The Calculus of Inductive Constructions
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Introduction to the Calculus of Inductive Constructions Christine Paulin-Mohring
Introduction to the Calculus of Inductive Constructions Christine Paulin-Mohring To cite this version: Christine Paulin-Mohring. Introduction to the Calculus of Inductive Constructions. Bruno Woltzen- logel Paleo; David Delahaye. All about Proofs, Proofs for All, 55, College Publications, 2015, Studies in Logic (Mathematical logic and foundations), 978-1-84890-166-7. hal-01094195 HAL Id: hal-01094195 https://hal.inria.fr/hal-01094195 Submitted on 11 Dec 2014 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. Introduction to the Calculus of Inductive Constructions Christine Paulin-Mohring1 LRI, Univ Paris-Sud, CNRS and INRIA Saclay - ˆIle-de-France, Toccata, Orsay F-91405 [email protected] 1 Introduction The Calculus of Inductive Constructions (CIC) is the formalism behind the interactive proof assis- tant Coq [24, 5]. It is a powerful language which aims at representing both functional programs in the style of the ML language and proofs in higher-order logic. Many data-structures can be repre- sented in this language: usual data-types like lists and binary trees (possibly polymorphic) but also infinitely branching trees. At the logical level, inductive definitions give a natural representation of notions like reachability and operational semantics defined using inference rules. -
A Core Language for Dependently Typed Programming
DRAFT ΠΣ: A Core Language for Dependently Typed Programming Thorsten Altenkirch Nicolas Oury University of Nottingham {txa,npo}@cs.nott.ac.uk Abstract recursion and hence is, without further restrictions, unsuitable as We introduce ΠΣ, a core language for dependently typed program- a logical system. It is, after all, intended to be primarily a core ming. Our intention is that ΠΣ should play the role extensions of language for programming, not for reasoning. Apart from Σ- and System F are playing for conventional functional languages with Π-types our language has only a few more, namely: polymorphism, like Haskell. The core language incorporates mu- Type : Type This is the simplest choice, the most general form of tual dependent recursive definitions, Type : Type, Π- and Σ-types, impredicative polymorphism possible. It is avoided in systems finite sets of labels and explicit constraints. We show that standard used for reasoning because it destroys logical consistency. We constructions in dependently typed programming can be easily en- don’t care because we have lost consistency already by allowing coded in our language. We address some important issues: having recursion. an equality checker which unfolds recursion only when needed, avoiding looping when typechecking sensible programs; the sim- Finite types A finite type is given by a collection of labels, e.g. { , } plification of type checking for eliminators like case by using equa- true false to define the type of Booleans. Our labels are tional constraints, allowing the flexible use of case expressions a special class and can be reused, opening the scope for a within dependently typed programming and the representation of hereditary definition of subtyping. -
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. -
A Calculus of Constructions with Explicit Subtyping Ali Assaf
A calculus of constructions with explicit subtyping Ali Assaf To cite this version: Ali Assaf. A calculus of constructions with explicit subtyping. 2014. hal-01097401v1 HAL Id: hal-01097401 https://hal.archives-ouvertes.fr/hal-01097401v1 Preprint submitted on 19 Dec 2014 (v1), last revised 14 Jan 2016 (v2) 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. A calculus of constructions with explicit subtyping Ali Assaf September 16, 2014 Abstract The calculus of constructions can be extended with an infinite hierar- chy of universes and cumulative subtyping. In this hierarchy, each uni- verse is contained in a higher universe. Subtyping is usually left implicit in the typing rules. We present an alternative version of the calculus of constructions where subtyping is explicit. This new system avoids prob- lems related to coercions and dependent types by using the Tarski style of universes and by introducing additional equations to reflect equality. 1 Introduction The predicative calculus of inductive constructions (PCIC), the theory behind the Coq proof system [15], contains an infinite hierarchy of predicative universes Type0 : Type1 : Type2 : ... and an impredicative universe Prop : Type1 for propositions, together with a cumulativity relation: Prop ⊆ Type0 ⊆ Type1 ⊆ Type2 ⊆ ... -
Type Theory and Applications
Type Theory and Applications Harley Eades [email protected] 1 Introduction There are two major problems growing in two areas. The first is in Computer Science, in particular software engineering. Software is becoming more and more complex, and hence more susceptible to software defects. Software bugs have two critical repercussions: they cost companies lots of money and time to fix, and they have the potential to cause harm. The National Institute of Standards and Technology estimated that software errors cost the United State's economy approximately sixty billion dollars annually, while the Federal Bureau of Investigations estimated in a 2005 report that software bugs cost U.S. companies approximately sixty-seven billion a year [90, 108]. Software bugs have the potential to cause harm. In 2010 there were a approximately a hundred reports made to the National Highway Traffic Safety Administration of potential problems with the braking system of the 2010 Toyota Prius [17]. The problem was that the anti-lock braking system would experience a \short delay" when the brakes where pressed by the driver of the vehicle [106]. This actually caused some crashes. Toyota found that this short delay was the result of a software bug, and was able to repair the the vehicles using a software update [91]. Another incident where substantial harm was caused was in 2002 where two planes collided over Uberlingen¨ in Germany. A cargo plane operated by DHL collided with a passenger flight holding fifty-one passengers. Air-traffic control did not notice the intersecting traffic until less than a minute before the collision occurred. -
Equality Is Typable in Semi-Full Pure Type Systems
Equality is typable in Semi-Full Pure Type Systems Vincent Siles Hugo Herbelin Ecole Polytechnique / INRIA / Laboratoire PPS INRIA / Laboratoire PPS Equipe πr2 Equipe πr2 23 avenue d’Italie, 75013 Paris, France 23 avenue d’Italie, 75013 Paris, France [email protected] [email protected] Abstract—There are two usual ways to describe equality in Surprisingly, showing the equivalence between those two a dependent typing system, one that uses an external notion definitions is difficult. Geuvers and Werner [7] early noticed of computation like beta-reduction, and one that introduces a that being able to lift an untyped equality to a typed one, typed judgement of beta-equality directly in the typing system. After being an open problem for some time, the general i.e. to turn a system with β-conversion into a system with equivalence between both approaches has been solved by judgmental equality requires to show Subject Reduction in Adams for a class of pure type systems (PTSs) called functional. the latter system: In this paper, we relax the functionality constraint and prove the equivalence for all semi-full PTSs by combining the ideas If Γ `e M : A and M β N then Γ `e M = N : A. of Adams with a study of the general shape of types in PTSs. As one application, an extension of this result to systems Subject Reduction requires the injectivity of dependent A with sub-typing would be a first step toward bringing closer products Πx :B : the theory behind a proof assistant such as Coq to its A C implementation. -
Polymorphism All the Way Up! from System F to the Calculus of Constructions
Programming = proving? The Curry-Howard correspondence today Second lecture Polymorphism all the way up! From System F to the Calculus of Constructions Xavier Leroy College` de France 2018-11-21 Curry-Howard in 1970 An isomorphism between simply-typed λ-calculus and intuitionistic logic that connects types and propositions; terms and proofs; reductions and cut elimination. This second lecture shows how: This correspondence extends to more expressive type systems and to more powerful logics. This correspondence inspired formal systems that are simultaneously a logic and a programming language (Martin-Lof¨ type theory, Calculus of Constructions, Pure Type Systems). 2 I Polymorphism and second-order logic Static typing vs. genericity Static typing with simple types (as in simply-typed λ-calculus but also as in Algol, Pascal, etc) sometimes forces us to duplicate code. Example A sorting algorithm applies to any list list(t) of elements of type t, provided it also receives then function t ! t ! bool that compares two elements of type t. With simple types, to sort lists of integers and lists of strings, we need two functions with dierent types, even if they implement the same algorithm: sort list int :(int ! int ! bool) ! list(int) ! list(int) sort list string :(string ! string ! bool) ! list(string) ! list(string) 4 Static typing vs. genericity There is a tension between static typing on the one hand and reusable implementations of generic algorithms on the other hand. Some languages elect to weaken static typing, e.g. by introducing a universal type “any” or “?” with run-time type checking, or even by turning typing o: void qsort(void * base, size_t nmemb, size_t size, int (*compar)(const void *, const void *)); Instead, polymorphic typing extends the algebra of types and the typing rules so as to give a precise type to a generic function. -
Investigation on the Typing of Equality in Type Systems Vincent Siles
Investigation on the typing of equality in type systems Vincent Siles To cite this version: Vincent Siles. Investigation on the typing of equality in type systems. Logic in Computer Science [cs.LO]. Ecole Polytechnique X, 2010. English. pastel-00556578 HAL Id: pastel-00556578 https://pastel.archives-ouvertes.fr/pastel-00556578 Submitted on 17 Jan 2011 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. Ecole´ Polytechnique Th`ese de Doctorat Sp´ecialit´eInformatique Investigation on the typing of equality in type systems Pr´esent´ee et soutenue publiquement par Vincent Siles le 25 Novembre 2010 devant le jury compos´ede Bruno Barras Co-directeur de th`ese Gilles Barthe Rapporteur Roberto Di Cosmo Herman Geuvers Rapporteur Hugo Herbelin Co-directeur de th`ese Alexandre Miquel Rapporteur Randy Pollack Benjamin Werner Abstract Pure Type Systems are a good way to factorize the questions of meta-theory about a large family of type systems. They have been introduced as a gener- alization of Barendregt’s λ-cube, an abstraction of several type systems like the Simply Typed λ-Calculus, System F or the Calculus of Constructions. One critical detail of the Pure Type Systems is their conversion rule that allows to do computation at the level of types. -
A Type System for Certified Runtime Type Analysis
A Type System For Certified Runtime Type Analysis A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Bratin Saha Dissertation Director: Professor Zhong Shao December 2002 Abstract A Type System For Certified Runtime Type Analysis Bratin Saha 2002 Modern programming paradigms are increasingly giving rise to applications that require type information at runtime. For example, services like garbage collection, marshalling, and serializa- tion need to analyze type information at runtime. When compiling code which uses runtime type inspections, most existing compilers use untyped intermediate languages and discard type informa- tion at an early stage. Unfortunately, such an approach is incompatible with type-based certifying compilation. A certifying compiler generates not only the code but also a proof that the code obeys a security policy. Therefore, one need not trust the correctness of a compiler generating certified code, instead one can verify the correctness of the generated code. This allows a code consumer to accept code from untrusted sources which is specially advantageous in a networked environment. In practice, most certifying compilers use a type system for generating and encoding the proofs. These systems are called type-based certifying compilers. This dissertation describes a type system that can be used for supporting runtime type analysis in type-based certifying compilers. The type system has two novel features. First, type analysis can be applied to the type of any runtime value. In particular quantified types such as polymorphic and existential types can also be analyzed, yet type-checking remains decidable. -
Notes: Barendregt's Cube and Programming with Dependent Types
Notes: Barendregt's cube and programming with dependent types Eric Lu I. ANNOTATED BIBLIOGRAPHY A. Lambda Calculi with Types @incollection{barendregt:1992 author = {Barendregt, Henk}, title = {Lambda Calculi with Types}, booktitle = {Handbook of Logic in Computer Science}, publisher = {Oxford University Press}, year = {1992}, editor = {Abramsky, S. and Gabbay, D.M. and Maibaum, T.S.E.}, volume = {2}, pages = {117--309}, address = {New York, New York} } Summary: As setup for introducing the lambda cube, Barendregt introduces the type-free lambda calculus and reviews some of its properties. Subsequently he presents accounts of typed lambda calculi from Curry and Church, including some Curry-style systems not belonging to the lambda cube (λU and λµ). He then describes the lambda cube construction that was first noted by Barendregt in 1991. The lambda cube describes an inclusion relation amongst eight typed lambda calculi. Moreover it explains a fine structure for the calculus of constructions arising from the presence or absence of three axiomatic additions to the simply-typed lambda calculus. The axes of the lambda cube are intuitively justified by the notion of dependency. After introducing the lambda cube, Barendregt gives properties of the languages belonging to the lambda cube, as well as some properties of pure type systems, which generalize the means of description of the axioms of the systems in the lambda cube. Evaluation: This work originates as a reference from the logical/mathematical community, but its content was important in providing a firm foundation for types in programming languages. The lambda cube explains prior work on a number of type systems by observing a taxonomy and relating them to logical systems via the Curry-Howard isomorphism. -
Basics of Type Theory and Coq
Basics of type theory and Coq Michael Shulman January 31, 2012 1 / 77 Type-theoretic foundations Set theory Type theory Logic Types ^; _; ); :; 8; 9 ×; +; !; Q; P Sets Logic ×; +; !; Q; P ^; _; ); :; 8; 9 x 2 A is a proposition x : A is a typing judgment 2 / 77 Type theory is programming For now, think of type theory as a programming language. • Closely related to functional programming languages like ML, Haskell, Lisp, Scheme. • More expressive and powerful. • Can manipulate “mathematical objects”. 4 / 77 Typing judgments Type theory consists of rules for manipulating judgments. The most important judgment is a typing judgment: x1 : A1; x2 : A2;::: xn : An ` b : B The turnstile ` binds most loosely, followed by commas. This should be read as: In the context of variables x1 of type A1, x2 of type A2,..., and xn of type An, the expression b has type B. Examples ` 0: N x : N; y : N ` x + y : N f : R ! R; x : R ` f (x): R 1 (n) 1 f : C (R; R); n: N ` f : C (R; R) 5 / 77 Type constructors The basic rules tell us how to construct valid typing judgments, i.e. how to write programs with given input and output types. This includes: 1 How to construct new types (judgments Γ ` A: Type). 2 How to construct terms of these types. 3 How to use such terms to construct terms of other types. Example (Function types) 1 If A: Type and B : Type, then A ! B : Type. 2 If x : A ` b : B, then λxA:b : A ! B. -
Interactive Theorem Proving in Coq and the Curry-Howard Isomorphism
Interactive Theorem Proving in Coq and the Curry-Howard Isomorphism Abhishek Kr Singh January 7, 2015 Abstract There seems to be a general consensus among mathematicians about the notion of a correct proof. Still, in mathematical literature, many invalid proofs remain accepted over a long period of time. It happens mostly because proofs are incomplete, and it is easy to make mistake while verifying an incomplete proof. However, writing a complete proof on paper consumes a lot of time and effort. Proof Assistants, such as Coq [8, 2], can minimize these overheads. It provides the user with lots of tools and tactics to interactively develop a proof. Most of the routine jobs, such as creation of proof-terms and type verification, are done internally by the proof assistant. Realization of such a computer based tool for representing proofs and propositions, is made possible because of the Curry- Howard Isomorphism [6]. It says that, proving and programming are essentially equivalent tasks. More precisely, we can use a programming paradigm, such as typed lambda calculus [1], to encode propositions and their proofs. This report tries to discuss curry-howard isomorphism at different levels. In particular, it discusses different features of Calculus of Inductive Constructions [3, 9], that makes it possible to encode different logics in Coq. Contents 1 Introduction 1 1.1 Proof Assistants . 1 1.2 The Coq Proof assistant . 2 1.3 An overview of the report . 2 2 Terms, Computations and Proofs in Coq 3 2.1 Terms and Types . 3 2.2 Computations . 5 2.3 Propositions and Proofs .