Le but principal de cette thèse est de définir une extension de la traduction de double-négation de Gödel à tous les types tronqués, dans le contexte de la théorie des types homotopique. Ce but utilisera des théories déjà existantes, comme la théorie des faisceaux de Lawvere-Tierney, quenous adapterons à la théorie des types homotopiques. En particulier, on définira le fonction de faisceautisation de Lawvere-Tierney, qui est le principal théorème présenté dans cette thèse.Pour le définir, nous aurons besoin de concepts soit déjà définis en théorie des types, soit non existants pour l’instant. En particulier, on définira une théorie des colimits sur des graphes, ainsi que leur version tronquée, et une notion de modalités tronquées basée sur la définition existante de modalité.Presque tous les résultats présentés dans cette thèse sont formalisée avec l’assistant de preuve Coq, muni de la librairie [HoTT/Coq] / The main goal of this thesis is to define an extension of Gödel not-not translation to all truncated types, in the setting of homotopy type theory. This goal will use some existing theories, like Lawvere-Tierney sheaves theory in toposes, we will adapt in the setting of homotopy type theory. In particular, we will define a Lawvere-Tierney sheafification functor, which is the main theorem presented in this thesis.To define it, we will need some concepts, either already defined in type theory, either not existing yet. In particular, we will define a theory of colimits over graphs as well as their truncated version, and the notion of truncated modalities, based on the existing definition of modalities.Almost all the result presented in this thesis are formalized with the proof assistant Coq together with the library [HoTT/Coq]

This thesis presents a characterization of those categories with weak factorization systems that can interpret the theory of intensional dependent type theory with Σ, Π, and identity types. We use display map categories to serve as models of intensional dependent type theory. If a display map category (C, D) models Σ and identity types, then this structure generates a weak factorization system (L, R). Moreover, we show that if the underlying category C is Cauchy complete, then (C, R) is also a display map category modeling Σ and identity types (as well as Π types if (C, D) models Π types). Thus, our main result is to characterize display map categories (C, R) which model Σ and identity types and where R is part of a weak factorization system (L, R) on the category C. We offer three such characterizations and show that they are all equivalent when C has all finite limits. The first is that the weak factorization system (L, R) has the properties that L is stable under pullback along R and all maps to a terminal object are in R. We call such weak factorization systems type theoretic. The second is that the weak factorization system has what we call an Id-presentation: it can be built from certain categorical structure in the same way that a model of Σ and identity types generates a weak factorization system. The third is that the weak factorization system (L, R) is generated by a Moore relation system. This is a technical tool used to establish the equivalence between the first and second characterizations described. To conclude the thesis, we describe a certain class of convenient categories of topological spaces (a generalization of compactly generated weak Hausdorff spaces). We then construct a Moore relation system within these categories (and also within the topological topos) and thus show that these form display map categories with Σ and identity types (as well as Π types in the topological topos).

Lambda encodings (such as Church encoding, Scott encoding and Parigot encoding) are methods to represent data in lambda calculus. Curry-Howard correspondence relates the formulas and proofs in intuitionistic logics to the types and programs in typed functional programming languages. Roughly speaking, Type theory (Intuitionistic Type Theory) formulates the intuitionistic logic in the style of typed functional programming language. This dissertation investigates the mechanisms to support lambda encodings in type theory. Type theory, for example, Calculus of Constructions(CC) does not directly support inductive data because the induction principle for the inductive data is proven to be not derivable. Thus inductive data together with inductive principle are added as primitive to CC, leading to several nontrivial extensions, e.g. Calculus of Inductive Constructions. In this dissertation, we explore alternatives to incorporate inductive data in type theory. We propose to consider adding an abstraction construct to the intuitionistic type to support lambda-encoded data, while still be able to derive the corresponding induction principle. The main benefit of this approach is that we obtain relatively simple systems, which are easier to analyze and implement.

No description available.

At the heart of the connections between Proof Theory and Type Theory, the Curry-Howard correspondence provides proof-terms with computational features and equational theories, i.e. notions of normalisation and equivalence. This dissertation contributes to extend its framework in the directions of proof-theoretic formalisms (such as sequent calculus) that are appealing for logical purposes like proof-search, powerful systems beyond propositional logic such as type theories, and classical (rather than intuitionistic) reasoning. Part I is entitled Proof-terms for Intuitionistic Implicational Logic. Its contributions use rewriting techniques on proof-terms for natural deduction (Lambda-calculus) and sequent calculus, and investigate normalisation and cut-elimination, with call-by-name and call-by-value semantics. In particular, it introduces proof-term calculi for multiplicative natural deduction and for the depth-bounded sequent calculus G4. The former gives rise to the calculus Lambdalxr with explicit substitutions, weakenings and contractions that refines the Lambda-calculus and Beta-reduction, and preserves strong normalisation with a full notion of composition of substitutions. The latter gives a new insight to cut-elimination in G4. Part II, entitled Type Theory in Sequent Calculus develops a theory of Pure Type Sequent Calculi (PTSC), which are sequent calculi that are equivalent (with respect to provability and normalisation) to Pure Type Systems but better suited for proof-search, in connection with proof-assistant tactics and proof-term enumeration algorithms. Part III, entitled Towards Classical Logic, presents some approaches to classical type theory. In particular it develops a sequent calculus for a classical version of System F_omega. Beyond such a type theory, the notion of equivalence of classical proofs becomes crucial and, with such a notion based on parallel rewriting in the Calculus of Structures, we compute canonical representatives of equivalent proofs.

Homotopy Type Theory (HoTT) is a quite recent branch of research in mathematical logic, which provides interesting connections among various areas of mathematics. It was first introduced by Vladimir Voevodsky as a means to develop synthetic homotopy theory, and further advancements suggested that it can be used as a formal foundation to mathematics. Among its notable features, inductive and higher inductive types are of great interest, e.g. allowing for the study of geometric entities (such as spheres) in the setting of HoTT. However, so far in most of the literature higher inductive types are treated in an ad-hoc way; there is no easy general schema stating what an higher inductive type is, thus hindering the study of the related proof theory. Moreover, although Martin-Löf Type Theory has been deeply and widely studied, many proof theoretic results about its specific variant used in HoTT are folklore, and the proofs are missing. In this final talk, we provide an overview on some results we obtained, aiming to address these problems. In the first part of the talk, we will discuss a normalization theorem for the type theory underlying HoTT. In the second part of the talk we will propose a general syntax schema to encapsulate a relevant class of higher inductive types, potentially allowing for future study of the proof theory of HoTT enriched with such types.

Cette thèse s'intéresse à la métathéorie de la théorie des types intuitionniste. Les systèmes que nous considérons sont des variantes de la théorie des types de Martin-Löf ou du Calcul des Constructions, et nous nous intéressons à la cohérence de ces systèmes ou encore à l'indépendance d'axiomes par rapport à ces systèmes. Le fil rouge de cette thèse est la construction de modèles syntaxiques, qui sont des modèles qui réutilisent la théorie des types pour interpréter la théorie des types. Dans une première partie, nous introduisons la théorie des types à l'aide d'un système minimal et de plusieurs extensions potentielles. Dans une seconde partie, nous introduisons les modèles syntaxiques donnés par traduction de programme et donnons plusieurs exemples. Dans une troisième partie, nous présentons Template-Coq, un plugin de métaprogrammation pour Coq. Nous montrons comment l'utiliser pour implémenter directement certains modèles syntaxiques. Enfin, dans une dernière partie, nous nous intéressons aux théories des types à deux égalités : une égalité stricte et une égalité univalente. Nous proposons une relecture des travaux de Coquand et. al. et Orton et Pitts sur le modèle cubique en introduisant la notion de fibrance dégénérée. / This thesis is about the metatheory of intuitionnistic type theory. The considered systems are variants of Martin-Löf type theory of Calculus of Constructions, and we are interested in the coherence of those systems and in the independence of axioms with respect to those systems. The common theme of this thesis is the construction of syntactic models, which are models reusing type theory to interpret type theory. In a first part, we introduce type theory by a minimal system and several possible extensions. In a second part, we introduce the syntactic models given by program translation and give several examples. In a third part, we present Template-Coq, a plugin for metaprogramming in Coq. We demonstrate how to use it to implement directly some syntactic models. Last, we consider type theories with two equalities: one strict and one univalent. We propose a re-reading of works of Coquand et.al. and of Orton and Pitts on the cubical model by introducing degenerate fibrancy.

L’objectif de cette thèse est de démontrer que π4(S3) ≃ Z/2Z en théorie des types homotopiques. En particulier, c’est une démonstration constructive et purement homotopique. On commence par rappeler les concepts de base de la théorie des types homotopiques et on démontre quelques résultats bien connus sur les groupes d’homotopie des sphères : le calcul des groupes d’homotopie du cercle, le fait que ceux de la forme πk(Sn) avec k < n sont triviaux et la construction de la fibration de Hopf. On passe ensuite à des outils plus avancés. En particulier, on définit la construction de James, ce qui nous permetde démontrer le théorème de suspension de Freudenthal et le fait qu’il existe un entier naturel n tel que π4(S3) ≃ Z/2Z. On étudie ensuite le produit smash des sphères, on construit l’anneau de cohomologie des espaces et on introduit l’invariant de Hopf, ce qui nous permet de montrer que n est égal soit à 1, soit à 2. L’invariant de Hopf nous permet également de montrer que tous les groupes de la forme π4n−1(S2n) sont infinis. Finalement, on construit la suite exacte de Gysin, ce qui nous permet de calculer la cohomologie de CP2 et de démontrer que π4(S3) ≃ Z/2Z, et que plus généralement on a πn+1(Sn) ≃ Z/2Z pour tout n ≥ 3 / The goal of this thesis is to prove that π4(S3) ≃ Z/2Z in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the homotopy groups of spheres: the computation of the homotopy groups of the circle, the triviality of those of the form πk(Sn) with k < n, and the construction of the Hopf fibration. We then move to more advanced tools. In particular, we define the James construction which allows us to prove the Freudenthal suspension theorem and the fact that there exists a natural number n such that π4(S3) ≃ Z/nZ. Then we study the smash product of spheres, we construct the cohomology ring of a space, and we introduce the Hopf invariant, allowing us to narrow down the n to either 1 or 2. The Hopf invariant also allows us to prove that all the groups of the form π4n−1(S2n) are infinite. Finally we construct the Gysin exact sequence, allowing us to compute the cohomology of CP2 and to prove that π4(S3) ≃ Z/2Z and that more generally πn+1(Sn) ≃ Z/2Z for every n ≥ 3

L'objectif de cette thèse est de développer une théorie de types pour le λ-calcul linéaire-algébrique, une extension du λ-calcul motivé par l'informatique quantique. Cette extension algébrique comprend tous les termes du λ-calcul plus leurs combinaisons linéaires, donc si t et r sont des termes, α.t+β.r est aussi un terme, avec α et β des scalaires pris dans un anneau. L'idée principale et le défi de cette thèse était d'introduire un système de types où les types, de la même façon que les termes, constituent un espace vectoriel, permettant la mise en évidence de la structure de la forme normale d'un terme. Cette thèse présente le système Lineal , ainsi que trois systèmes intermédiaires, également intéressants en eux-même : Scalar, Additive et λCA, chacun avec leurs preuves de préservation de type et de normalisation forte. / The objective of this thesis is to develop a type theory for the linear-algebraic λ-calculus, an extension of λ-calculus motivated by quantum computing. This algebraic extension encompass all the terms of λ-calculus together with their linear combinations, so if t and r are two terms, so is α.t + β.r, with α and β being scalars from a given ring. The key idea and challenge of this thesis was to introduce a type system where the types, in the same way as the terms, form a vectorial space, providing the information about the structure of the normal form of the terms. This thesis presents the system Lineal, and also three intermediate systems, however interesting by themselves: Scalar, Additive and λCA, all of them with their subject reduction and strong normalisation proofs.

This thesis explores the impact of mentor and mentee personality type in formal mentoring relationships. The research sought to identify whether there were individual personality characteristics which impact on relationship dynamics and the learning derived from these relationships. The Myers Briggs Type Indicator (MBTI) was used to identify personality type thus ensuring that the research had practical utility in organisations. Twelve mentoring dyads from public, private and third sector mentoring initiatives participated in the study which adopted an exploratory and qualitative methodology. Multiple methods were used to collect data and an analysis framework was developed, using Activity Theory tenets, to synthesise the different data sets and create narratives of each mentoring relationship. The thesis argues that by enhancing understanding of Type Theory in mentoring relationships, informal learning can be enhanced for mentors and mentees. The research shows how informal learning within mentoring dyads often stems from social comparison and thus differences between mentor and mentee can provide a medium for learning in the workplace. The findings suggest that this will be particularly pertinent for mentors. In addition, the study conclusions highlight the value of using the MBTI to support mentoring relationship development thus enhancing the potential for further learning. The research finds that individual differences will determine the extent to which relationships operate on a traditional, peer or reverse level and not demographic differences as suggested in the extant literature. Furthermore, common personality preferences were identified in individuals who are drawn to the role of mentor and an initial framework for a typology of mentoring relationships was developed. There were two main limitations of the research. First, the study employed a cross-sectional design which resulted in data being collected from participants at different stages of the mentoring relationship. The second limitation concerned the small sample size. Whilst sample size is less relevant in qualitative research, the study sample cannot be considered representative of all formal mentoring programmes or even the programmes studied. The intention was to identify informative cases which would address the research objectives and this was subsequently achieved. The research has contributed to the body of mentoring knowledge by drawing theory from one academic field into another. The findings provide new insights into individual differences and mentoring relationship dynamics thus adding to a sparse area of knowledge in mentoring research. Further, the findings challenge some of the assumptions implicit in the extant literature and highlight the need to examine the construct of mentoring from a broader social science perspective.