Agda (“Dependent Types at Work”)
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Experience Implementing a Performant Category-Theory Library in Coq
Experience Implementing a Performant Category-Theory Library in Coq Jason Gross, Adam Chlipala, and David I. Spivak Massachusetts Institute of Technology, Cambridge, MA, USA [email protected], [email protected], [email protected] Abstract. We describe our experience implementing a broad category- theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed sub- stantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to repre- sent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such rea- soning. One particular encoding trick to which we draw attention al- lows category-theoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful. Keywords: Coq · category theory · homotopy type theory · duality · performance 1 Introduction Category theory [36] is a popular all-encompassing mathematical formalism that casts familiar mathematical ideas from many domains in terms of a few unifying concepts. A category can be described as a directed graph plus algebraic laws stating equivalences between paths through the graph. Because of this spar- tan philosophical grounding, category theory is sometimes referred to in good humor as \formal abstract nonsense." Certainly the popular perception of cat- egory theory is quite far from pragmatic issues of implementation. -
Countability of Inductive Types Formalized in the Object-Logic Level
Countability of Inductive Types Formalized in the Object-Logic Level QinxiangCao XiweiWu * Abstract: The set of integer number lists with finite length, and the set of binary trees with integer labels are both countably infinite. Many inductively defined types also have countably many ele- ments. In this paper, we formalize the syntax of first order inductive definitions in Coq and prove them countable, under some side conditions. Instead of writing a proof generator in a meta language, we develop an axiom-free proof in the Coq object logic. In other words, our proof is a dependently typed Coq function from the syntax of the inductive definition to the countability of the type. Based on this proof, we provide a Coq tactic to automatically prove the countability of concrete inductive types. We also developed Coq libraries for countability and for the syntax of inductive definitions, which have value on their own. Keywords: countable, Coq, dependent type, inductive type, object logic, meta logic 1 Introduction In type theory, a system supports inductive types if it allows users to define new types from constants and functions that create terms of objects of that type. The Calculus of Inductive Constructions (CIC) is a powerful language that aims to represent both functional programs in the style of the ML language and proofs in higher-order logic [16]. Its extensions are used as the kernel language of Coq [5] and Lean [14], both of which are widely used, and are widely considered to be a great success. In this paper, we focus on a common property, countability, of all first-order inductive types, and provide a general proof in Coq’s object-logic. -
Proving Termination of Evaluation for System F with Control Operators
Proving termination of evaluation for System F with control operators Małgorzata Biernacka Dariusz Biernacki Sergue¨ıLenglet Institute of Computer Science Institute of Computer Science LORIA University of Wrocław University of Wrocław Universit´ede Lorraine [email protected] [email protected] [email protected] Marek Materzok Institute of Computer Science University of Wrocław [email protected] We present new proofs of termination of evaluation in reduction semantics (i.e., a small-step opera- tional semantics with explicit representation of evaluation contexts) for System F with control oper- ators. We introduce a modified version of Girard’s proof method based on reducibility candidates, where the reducibility predicates are defined on values and on evaluation contexts as prescribed by the reduction semantics format. We address both abortive control operators (callcc) and delimited- control operators (shift and reset) for which we introduce novel polymorphic type systems, and we consider both the call-by-value and call-by-name evaluation strategies. 1 Introduction Termination of reductions is one of the crucial properties of typed λ-calculi. When considering a λ- calculus as a deterministic programming language, one is usually interested in termination of reductions according to a given evaluation strategy, such as call by value or call by name, rather than in more general normalization properties. A convenient format to specify such strategies is reduction semantics, i.e., a form of operational semantics with explicit representation of evaluation (reduction) contexts [14], where the evaluation contexts represent continuations [10]. Reduction semantics is particularly convenient for expressing non-local control effects and has been most successfully used to express the semantics of control operators such as callcc [14], or shift and reset [5]. -
Types Are Internal Infinity-Groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau
Types are internal infinity-groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau To cite this version: Antoine Allioux, Eric Finster, Matthieu Sozeau. Types are internal infinity-groupoids. 2021. hal- 03133144 HAL Id: hal-03133144 https://hal.inria.fr/hal-03133144 Preprint submitted on 5 Feb 2021 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 are Internal 1-groupoids Antoine Allioux∗, Eric Finstery, Matthieu Sozeauz ∗Inria & University of Paris, France [email protected] yUniversity of Birmingham, United Kingdom ericfi[email protected] zInria, France [email protected] Abstract—By extending type theory with a universe of defini- attempts to import these ideas into plain homotopy type theory tionally associative and unital polynomial monads, we show how have, so far, failed. This appears to be a result of a kind of to arrive at a definition of opetopic type which is able to encode circularity: all of the known classical techniques at some point a number of fully coherent algebraic structures. In particular, our approach leads to a definition of 1-groupoid internal to rely on set-level algebraic structures themselves (presheaves, type theory and we prove that the type of such 1-groupoids is operads, or something similar) as a means of presenting or equivalent to the universe of types. -
Sequent Calculus and Equational Programming (Work in Progress)
Sequent Calculus and Equational Programming (work in progress) Nicolas Guenot and Daniel Gustafsson IT University of Copenhagen {ngue,dagu}@itu.dk Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums. 1 Programming with Equations Functional programming has proved extremely useful in making the task of writing correct software more abstract and thus less tied to the specific, and complex, architecture of modern computers. This, is in a large part, due to its extensive use of types as an abstraction mechanism, specifying in a crisp way the intended behaviour of a program, but it also relies on its declarative style, as a mathematical approach to functions and data structures. However, the vast gain in expressivity obtained through the development of dependent types makes the programming task more challenging, as it amounts to the question of proving complex theorems — as illustrated by the double nature of proof assistants such as Coq [11] and Agda [18]. Keeping this task as simple as possible is then of the highest importance, and it requires the use of a clear declarative style. -
On Modeling Homotopy Type Theory in Higher Toposes
Review: model categories for type theory Left exact localizations Injective fibrations On modeling homotopy type theory in higher toposes Mike Shulman1 1(University of San Diego) Midwest homotopy type theory seminar Indiana University Bloomington March 9, 2019 Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Here we go Theorem Every Grothendieck (1; 1)-topos can be presented by a model category that interprets \Book" Homotopy Type Theory with: • Σ-types, a unit type, Π-types with function extensionality, and identity types. • Strict universes, closed under all the above type formers, and satisfying univalence and the propositional resizing axiom. Review: model categories for type theory Left exact localizations Injective fibrations Some caveats 1 Classical metatheory: ZFC with inaccessible cardinals. 2 Classical homotopy theory: simplicial sets. (It's not clear which cubical sets can even model the (1; 1)-topos of 1-groupoids.) 3 Will not mention \elementary (1; 1)-toposes" (though we can deduce partial results about them by Yoneda embedding). 4 Not the full \internal language hypothesis" that some \homotopy theory of type theories" is equivalent to the homotopy theory of some kind of (1; 1)-category. Only a unidirectional interpretation | in the useful direction! 5 We assume the initiality hypothesis: a \model of type theory" means a CwF. -
Irrelevance, Heterogeneous Equality, and Call-By-Value Dependent Type Systems
Irrelevance, Heterogeneous Equality, and Call-by-value Dependent Type Systems Vilhelm Sjoberg¨ Chris Casinghino Ki Yung Ahn University of Pennsylvania University of Pennsylvania Portland State University [email protected] [email protected] [email protected] Nathan Collins Harley D. Eades III Peng Fu Portland State University University of Iowa University of Iowa [email protected] [email protected] [email protected] Garrin Kimmell Tim Sheard Aaron Stump University of Iowa Portland State University University of Iowa [email protected] [email protected] [email protected] Stephanie Weirich University of Pennsylvania [email protected] We present a full-spectrum dependently typed core language which includes both nontermination and computational irrelevance (a.k.a. erasure), a combination which has not been studied before. The two features interact: to protect type safety we must be careful to only erase terminating expressions. Our language design is strongly influenced by the choice of CBV evaluation, and by our novel treatment of propositional equality which has a heterogeneous, completely erased elimination form. 1 Introduction The Trellys project is a collaborative effort to design a new dependently typed programming language. Our goal is to bridge the gap between ordinary functional programming and program verification with dependent types. Programmers should be able to port their existing functional programs to Trellys with minor modifications, and then gradually add more expressive types as appropriate. This goal has implications for our design. First, and most importantly, we must consider nonter- mination and other effects. Unlike Coq and Agda, functional programming languages like OCaml and Haskell allow general recursive functions, so to accept functional programs ‘as-is’ we must be able to turn off the termination checker. -
Lecture Notes on Types for Part II of the Computer Science Tripos
Q Lecture Notes on Types for Part II of the Computer Science Tripos Prof. Andrew M. Pitts University of Cambridge Computer Laboratory c 2016 A. M. Pitts Contents Learning Guide i 1 Introduction 1 2 ML Polymorphism 6 2.1 Mini-ML type system . 6 2.2 Examples of type inference, by hand . 14 2.3 Principal type schemes . 16 2.4 A type inference algorithm . 18 3 Polymorphic Reference Types 25 3.1 The problem . 25 3.2 Restoring type soundness . 30 4 Polymorphic Lambda Calculus 33 4.1 From type schemes to polymorphic types . 33 4.2 The Polymorphic Lambda Calculus (PLC) type system . 37 4.3 PLC type inference . 42 4.4 Datatypes in PLC . 43 4.5 Existential types . 50 5 Dependent Types 53 5.1 Dependent functions . 53 5.2 Pure Type Systems . 57 5.3 System Fw .............................................. 63 6 Propositions as Types 67 6.1 Intuitionistic logics . 67 6.2 Curry-Howard correspondence . 69 6.3 Calculus of Constructions, lC ................................... 73 6.4 Inductive types . 76 7 Further Topics 81 References 84 Learning Guide These notes and slides are designed to accompany 12 lectures on type systems for Part II of the Cambridge University Computer Science Tripos. The course builds on the techniques intro- duced in the Part IB course on Semantics of Programming Languages for specifying type systems for programming languages and reasoning about their properties. The emphasis here is on type systems for functional languages and their connection to constructive logic. We pay par- ticular attention to the notion of parametric polymorphism (also known as generics), both because it has proven useful in practice and because its theory is quite subtle. -
Certification of a Tool Chain for Deductive Program Verification Paolo Herms
Certification of a Tool Chain for Deductive Program Verification Paolo Herms To cite this version: Paolo Herms. Certification of a Tool Chain for Deductive Program Verification. Other [cs.OH]. Université Paris Sud - Paris XI, 2013. English. NNT : 2013PA112006. tel-00789543 HAL Id: tel-00789543 https://tel.archives-ouvertes.fr/tel-00789543 Submitted on 18 Feb 2013 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. UNIVERSITÉ DE PARIS-SUD École doctorale d’Informatique THÈSE présentée pour obtenir le Grade de Docteur en Sciences de l’Université Paris-Sud Discipline : Informatique PAR Paolo HERMS −! − SUJET : Certification of a Tool Chain for Deductive Program Verification soutenue le 14 janvier 2013 devant la commission d’examen MM. Roberto Di Cosmo Président du Jury Xavier Leroy Rapporteur Gilles Barthe Rapporteur Emmanuel Ledinot Examinateur Burkhart Wolff Examinateur Claude Marché Directeur de Thèse Benjamin Monate Co-directeur de Thèse Jean-François Monin Invité Résumé Cette thèse s’inscrit dans le domaine de la vérification du logiciel. Le but de la vérification du logiciel est d’assurer qu’une implémentation, un programme, répond aux exigences, satis- fait sa spécification. Cela est particulièrement important pour le logiciel critique, tel que des systèmes de contrôle d’avions, trains ou centrales électriques, où un mauvais fonctionnement pendant l’opération aurait des conséquences catastrophiques. -
Testing Noninterference, Quickly
Testing Noninterference, Quickly Cat˘ alin˘ Hrit¸cu1 John Hughes2 Benjamin C. Pierce1 Antal Spector-Zabusky1 Dimitrios Vytiniotis3 Arthur Azevedo de Amorim1 Leonidas Lampropoulos1 1University of Pennsylvania 2Chalmers University 3Microsoft Research Abstract To answer this question, we take as a case study the task of Information-flow control mechanisms are difficult to design and extending a simple abstract stack-and-pointer machine to track dy- labor intensive to prove correct. To reduce the time wasted on namic information flow and enforce termination-insensitive nonin- proof attempts doomed to fail due to broken definitions, we ad- terference [23]. Although our machine is simple, this exercise is vocate modern random testing techniques for finding counterexam- both nontrivial and novel. While simpler notions of dynamic taint ples during the design process. We show how to use QuickCheck, tracking are well studied for both high- and low-level languages, a property-based random-testing tool, to guide the design of a sim- it has only recently been shown [1, 24] that dynamic checks are ple information-flow abstract machine. We find that both sophis- capable of soundly enforcing strong security properties. Moreover, ticated strategies for generating well-distributed random programs sound dynamic IFC has been studied only in the context of lambda- and readily falsifiable formulations of noninterference properties calculi [1, 16, 25] and While programs [24]; the unstructured con- are critically important. We propose several approaches and eval- trol flow of a low-level machine poses additional challenges. (Test- uate their effectiveness on a collection of injected bugs of varying ing of static IFC mechanisms is left as future work.) subtlety. -
Inductive Types an Proof Assistants
Inductive Types Proof Assistants Homotopy Type Theory Radboud University Lecture 4: Inductive types an Proof Assistants H. Geuvers Radboud University Nijmegen, NL 21st Estonian Winter School in Computer Science Winter 2016 H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 1 / 58 Inductive Types Proof Assistants Homotopy Type Theory Radboud University Outline Inductive Types Proof Assistants Homotopy Type Theory H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 2 / 58 Inductive Types Proof Assistants Homotopy Type Theory Radboud University Curry-Howard-de Bruijn logic ∼ type theory formula ∼ type proof ∼ term detour elimination ∼ β-reduction proposition logic ∼ simply typed λ-calculus predicate logic ∼ dependently typed λ-calculus λP intuitionistic logic ∼ . + inductive types higher order logic ∼ . + higher types and polymorphism classical logic ∼ . + exceptions H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 3 / 58 Inductive Types Proof Assistants Homotopy Type Theory Radboud University Inductive types by examples from the Coq system • booleans • natural numbers • integers • pairs • linear lists • binary trees • logical operations H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 5 / 58 Inductive Types Proof Assistants Homotopy Type Theory Radboud University Inductive types • types • recursive functions • definition using pattern matching • ι-reduction = evaluation of recursive functions • recursion principle • proof by cases proof by induction • induction principle H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 6 / 58 Inductive Types Proof Assistants Homotopy Type Theory Radboud University Booleans: type Coq definition: Inductive bool : Set := | true : bool | false : bool. Coq produces: bool rec. bool rect. bool ind. H. Geuvers - Radboud Univ. EWSCS 2016 Typed λ-calculus 7 / 58 Inductive Types Proof Assistants Homotopy Type Theory Radboud University Booleans: elimination Check bool ind. -
An Anti-Locally-Nameless Approach to Formalizing Quantifiers Olivier Laurent
An Anti-Locally-Nameless Approach to Formalizing Quantifiers Olivier Laurent To cite this version: Olivier Laurent. An Anti-Locally-Nameless Approach to Formalizing Quantifiers. Certified Programs and Proofs, Jan 2021, virtual, Denmark. pp.300-312, 10.1145/3437992.3439926. hal-03096253 HAL Id: hal-03096253 https://hal.archives-ouvertes.fr/hal-03096253 Submitted on 4 Jan 2021 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. An Anti-Locally-Nameless Approach to Formalizing Quantifiers Olivier Laurent Univ Lyon, EnsL, UCBL, CNRS, LIP LYON, France [email protected] Abstract all cases have been covered. This works perfectly well for We investigate the possibility of formalizing quantifiers in various propositional systems, but as soon as (first-order) proof theory while avoiding, as far as possible, the use of quantifiers enter the picture, we have to face one ofthe true binding structures, α-equivalence or variable renam- nightmares of formalization: quantifiers are binders... The ings. We propose a solution with two kinds of variables in question of the formalization of binders does not yet have an 1 terms and formulas, as originally done by Gentzen. In this answer which makes perfect consensus .