Decidable Subtyping for Path Dependent Types
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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. -
Design and Implementation of Generics for the .NET Common Language Runtime
Design and Implementation of Generics for the .NET Common Language Runtime Andrew Kennedy Don Syme Microsoft Research, Cambridge, U.K. fakeÒÒ¸d×ÝÑeg@ÑicÖÓ×ÓfغcÓÑ Abstract cally through an interface definition language, or IDL) that is nec- essary for language interoperation. The Microsoft .NET Common Language Runtime provides a This paper describes the design and implementation of support shared type system, intermediate language and dynamic execution for parametric polymorphism in the CLR. In its initial release, the environment for the implementation and inter-operation of multiple CLR has no support for polymorphism, an omission shared by the source languages. In this paper we extend it with direct support for JVM. Of course, it is always possible to “compile away” polymor- parametric polymorphism (also known as generics), describing the phism by translation, as has been demonstrated in a number of ex- design through examples written in an extended version of the C# tensions to Java [14, 4, 6, 13, 2, 16] that require no change to the programming language, and explaining aspects of implementation JVM, and in compilers for polymorphic languages that target the by reference to a prototype extension to the runtime. JVM or CLR (MLj [3], Haskell, Eiffel, Mercury). However, such Our design is very expressive, supporting parameterized types, systems inevitably suffer drawbacks of some kind, whether through polymorphic static, instance and virtual methods, “F-bounded” source language restrictions (disallowing primitive type instanti- type parameters, instantiation at pointer and value types, polymor- ations to enable a simple erasure-based translation, as in GJ and phic recursion, and exact run-time types. -
G22.2110-003 Programming Languages - Fall 2012 Week 14 - Part 1
G22.2110-003 Programming Languages - Fall 2012 Week 14 - Part 1 Thomas Wies New York University Review Last lecture I Exceptions Outline Today: I Generic Programming Sources for today's lecture: I PLP, ch. 8.4 I Programming in Scala, ch. 19, 20.6 Generic programming Subroutines provide a way to abstract over values. Generic programming lets us abstract over types. Examples: I A sorting algorithm has the same structure, regardless of the types being sorted I Stack primitives have the same semantics, regardless of the objects stored on the stack. One common use: I algorithms on containers: updating, iteration, search Language models: I C: macros (textual substitution) or unsafe casts I Ada: generic units and instantiations I C++, Java, C#, Scala: generics (also called templates) I ML: parametric polymorphism, functors Parameterizing software components Construct Parameter(s): array bounds, element type subprogram values (arguments) Ada generic package values, types, packages Ada generic subprogram values, types C++ class template values, types C++ function template values, types Java generic classes Scala generic types (and implicit values) ML function values (including other functions) ML type constructor types ML functor values, types, structures Templates in C++ template <typename T> class Array { public : explicit Array (size_t); // constructor T& operator[] (size_t); // subscript operator ... // other operations private : ... // a size and a pointer to an array }; Array<int> V1(100); // instantiation Array<int> V2; // use default constructor -
Proof-Assistants Using Dependent Type Systems
CHAPTER 18 Proof-Assistants Using Dependent Type Systems Henk Barendregt Herman Geuvers Contents I Proof checking 1151 2 Type-theoretic notions for proof checking 1153 2.1 Proof checking mathematical statements 1153 2.2 Propositions as types 1156 2.3 Examples of proofs as terms 1157 2.4 Intermezzo: Logical frameworks. 1160 2.5 Functions: algorithms versus graphs 1164 2.6 Subject Reduction . 1166 2.7 Conversion and Computation 1166 2.8 Equality . 1168 2.9 Connection between logic and type theory 1175 3 Type systems for proof checking 1180 3. l Higher order predicate logic . 1181 3.2 Higher order typed A-calculus . 1185 3.3 Pure Type Systems 1196 3.4 Properties of P ure Type Systems . 1199 3.5 Extensions of Pure Type Systems 1202 3.6 Products and Sums 1202 3.7 E-typcs 1204 3.8 Inductive Types 1206 4 Proof-development in type systems 1211 4.1 Tactics 1212 4.2 Examples of Proof Development 1214 4.3 Autarkic Computations 1220 5 P roof assistants 1223 5.1 Comparing proof-assistants . 1224 5.2 Applications of proof-assistants 1228 Bibliography 1230 Index 1235 Name index 1238 HANDBOOK OF AUTOMAT8D REASONING Edited by Alan Robinson and Andrei Voronkov © 2001 Elsevier Science Publishers 8.V. All rights reserved PROOF-ASSISTANTS USING DEPENDENT TYPE SYSTEMS 1151 I. Proof checking Proof checking consists of the automated verification of mathematical theories by first fully formalizing the underlying primitive notions, the definitions, the axioms and the proofs. Then the definitions are checked for their well-formedness and the proofs for their correctness, all this within a given logic. -
Advanced-Java.Pdf
Advanced java i Advanced java Advanced java ii Contents 1 How to create and destroy objects 1 1.1 Introduction......................................................1 1.2 Instance Construction.................................................1 1.2.1 Implicit (Generated) Constructor.......................................1 1.2.2 Constructors without Arguments.......................................1 1.2.3 Constructors with Arguments........................................2 1.2.4 Initialization Blocks.............................................2 1.2.5 Construction guarantee............................................3 1.2.6 Visibility...................................................4 1.2.7 Garbage collection..............................................4 1.2.8 Finalizers...................................................5 1.3 Static initialization..................................................5 1.4 Construction Patterns.................................................5 1.4.1 Singleton...................................................6 1.4.2 Utility/Helper Class.............................................7 1.4.3 Factory....................................................7 1.4.4 Dependency Injection............................................8 1.5 Download the Source Code..............................................9 1.6 What’s next......................................................9 2 Using methods common to all objects 10 2.1 Introduction...................................................... 10 2.2 Methods equals and hashCode........................................... -
Java Generics Adoption: How New Features Are Introduced, Championed, Or Ignored
Java Generics Adoption: How New Features are Introduced, Championed, or Ignored Chris Parnin Christian Bird Emerson Murphy-Hill College of Computing Microsoft Research Dept. of Computer Science Georgia Institute of Redmond, Washington North Carolina State Technology [email protected] University Atlanta, Georgia Raleigh, North Carolina [email protected] [email protected] Far too often, greatly heralded claims and visions of new language features fail to hold or persist in practice. Discus- ABSTRACT sions of the costs and benefits of language features can easily devolve into a religious war with both sides armed with little Support for generic programming was added to the Java more than anecdotes [13]. Empirical evidence about the language in 2004, representing perhaps the most significant adoption and use of past language features should inform change to one of the most widely used programming lan- and encourage a more rational discussion when designing guages today. Researchers and language designers antici- language features and considering how they should be de- pated this addition would relieve many long-standing prob- ployed. Collecting this evidence is not just sensible but a lems plaguing developers, but surprisingly, no one has yet responsibility of our community. measured whether generics actually provide such relief. In In this paper, we examine the adoption and use of generics, this paper, we report on the first empirical investigation into which were introduced into the Java language in 2004. When how Java generics have been integrated into open source Sun introduced generics, they claimed that the language software by automatically mining the history of 20 popular feature was \a long-awaited enhancement to the type system" open source Java programs, traversing more than 500 million that \eliminates the drudgery of casting." Sun recommended lines of code in the process. -
Proof Theory of Constructive Systems: Inductive Types and Univalence
Proof Theory of Constructive Systems: Inductive Types and Univalence Michael Rathjen Department of Pure Mathematics University of Leeds Leeds LS2 9JT, England [email protected] Abstract In Feferman’s work, explicit mathematics and theories of generalized inductive definitions play a cen- tral role. One objective of this article is to describe the connections with Martin-L¨of type theory and constructive Zermelo-Fraenkel set theory. Proof theory has contributed to a deeper grasp of the rela- tionship between different frameworks for constructive mathematics. Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky’s univalence axiom. A further goal is to investigate the strength of intuitionistic theories of generalized inductive definitions in the framework of intuitionistic explicit mathematics that lie beyond the reach of Martin-L¨of type theory. Key words: Explicit mathematics, constructive Zermelo-Fraenkel set theory, Martin-L¨of type theory, univalence axiom, proof-theoretic strength MSC 03F30 03F50 03C62 1 Introduction Intuitionistic systems of inductive definitions have figured prominently in Solomon Feferman’s program of reducing classical subsystems of analysis and theories of iterated inductive definitions to constructive theories of various kinds. In the special case of classical theories of finitely as well as transfinitely iterated inductive definitions, where the iteration occurs along a computable well-ordering, the program was mainly completed by Buchholz, Pohlers, and Sieg more than 30 years ago (see [13, 19]). For stronger theories of inductive 1 i definitions such as those based on Feferman’s intutitionic Explicit Mathematics (T0) some answers have been provided in the last 10 years while some questions are still open. -
Addressing Common Crosscutting Problems with Arcum∗
Addressing Common Crosscutting Problems with Arcum∗ Macneil Shonle William G. Griswold Sorin Lerner Computer Science & Engineering, UC San Diego La Jolla, CA 92093-0404 {mshonle, wgg, lerner}@cs.ucsd.edu ABSTRACT 1. INTRODUCTION Crosscutting is an inherent part of software development and can Arcum is a framework for declaring and performing user-defined typically be managed through modularization: A module’s stable program checks and transformations, with the goal of increasing properties are defined in an interface while its likely-to-change automated refactoring opportunities for the user [21]. By using Ar- properties are encapsulated within the module [19]. The cross- cum, a programmer can view the implementation of a crosscutting cutting of the stable properties, such as class and method names, design idiom as a form of module. Arcum uses a declarative lan- can be mitigated with automated refactoring tools that allow, for guage to describe the idiom’s implementation, where descriptions example, the interface’s elements to be renamed [9, 18]. However, are composed of Arcum interface and Arcum option constructs. An often the crosscutting from design idioms (such as design patterns option describes one possible implementation of a crosscutting de- and coding styles) are so specific to the program’s domain that sign idiom, and a set of options are related to each other when they their crosscutting would not likely have been anticipated by the all implement the same Arcum interface. developers of an automated refactoring system. Arcum’s declarative language uses a Java-like syntax for first- The Arcum plug-in for Eclipse enables programmers to describe order logic predicate statements, including a special pattern nota- the implementation of a crosscutting design idiom as a set of syn- tion for expressing Java code. -
CS 6110 S18 Lecture 32 Dependent Types 1 Typing Lists with Lengths
CS 6110 S18 Lecture 32 Dependent Types When we added kinding last time, part of the motivation was to complement the functions from types to terms that we had in System F with functions from types to types. Of course, all of these languages have plain functions from terms to terms. So it's natural to wonder what would happen if we added functions from values to types. The result is a language with \compile-time" types that can depend on \run-time" values. While this arrangement might seem paradoxical, it is a very powerful way to express the correctness of programs; and via the propositions-as-types principle, it also serves as the foundation for a modern crop of interactive theorem provers. The language feature is called dependent types (i.e., types depend on terms). Prominent dependently-typed languages include Coq, Nuprl, Agda, Lean, F*, and Idris. Some of these languages, like Coq and Nuprl, are more oriented toward proving things using propositions as types, and others, like F* and Idris, are more oriented toward writing \normal programs" with strong correctness guarantees. 1 Typing Lists with Lengths Dependent types can help avoid out-of-bounds errors by encoding the lengths of arrays as part of their type. Consider a plain recursive IList type that represents a list of any length. Using type operators, we might use a general type List that can be instantiated as List int, so its kind would be type ) type. But let's use a fixed element type for now. With dependent types, however, we can make IList a type constructor that takes a natural number as an argument, so IList n is a list of length n. -
REVERSE GENERICS Parametrization After the Fact
REVERSE GENERICS Parametrization after the Fact Alexandre Bergel PLEIAD Laboratory, Computer Science Department (DCC), University of Chile, Santiago, Chile Lorenzo Bettini Dipartimento di Informatica, Universit`adi Torino, Italy Keywords: Generic programming, Java generics, C++ templates. Abstract: By abstracting over types, generic programming enables one to write code that is independent from specific data type implementation. This style is supported by most mainstream languages, including C++ with tem- plates and Java with generics. If some code is not designed in a generic way from the start, a major effort is required to convert this code to use generic types. This conversion is manually realized which is known to be tedious and error-prone. We propose Reverse Generics, a general linguistic mechanism to define a generic class from a non-generic class. For a given set of types, a generic is formed by unbinding static dependencies contained in these types. This generalization and generic type instantiation may be done incrementally. This paper studies the possible application of this linguistic mechanism to C++ and Java and, in particular, it reviews limitations of Java generics against our proposal. 1 INTRODUCTION more than 100 down-casts and up-casts and 70 uses of instanceof. This examination reveals that in many The concept of generic programming (Dos Reis and places the amount of up-casting subsequent down- J¨arvi, 2005), which has characterized functional pro- casting that is used almost makes the programs be- gramming for several decades, appeared in main- have like dynamically typed code. stream programming object-oriented languages such Note, that the need to make existing code generic as C++, only in the late 80s, where it motivated from may arise also in languages where generic types were the beginning the design of the Standard Template Li- already available.