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Combining Union, Intersection and Dependent Types in an Explicitely Typed Lambda-Calculus Claude Stolze

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Combining Union, Intersection and Dependent Types in an Explicitely Typed Lambda-Calculus Claude Stolze Combining union, intersection and dependent types in an explicitely typed lambda-calculus Claude Stolze To cite this version: Claude Stolze. Combining union, intersection and dependent types in an explicitely typed lambda- calculus. Logic [math.LO]. COMUE Université Côte d’Azur (2015 - 2019), 2019. English. NNT : 2019AZUR4104. tel-02406953v2 HAL Id: tel-02406953 https://hal.archives-ouvertes.fr/tel-02406953v2 Submitted on 26 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Types union, intersection, et dépendants dans le lambda-calcul explicitement typé Claude STOLZE INRIA Sophia-Antipolis Méditerranée Présentée en vue de l’obtention du Devant le jury, composé de : grade de docteur en Informatique de Giuseppe Castagna l’Université Côte d’Azur. Ugo de’Liguoro Silvia Ghilezan Furio Honsell Dirigée par : Luigi Liquori Delia Kesner Bruno Martin Jakob Rehof Soutenue le : 16 décembre 2019 Luigi Liquori Types union, intersection, et dépendants dans le lambda-calcul explicitement typé Jury Président du jury Bruno Martin, Professeur, Université Côte d’Azur Rapporteurs Silvia Ghilezan, Professeur, University of Novi Sad, Serbie Delia Kesner, Professeur, Université de Paris Examinateurs Giuseppe Castagna, Directeur de recherche, CNRS, Université de Paris Ugo de’Liguoro, Professeur, Università degli Studi di Torino, Italie Furio Honsell, Professeur, Università degli Studi di Udine, Italie Jakob Rehof, Professeur, Technische Universität Dortmund, Allemagne Encadrant Luigi Liquori, Directeur de recherche, INRIA Sophia-Antipolis Méditerranée Types union, intersection, et dépendants dans le lambda-calcul explicitement typé Résumé : Le sujet de cette thèse est sur le lambda-calcul décoré avec des types, communément appelé « lambda-calcul typé à la Church ». Nous étudions des versions de ce lambda- calcul muni de types intersections, tels que ceux décrits dans le livre « Lambda-calculus with types » de Barendregt, Dekkers et Statman ; les types unions, qui ont été introduits par Plotkin, MacQueen et Sethi ; et les types dépendants, tels qu’ils ont été décrits par Plotkin, Harper et Honsell lorsqu’ils ont introduit le Logical Framework d’Edinbourgh LF. Les types intersections et unions sont un moyen d’exprimer du polymorphisme ad hoc et sont une alternative au polymorphisme paramétrique de Girard. Les types dépendants ont été introduits pour formaliser la logique intuitionniste avec la correspondance de Curry- Howard. Le système de types obtenu peut être enrichi avec une relation de soutypage décidable. La combinaison de ces trois disciplines de type donne lieu à une famille de calculs qui peuvent être paramétrés et classifiés. Nous appelons le système générique le Delta-calcul. Nous discutons ensuite des décisions de conception qui nous ont amené à la formulation de ces calculs, nous étudions leur métathéorie, et nous présentons divers exemples d’applications avant de présenter une implémentation logicielle du Delta-calcul, avec une description des algorithmes de vérification de type, de raffinement, de soutypage, d’évaluation, ainsi que de l’interface en ligne de commande. Ce travail de recherche peut être vu comme un petit pas franchi dans la direction d’une théorie des types alternative pour définir du polymorphisme dans les langages de programmation et dans les assistants de preuve interactifs. Mots clés : lambda-calcul, théorie des types, correspondance de Curry-Howard. Combining union, intersection and dependent types in an explicitely typed lambda-calculus Abstract: The subject of this thesis is about lambda-calculus decorated with types, usually called “Church-style typed lambda-calculus”. We study this lambda-calculus enhanced with Intersection types, as described by Barendregt, Dekkers and Statman in the book “Lambda-calculus with Types”; Union types, as introduced by Plotkin, MacQueen and Sethi; and Dependent types, as described by Plotkin, Harper and Honsell when they introduced the Edinburgh Logical Framework LF. Intersection and union types are a way to express ad hoc polymorphism and are an alternative to the parametric polymorphism of Girard. Dependent types were introduced as a way to formalize intuitionistic logic using the “proofs-as-lambda-terms / formulae-as-types” Curry-Howard principle. The resulting type system can be enriched with a decidable subtyping relation. Combining these three type disciplines gives rise to a family of calculi that can be parametrized and classified: we call the resulting system the Delta-calculus. We then discuss the design decisions which have led us to the formulation of these calculi, study their metatheory, and provide various examples of applications; and we finally present a software implementation of the Delta- calculus, with a description of the type checker, the refinement algorithm, the subtyping algorithm, the evaluation algorithm and the command-line interface. This work can be understood as a little step toward an alternative type theory to defining polymorphic programming languages and interactive proof assistants. Keywords: lambda-calculus, type theory, Curry-Howard correspondence. ix Remerciements – Acknowledgements Je ne peux commencer cette thèse sans remercier mon directeur, Luigi Liquori, pour tout le soutien et tous les conseils qu’il m’a prodigués au cours de ces trois années de travail ensemble. Merci en particulier pour la confiance que tu m’as accordé. Tu m’as appris l’exigeant métier de chercheur, notamment qu’un article n’a fini d’être écrit que lorsque la deadline est passée, et que les détails mathématiques et l’esthétique ont une importance cruciale dans la science. J’aimerais aussi remercier ma mère pour son soutien à distance de l’autre bout de la France, et qui me réconfortait quand j’avais le mal du pays. Je voudrais remercier Enrico Tassi qui m’a détaillé les entrailles du code de Coq. Et sans oublier mes camarades Siargey, Owen, Sophie, Cécile, Damien, Carsten, Giovanni, et tant d’autres. I would also like to thank my jury for accepting to review my work. I am particularly thankful to Delia Kesner and Silvia Ghilezan, who accepted to be my referees. Many thanks to Bruno Martin, Giuseppe Castagna, Ugo de’Liguoro, Furio Honsell, and Jakob Rehof, for showing interest in my work and for accepting to participate in this jury. x Contents 1 Introduction 1 1.1 Prolegomenon . .1 1.2 Contributions . .3 1.3 Related works . .5 2 A typed calculus with intersection types 9 2.1 Syntax, Reduction and Types . 10 2.1.1 The ∆-calculi . 11 2.1.2 The ∆-chair . 14 2.2 Examples . 15 2.2.1 On synchronization and subject reduction . 18 T 2.3 Metatheory of ∆R ................................ 19 2.3.1 General properties . 19 2.3.2 Synchronous reduction . 25 2.3.3 Strong normalization of the generic ∆-calculus . 29 2.4 Church-style vs. Curry-style λ-calculus . 31 T T 2.4.1 Relation between type assignment systems λ\ and typed systems ∆R 31 2.4.2 Subtyping and explicit coercions . 34 3 Adding union types 39 3.1 Syntax and semantics of ∆BDdL and λ@BDdL .................. 40 3.2 Metatheory of ∆BDdL and λ@BDdL ....................... 43 4 On Mints’ realizability 45 4.1 Presentation of NJ(β) .............................. 46 4.2 Soundness of NJ(β) ............................... 47 4.3 Completeness of NJ(β) ............................. 49 4.3.1 Failure of completeness of NJ(β) without subtyping . 49 4.3.2 Counter-example . 50 5 Subtyping algorithm 51 5.1 The algorithm, shortly explained . 51 5.1.1 Soundness and correctness of the algorithm . 54 5.2 Coq implementation of the theory Ξ ..................... 55 5.2.1 Definition of normal forms . 57 5.2.2 Filters and ideals . 59 xi xii CONTENTS 5.2.3 Induction scheme for filters and ideals . 60 5.2.4 Properties of filters and ideals . 62 5.3 Coq implementation of the subtyping algorithm . 63 5.4 Extracting the subtyping algorithm in OCaml . 69 5.5 The preorder tactic . 70 5.5.1 Denotation . 71 5.5.2 Implementation of the heuristic function . 72 5.5.3 Reification . 74 6 Dependent types 77 6.1 The ∆-framework: LF with proof-functional operators . 80 BDdL 6.2 Relating LF∆ to λ ............................. 84 6.2.1 Typed derivation of Pierce’s example . 85 6.2.2 LF∆ metatheory . 86 6.3 Minimal relevant implications and type inclusion . 89 6.4 Pure Type System presentation of LF∆ .................... 91 6.5 Future Work . 92 7 Implementation of the theorem prover Bull 95 7.1 de Bruijn indices in Bull . 96 7.2 Syntax of terms . 97 7.2.1 Concrete syntax . 98 7.2.2 Implementation of the syntax . 99 7.2.3 Environments . 100 7.2.4 Suspended substitution . 101 7.3 The evaluator of Bull . 102 7.3.1 Reduction rules . 102 7.3.2 Implementation . 102 7.4 The subtyping algorithm of Bull . 104 7.5 The unification algorithm of Bull . 105 7.6 The refinement algorithm of Bull . 105 7.7 The Read-Eval-Print-Loop of Bull . 108 7.8 Future work . 109 8 Examples in Bull 115 8.1 Encodings in LF∆ ................................ 115 8.1.1 Pierce’s code . 116 8.1.2 Hereditary Harrop formulæ . 116 8.1.3 Natural deductions in normal form . 118 8.2 Encodings in LF . 121 List of Figures 1.1 Pierce’s code . .7 2.1 Minimal type theory 6min, axioms and rule schemes (from Figure 13.2 and 13.3 of [12]) . 10 T 2.2 Generic intersection type assignment system λ\ (from Figure 13.8 of [12]) . 10 CD CDS CDV BCD 2.3 Type theories λ\ , λ\ , λ\ , and λ\ .
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