On Choice Rules in Dependent Type Theory
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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. -
Type Safety of Rewrite Rules in Dependent Types
Type safety of rewrite rules in dependent types Frédéric Blanqui Université Paris-Saclay, ENS Paris-Saclay, CNRS, Inria Laboratoire Spécification et Vérification, 94235, Cachan, France Abstract The expressiveness of dependent type theory can be extended by identifying types modulo some additional computation rules. But, for preserving the decidability of type-checking or the logical consistency of the system, one must make sure that those user-defined rewriting rules preserve typing. In this paper, we give a new method to check that property using Knuth-Bendix completion. 2012 ACM Subject Classification Theory of computation → Type theory; Theory of computation → Equational logic and rewriting; Theory of computation → Logic and verification Keywords and phrases subject-reduction, rewriting, dependent types Digital Object Identifier 10.4230/LIPIcs.FSCD.2020.11 Supplement Material https://github.com/wujuihsuan2016/lambdapi/tree/sr Acknowledgements The author thanks Jui-Hsuan Wu for his prototype implementation and his comments on a preliminary version of this work. 1 Introduction The λΠ-calculus, or LF [12], is an extension of the simply-typed λ-calculus with dependent types, that is, types that can depend on values like, for instance, the type V n of vectors of dimension n. And two dependent types like V n and V p are identified as soon as n and p are two expressions having the same value (modulo the evaluation rule of λ-calculus, β-reduction). In the λΠ-calculus modulo rewriting, function and type symbols can be defined not only by using β-reduction but also by using rewriting rules [19]. Hence, types are identified modulo β-reduction and some user-defined rewriting rules. -
A Certified Study of a Reversible Programming Language
A Certified Study of a Reversible Programming Language∗ Luca Paolini1, Mauro Piccolo2, and Luca Roversi3 1 Dipartimento di Informatica, Università degli Studi di Torino, corso Svizzera 185, 10149 Torino, Italy [email protected] 1 Dipartimento di Informatica, Università degli Studi di Torino, corso Svizzera 185, 10149 Torino, Italy [email protected] 1 Dipartimento di Informatica, Università degli Studi di Torino, corso Svizzera 185, 10149 Torino, Italy [email protected] Abstract We advance in the study of the semantics of Janus, a C-like reversible programming language. Our study makes utterly explicit some backward and forward evaluation symmetries. We want to deepen mathematical knowledge about the foundations and design principles of reversible computing and programming languages. We formalize a big-step operational semantics and a denotational semantics of Janus. We show a full abstraction result between the operational and denotational semantics. Last, we certify our results by means of the proof assistant Matita. 1998 ACM Subject Classification F.1.2 Computation by Abstract Devices: Modes of Compu- tation, F.3.2 Logics and Meanings of Programs: Semantics of Programming Languages Keywords and phrases reversible computing, Janus, operational semantics, bi-deterministic eval- uation, categorical semantics Digital Object Identifier 10.4230/LIPIcs.TYPES.2015.7 1 Introduction Reversible computing is an alternative form of computing: the isentropic core of classical computing [14] and quantum computing [21]. A classic computation is (forward-)deterministic, -
Performance Engineering of Proof-Based Software Systems at Scale by Jason S
Performance Engineering of Proof-Based Software Systems at Scale by Jason S. Gross B.S., Massachusetts Institute of Technology (2013) S.M., Massachusetts Institute of Technology (2015) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2021 © Massachusetts Institute of Technology 2021. All rights reserved. Author............................................................. Department of Electrical Engineering and Computer Science January 27, 2021 Certified by . Adam Chlipala Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by . Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 2 Performance Engineering of Proof-Based Software Systems at Scale by Jason S. Gross Submitted to the Department of Electrical Engineering and Computer Science on January 27, 2021, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Formal verification is increasingly valuable as our world comes to rely more onsoft- ware for critical infrastructure. A significant and understudied cost of developing mechanized proofs, especially at scale, is the computer performance of proof gen- eration. This dissertation aims to be a partial guide to identifying and resolving performance bottlenecks in dependently typed tactic-driven proof assistants like Coq. We present a survey of the landscape of performance issues in Coq, with micro- and macro-benchmarks. We describe various metrics that allow prediction of performance, such as term size, goal size, and number of binders, and note the occasional surprising lack of a bottleneck for some factors, such as total proof term size. -
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. -
A Survey of Engineering of Formally Verified Software
The version of record is available at: http://dx.doi.org/10.1561/2500000045 QED at Large: A Survey of Engineering of Formally Verified Software Talia Ringer Karl Palmskog University of Washington University of Texas at Austin [email protected] [email protected] Ilya Sergey Milos Gligoric Yale-NUS College University of Texas at Austin and [email protected] National University of Singapore [email protected] Zachary Tatlock University of Washington [email protected] arXiv:2003.06458v1 [cs.LO] 13 Mar 2020 Errata for the present version of this paper may be found online at https://proofengineering.org/qed_errata.html The version of record is available at: http://dx.doi.org/10.1561/2500000045 Contents 1 Introduction 103 1.1 Challenges at Scale . 104 1.2 Scope: Domain and Literature . 105 1.3 Overview . 106 1.4 Reading Guide . 106 2 Proof Engineering by Example 108 3 Why Proof Engineering Matters 111 3.1 Proof Engineering for Program Verification . 112 3.2 Proof Engineering for Other Domains . 117 3.3 Practical Impact . 124 4 Foundations and Trusted Bases 126 4.1 Proof Assistant Pre-History . 126 4.2 Proof Assistant Early History . 129 4.3 Proof Assistant Foundations . 130 4.4 Trusted Computing Bases of Proofs and Programs . 137 5 Between the Engineer and the Kernel: Languages and Automation 141 5.1 Styles of Automation . 142 5.2 Automation in Practice . 156 The version of record is available at: http://dx.doi.org/10.1561/2500000045 6 Proof Organization and Scalability 162 6.1 Property Specification and Encodings . -
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. -
Analyzing Individual Proofs As the Basis of Interoperability Between Proof Systems
Analyzing Individual Proofs as the Basis of Interoperability between Proof Systems Gilles Dowek Inria and Ecole´ normale sup´erieure de Paris-Saclay, 61, avenue du Pr´esident Wilson, 94235 Cachan Cedex, France [email protected] We describe the first results of a project of analyzing in which theories formal proofs can be ex- pressed. We use this analysis as the basis of interoperability between proof systems. 1 Introduction Sciences study both individual objects and generic ones. For example, Astronomy studies both the individual planets of the Solar system: Mercury, Venus, etc. determining their radius, mass, composition, etc., but also the motion of generic planets: Kepler’s laws, that do not just apply to the six planets known at the time of Kepler, but also to those that have been discovered after, and those that may be discovered in the future. Computer science studies both algorithms that apply to generic data, but also specific pieces of data. Mathematics mostly studies generic objects, but sometimes also specific ones, such as the number π or the function ζ. Proof theory mostly studies generic proofs. For example, Gentzen’s cut elimination theorem for Predicate logic applies to any proof expressed in Predicate logic, those that were known at the time of Gentzen, those that have been constructed after, and those that will be constructed in the future. Much less effort is dedicated to studying the individual mathematical proofs, with a few exceptions, for example [29]. Considering the proofs that we have, instead of all those that we may build in some logic, sometimes changes the perspective.