Encoding a Dependent-Type Lambda-Calculus in a Logic Programming Language
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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. -
Category Theory and Lambda Calculus
Category Theory and Lambda Calculus Mario Román García Trabajo Fin de Grado Doble Grado en Ingeniería Informática y Matemáticas Tutores Pedro A. García-Sánchez Manuel Bullejos Lorenzo Facultad de Ciencias E.T.S. Ingenierías Informática y de Telecomunicación Granada, a 18 de junio de 2018 Contents 1 Lambda calculus 13 1.1 Untyped λ-calculus . 13 1.1.1 Untyped λ-calculus . 14 1.1.2 Free and bound variables, substitution . 14 1.1.3 Alpha equivalence . 16 1.1.4 Beta reduction . 16 1.1.5 Eta reduction . 17 1.1.6 Confluence . 17 1.1.7 The Church-Rosser theorem . 18 1.1.8 Normalization . 20 1.1.9 Standardization and evaluation strategies . 21 1.1.10 SKI combinators . 22 1.1.11 Turing completeness . 24 1.2 Simply typed λ-calculus . 24 1.2.1 Simple types . 25 1.2.2 Typing rules for simply typed λ-calculus . 25 1.2.3 Curry-style types . 26 1.2.4 Unification and type inference . 27 1.2.5 Subject reduction and normalization . 29 1.3 The Curry-Howard correspondence . 31 1.3.1 Extending the simply typed λ-calculus . 31 1.3.2 Natural deduction . 32 1.3.3 Propositions as types . 34 1.4 Other type systems . 35 1.4.1 λ-cube..................................... 35 2 Mikrokosmos 38 2.1 Implementation of λ-expressions . 38 2.1.1 The Haskell programming language . 38 2.1.2 De Bruijn indexes . 40 2.1.3 Substitution . 41 2.1.4 De Bruijn-terms and λ-terms . 42 2.1.5 Evaluation . -
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. -
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. -
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. -
Reasoning in an ∞-Topos with Homotopy Type Theory
Reasoning in an 1-topos with homotopy type theory Dan Christensen University of Western Ontario 1-Topos Institute, April 1, 2021 Outline: Introduction to homotopy type theory Examples of applications to 1-toposes 1 / 23 2 / 23 Motivation for Homotopy Type Theory People study type theory for many reasons. I'll highlight two: Its intrinsic homotopical/topological content. Things we prove in type theory are true in any 1-topos. Its suitability for computer formalization. This talk will focus on the first point. So, what is an 1-topos? 3 / 23 1-toposes An 1-category has objects, (1-)morphisms between objects, 2-morphisms between 1-morphisms, etc., with all k-morphisms invertible for k > 1. Alternatively: a category enriched over spaces. The prototypical example is the 1-category of spaces. An 1-topos is an 1-category C that shares many properties of the 1-category of spaces: C is presentable. In particular, C is cocomplete. C satisfies descent. In particular, C is locally cartesian closed. Examples include Spaces, slice categories such as Spaces/S1, and presheaves of Spaces, such as G-Spaces for a group G. If C is an 1-topos, so is every slice category C=X. 4 / 23 History of (Homotopy) Type Theory Dependent type theory was introduced in the 1970's by Per Martin-L¨of,building on work of Russell, Church and others. In 2006, Awodey, Warren, and Voevodsky discovered that dependent type theory has homotopical models, extending 1998 work of Hofmann and Streicher. At around this time, Voevodsky discovered his univalance axiom. -
Dependent Types: Easy As PIE
Dependent Types: Easy as PIE Work-In-Progress Project Description Dimitrios Vytiniotis and Stephanie Weirich University of Pennsylvania Abstract Dependent type systems allow for a rich set of program properties to be expressed and mechanically verified via type checking. However, despite their significant expres- sive power, dependent types have not yet advanced into mainstream programming languages. We believe the reason behind this omission is the large design space for dependently typed functional programming languages, and the consequent lack of ex- perience in dependently-typed programming and language implementations. In this newly-started project, we lay out the design considerations for a general-purpose, ef- fectful, functional, dependently-typed language, called PIE. The goal of this project is to promote dependently-typed programming to a mainstream practice. 1 INTRODUCTION The goal of static type systems is to identify programs that contain errors. The type systems of functional languages, such as ML and Haskell, have been very successful in this respect. However, these systems can ensure only relatively weak safety properties—there is a wide range of semantic errors that these systems can- not eliminate. As a result, there is growing consensus [13, 15, 28, 27, 26, 6, 7, 20, 21, 16, 11, 2, 25, 19, 1, 22] that the right way to make static type systems more expressive in this respect is to adopt ideas from dependent type theory [12]. Types in dependent type systems can carry significant information about program values. For example, a datatype for network packets may describe the internal byte align- ment of variable-length subfields. In such precise type systems, programmers can express many of complex properties about programs—e.g. -
Constructive Set Theory July, 2008, Mathlogaps Workshop, Manchester
Constructive Set Theory July, 2008, Mathlogaps workshop, Manchester . Peter Aczel [email protected] Manchester University Constructive Set Theory – p.1/88 Plan of lectures Lecture 1 1: Background to CST 2: The axiom system CZF Lecture 2 3: The number systems in CZF 4: The constructive notion of set Lectures 3,4 ? 5: Inductive definitions 6: Locales and/or 7: Coinductive definitions Constructive Set Theory – p.2/88 1: Background to CST Constructive Set Theory – p.3/88 Some brands of constructive mathematics B1: Intuitionism (Brouwer, Heyting, ..., Veldman) B2: ‘Russian’ constructivism (Markov,...) B3: ‘American’ constructivism (Bishop, Bridges,...) B4: ‘European’ constructivism (Martin-Löf, Sambin,...) B1,B2 contradict classical mathematics; e.g. B1 : All functions R → R are continuous, B2 : All functions N → N are recursive (i.e. CT). B3 is compatible with each of classical maths, B1,B2 and forms their common core. B4 is a more philosophical foundational approach to B3. All B1-B4 accept RDC and so DC and CC. Constructive Set Theory – p.4/88 Some liberal brands of mathematics using intuitionistic logic B5: Topos mathematics (Lawvere, Johnstone,...) B6: Liberal Intuitionism (Mayberry,...) B5 does not use any choice principles. B6 accepts Restricted EM. B7: A minimalist, non-ideological approach: The aim is to do as much mainstream constructive mathematics as possible in a weak framework that is common to all brands, and explore the variety of possible extensions. Constructive Set Theory – p.5/88 Some settings for constructive mathematics type theoretical category theoretical set theoretical Constructive Set Theory – p.6/88 Some contrasts classical logic versus intuitionistic logic impredicative versus predicative some choice versus no choice intensional versus extensional consistent with EM versus inconsistent with EM Constructive Set Theory – p.7/88 Mathematical Taboos A mathematical taboo is a statement that we may not want to assume false, but we definately do not want to be able to prove. -
Collection Principles in Dependent Type Theory
This is a repository copy of Collection Principles in Dependent Type Theory. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/113157/ Version: Accepted Version Proceedings Paper: Azcel, P and Gambino, N orcid.org/0000-0002-4257-3590 (2002) Collection Principles in Dependent Type Theory. In: Lecture Notes in Computer Science: TYPES 2000. International Workshop on Types for Proofs and Programs (TYPES) 2000, 08-12 Dec 2000, Durham, UK. Springer Verlag , pp. 1-23. ISBN 3-540-43287-6 © 2002, Springer-Verlag Berlin Heidelberg. This is an author produced version of a conference paper published in Lecture Notes in Computer Science: Types for Proofs and Programs. Uploaded in accordance with the publisher's self-archiving policy. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Collection Principles in Dependent Type Theory⋆ Peter Aczel1 and Nicola Gambino2 1 Departments of Mathematics and Computer Science, University of Manchester, e-mail: [email protected] 2 Department of Computer Science, University of Manchester, e-mail: [email protected] Abstract. -
A Concurrent Logical Framework I: Judgments and Properties
A concurrent logical framework I: Judgments and properties Kevin Watkins Iliano Cervesato Frank Pfenning David Walker March 2002, revised May 2003 CMU-CS-02-101 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 This research was sponsored in part by the National Science Foundation under the title “Meta-Logical Frameworks,” grants CCR-9619584 and CCR-9988281; and in part by NRL under grant N00173-00-C- 2086. Iliano Cervesato was partially supported by NRL under contract N00173-00-C-2086 and by NSF grant INT98-15731. David Walker was partially supported by the Office of Naval Research under the title “Efficient Logics for Network Security,” grant N00014-01-1-0432; by DARPA under the title “Scaling Proof- Carrying Code to Production Compilers and Security Policies,” Contract No. F30603-99-1-0519; and by Microsoft Research. A part of this work was completed while David Walker was a post-doctoral fellow at Carnegie Mellon University. Abstract The Concurrent Logical Framework, or CLF, is a new logical framework in which concurrent computations can be represented as monadic objects, for which there is an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous connectives ⊗,1,!,and∃of intuitionistic linear logic, encapsulated in a monad. LLF is itself a conservative extension of LF with the asynchronous connectives −◦ ,&and>. The present report, the first of two technical reports describing CLF, presents the frame- work itself and its meta-theory. A novel, algorithmic formulation of the underlying type theory concentrating on canonical forms leads to a simple notion of definitional equality for concurrent computations in which the order of independent steps cannot be distinguished.