An Introduction to Lean
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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. -
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. -
Chapter 5 Type Declarations
Ch.5: Type Declarations Plan Chapter 5 Type Declarations (Version of 27 September 2004) 1. Renaming existing types . 5.2 2. Enumeration types . 5.3 3. Constructed types . 5.5 4. Parameterised/polymorphic types . 5.10 5. Exceptions, revisited . 5.12 6. Application: expression evaluation . 5.13 °c P. Flener/IT Dept/Uppsala Univ. FP 5.1 Ch.5: Type Declarations 5.1. Renaming existing types 5.1. Renaming existing types Example: polynomials, revisited Representation of a polynomial by a list of integers: type poly = int list ² Introduction of an abbreviation for the type int list ² The two names denote the same type ² The object [4,2,8] is of type poly and of type int list - type poly = int list ; type poly = int list - type poly2 = int list ; type poly2 = int list - val p:poly = [1,2] ; val p = [1,2] : poly - val p2:poly2 = [1,2] ; val p2 = [1,2] : poly2 - p = p2 ; val it = true : bool °c P. Flener/IT Dept/Uppsala Univ. FP 5.2 Ch.5: Type Declarations 5.2. Enumeration types 5.2. Enumeration types Declaration of a new type having a finite number of values Example (weekend.sml) datatype months = Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec datatype days = Mon | Tue | Wed | Thu | Fri | Sat | Sun fun weekend Sat = true | weekend Sun = true | weekend d = false - datatype months = Jan | ... | Dec ; datatype months = Jan | ... | Dec - datatype days = Mon | ... | Sun ; datatype days = Mon | ... | Sun - fun weekend ... ; val weekend = fn : days -> bool °c P. Flener/IT Dept/Uppsala Univ. FP 5.3 Ch.5: Type Declarations 5.2. -
Nominal Wyvern: Employing Semantic Separation for Usability Yu Xiang Zhu CMU-CS-19-105 April 2019
Nominal Wyvern: Employing Semantic Separation for Usability Yu Xiang Zhu CMU-CS-19-105 April 2019 School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Thesis Committee: Jonathan Aldrich, Chair Heather Miller Alex Potanin, Victoria University of Wellington, NZ Submitted in partial fulfillment of the requirements for the degree of Master of Science. Copyright c 2019 Yu Xiang Zhu Keywords: Nominality, Wyvern, Dependent Object Types, Subtype Decidability Abstract This thesis presents Nominal Wyvern, a nominal type system that empha- sizes semantic separation for better usability. Nominal Wyvern is based on the dependent object types (DOT) calculus, which provides greater expressiv- ity than traditional object-oriented languages by incorporating concepts from functional languages. Although DOT is generally perceived to be nominal due to its path-dependent types, it is still a mostly structural system and relies on the free construction of types. This can present usability issues in a subtyping- based system where the semantics of a type are as important as its syntactic structure. Nominal Wyvern overcomes this problem by semantically separat- ing structural type/subtype definitions from ad hoc type refinements and type bound declarations. In doing so, Nominal Wyvern is also able to overcome the subtype undecidability problem of DOT by adopting a semantics-based sep- aration between types responsible for recursive subtype definitions and types that represent concrete data. The result is a more intuitive type system that achieves nominality and decidability while maintaining the expressiveness of F-bounded polymorphism that is used in practice. iv Acknowledgments This research would not have been possible without the support from the following in- dividuals. -
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. -
Cablelabs® Specifications Cablelabs' DHCP Options Registry CL-SP-CANN-DHCP-Reg-I13-160317
CableLabs® Specifications CableLabs' DHCP Options Registry CL-SP-CANN-DHCP-Reg-I13-160317 ISSUED Notice This CableLabs specification is the result of a cooperative effort undertaken at the direction of Cable Television Laboratories, Inc. for the benefit of the cable industry and its customers. You may download, copy, distribute, and reference the documents herein only for the purpose of developing products or services in accordance with such documents, and educational use. Except as granted by CableLabs in a separate written license agreement, no license is granted to modify the documents herein (except via the Engineering Change process), or to use, copy, modify or distribute the documents for any other purpose. This document may contain references to other documents not owned or controlled by CableLabs. Use and understanding of this document may require access to such other documents. Designing, manufacturing, distributing, using, selling, or servicing products, or providing services, based on this document may require intellectual property licenses from third parties for technology referenced in this document. To the extent this document contains or refers to documents of third parties, you agree to abide by the terms of any licenses associated with such third-party documents, including open source licenses, if any. Cable Television Laboratories, Inc. 2006-2016 CL-SP-CANN-DHCP-Reg-I13-160317 CableLabs® Specifications DISCLAIMER This document is furnished on an "AS IS" basis and neither CableLabs nor its members provides any representation or warranty, express or implied, regarding the accuracy, completeness, noninfringement, or fitness for a particular purpose of this document, or any document referenced herein. Any use or reliance on the information or opinion in this document is at the risk of the user, and CableLabs and its members shall not be liable for any damage or injury incurred by any person arising out of the completeness, accuracy, or utility of any information or opinion contained in the document. -
The Strength of Some Martin–Löf Type Theories
The Strength of Some Martin{LÄofType Theories Michael Rathjen¤y Abstract One objective of this paper is the determination of the proof{theoretic strength of Martin{ LÄof's type theory with a universe and the type of well{founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order 1 arithmetic, namely the one with ¢2 comprehension and bar induction. As Martin-LÄofintended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the proof- theoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts. Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman. 0 Introduction The intuitionistic theory of types as developed by Martin-LÄofis intended to be a system for formalizing intuitionistic mathematics together with an informal semantics, called \meaning expla- nation", which enables him to justify the rules of his type theory by showing their validity with respect to that semantics. Martin-LÄof's theory gives rise to a full scale philosophy of constructivism. It is a typed theory of constructions containing types which themselves depend on the constructions contained in previously constructed types. These dependent types enable one to express the general Cartesian product of the resulting families of types, as well as the disjoint union of such a family. -
Project Overview 1 Introduction 2 a Quick Introduction to SOOL
CMSC 22600 Compilers for Computer Languages Handout 1 Autumn 2016 October 6, 2016 Project Overview 1 Introduction The project for the course is to implement a small object-oriented programming language, called SOOL. The project will consist of four parts: 1. the parser and scanner, which coverts a textual representation of the program into a parse tree representation. 2. the type checker, which checks the parse tree for type correctness and produces a typed ab- stract syntax tree (AST) 3. the normalizer, which converts the typed AST representation into a static single-assignment (SSA) representation. 4. the code generator, which takes the SSA representation and generates LLVM assembly code. Each part of the project builds upon the previous parts, but we will provide reference solutions for previous parts. You will implement the project in the Standard ML programming language and submission of the project milestones will be managed using Phoenixforge. Important note: You are expected to submit code that compiles and that is well documented. Points for project code are assigned 30% for coding style (documentation, choice of variable names, and program structure), and 70% for correctness. Code that does not compile will not receive any points for correctness. 2 A quick introduction to SOOL SOOL is a statically-typed object-oriented language. It supports single inheritance with nominal subtyping and interfaces with structural subtyping. The SOOL version of the classic Hello World program is class main () { meth run () -> void { system.print ("hello world\n"); } } Here we are using the predefined system object to print the message. Every SOOL program has a main class that must have a run function. -
Formats and Protocols for Continuous Data CD-1.1
IDC Software documentation Formats and Protocols for Continuous Data CD-1.1 Document Version: 0.3 Document Edition: English Document Status: Final Document ID: IDC 3.4.3 Document Date: 18 December 2002 IDC Software documentation Formats and Protocols for Continuous Data CD-1.1 Version: 0.3 Notice This document was published December 2002 as Revision 0.3, using layout templates established by the IPT Group of CERN and IDC Corporate Style guidelines. This document is a re-casting of Revision 0.2published in December 2001 as part of the International Data Centre (IDC) Documentation. It was first published in July 2000 and was repub- lished electronically as Revision 0.1 in September 2001 and as Revision 0.2 in December 2001 to include minor changes (see the following Change Page for Revision 0.2). IDC documents that have been changed substantially are indicated by a whole revision number (for example, Revision 1). Trademarks IEEE is a registered trademark of the Institute of Electrical and Electronics Engineers, Inc. Motorola is a registered trademark of Motorola Inc. SAIC is a trademark of Science Applications International Corporation. Sun is a registered trademark of Sun Microsystems. UNIX is a registered trademark of UNIX System Labs, Inc. This document has been prepared using the Software Documentation Layout Templates that have been prepared by the IPT Group (Information, Process and Technology), IT Division, CERN (The European Laboratory for Particle Physics). For more information, go to http://framemaker.cern.ch/. page ii Final Abstract Version: 0.3 Abstract This document describes the formats and protocols of Continuous Data (CD-1.1), a standard to be used for transmitting continuous, time series data from stations of the International Monitoring System (IMS) to the International Data Centre (IDC) and for transmitting these data from the IDC to National Data Centers (NDCs). -
Parts 43 and 45 Technical Specifications
CFTC Technical Specification Parts 43 and 45 swap data reporting and public dissemination requirements September 17, 2020 Version 2.0 This Technical Specification document (“Tech Spec”) provides technical specifications for reporting swap data to Swap Data Repositories (SDRs) under parts 43 and 45. i TABLE OF CONTENTS INDEX OF DATA ELEMENTS ........................................................................................................................................................................................... III 1 INTRODUCTION ................................................................................................................................................................................................. VI 1.1 BACKGROUND .................................................................................................................................................................................................................. VI 1.2 STRUCTURE AND DESCRIPTION OF COLUMN HEADINGS ............................................................................................................................................................ VI 1.3 EXPLANATION OF DATA ELEMENT OR CATEGORY .................................................................................................................................................................... IX 1.4 UNIQUE PRODUCT IDENTIFIER (UPI) .................................................................................................................................................................................... -
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. -
Datatypes in Isabelle/HOL
Defining (Co)datatypes in Isabelle/HOL Jasmin Christian Blanchette, Lorenz Panny, Andrei Popescu, and Dmitriy Traytel Institut f¨urInformatik, Technische Universit¨atM¨unchen 5 December 2013 Abstract This tutorial describes how to use the new package for defining data- types and codatatypes in Isabelle/HOL. The package provides five main commands: datatype new, codatatype, primrec new, prim- corecursive, and primcorec. The commands suffixed by new are intended to subsume, and eventually replace, the corresponding com- mands from the old datatype package. Contents 1 Introduction3 2 Defining Datatypes5 2.1 Introductory Examples.....................5 2.1.1 Nonrecursive Types...................5 2.1.2 Simple Recursion.....................6 2.1.3 Mutual Recursion....................6 2.1.4 Nested Recursion.....................6 2.1.5 Custom Names and Syntaxes..............7 2.2 Command Syntax........................8 2.2.1 datatype new .....................8 2.2.2 datatype new compat ................ 11 2.3 Generated Constants....................... 12 2.4 Generated Theorems....................... 12 2.4.1 Free Constructor Theorems............... 13 1 CONTENTS 2 2.4.2 Functorial Theorems................... 15 2.4.3 Inductive Theorems................... 15 2.5 Compatibility Issues....................... 16 3 Defining Recursive Functions 17 3.1 Introductory Examples..................... 17 3.1.1 Nonrecursive Types................... 17 3.1.2 Simple Recursion..................... 17 3.1.3 Mutual Recursion.................... 18 3.1.4 Nested Recursion..................... 19 3.1.5 Nested-as-Mutual Recursion............... 20 3.2 Command Syntax........................ 20 3.2.1 primrec new ...................... 20 3.3 Recursive Default Values for Selectors............. 21 3.4 Compatibility Issues....................... 22 4 Defining Codatatypes 22 4.1 Introductory Examples..................... 22 4.1.1 Simple Corecursion................... 22 4.1.2 Mutual Corecursion..................