these instances. All formalisations in this thesis have been made in Constructive Type Theory and completely checked using the Agda proof assistan

Beyond Logic. Proceedings of the Conference held in Cerisy-la-Salle, 22-27 May 2017

The project "Beyond Logic" is devoted to what hypothetical reasoning is all about when we go beyond the realm of "pure" logic into the world where logic is applied. As such extralogical areas we have chosen philosophy of science as an application within philosophy, informatics as an application within the formal sciences, and law as an application within the field of social interaction. The aim of the conference was to allow philosophers, logicians and computer scientists to present their work in connection with these three areas. The conference took place 22-27 May, 2017 in Cerisy-la-Salle at the Centre Culturel International de Cerisy. The proceedings collect abstracts, slides and papers of the presentations given, as well as a contribution from a speaker who was unable to attend

Towards a formally verified functional quantum programming language

This thesis looks at the development of a framework for a functional quantum programming language. The framework is first developed in Haskell, looking at how a monadic structure can be used to explicitly deal with the side-effects inherent in the measurement of quantum systems, and goes on to look at how a dependently-typed reimplementation in Agda gives us the basis for a formally verified quantum programming language. The two implementations are not in themselves fully developed quantum programming languages, as they are embedded in their respective parent languages, but are a major step towards the development of a full formally verified, functional quantum programming language. Dubbed the “Quantum IO Monad”, this framework is designed following a structural approach as given by a categorical model of quantum computation