DOCSLIB.ORG

Reporte Técnico RT 10-01 Representation of Metamodels

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Reporte Técnico RT 10-01 Representation of Metamodels PEDECIBA Informática Instituto de Computación – Facultad de Ingeniería Universidad de la República Montevideo, Uruguay Reporte Técnico RT 10-01 Representation of metamodels using inductive types in a Type-Theoretic Framework for MDE Daniel Calegari Carlos Luna Nora Szasz Alvaro Tasistro 2010 Representation of metamodels using inductive types in a Type-Theoretic Framework for MDE Calegari, Daniel; Luna, Carlos; Szasz, Nora; Tasistro, Alvaro ISSN 0797-6410 Reporte Técnico RT 10-01 PEDECIBA Instituto de Computación – Facultad de Ingeniería Universidad de la República Montevideo, Uruguay, 2010 Representation of Metamodels using Inductive Types in a Type-Theoretic Framework for MDE Daniel Calegari∗, Carlos Lunay, Nora Szaszy, Alvaro´ Tasistroy ∗Instituto de Computacion,´ Universidad de la Republica,´ Uruguay Email: dcalegar@fing.edu.uy yFacultad de Ingenier´ıa, Universidad ORT Uruguay Email: fluna,szasz,[email protected] Abstract—We present discussions on how to apply a type- (type) and satisfying a certain (pre-)condition, an output theoretic framework –composed out by the Calculus of In- model exists which conforms to a given metamodel and ductive Constructions and its associated tool the Coq proof which stands in a certain relation with the input model. assistant– to the formal treatment of model transformations in the context of Model-Driven Engineering. We start by studying Proofs of such propositions in the CIC are constructive and how to represent models and metamodels in the mentioned therefore it is possible to automatically extract from each theory, which leads us to a formalization in which a metamodel of them a provably correct (functional) program computing is a collection of mutually defined inductive types representing the output model from any appropriate input. Alternatively, its various classes and associations. This representation has one can specify the transformation by asserting and proving been put into use for carrying out and verifying on machine the well-known case study of the Class to Relational model directly the existence of such a function. These are thus two transformation. We finally end up discussing ways in which ways in which it is possible to obtain certified zero-fault the framework can be used to obtain provably correct model transformations. transformations. The aim of this paper is to discuss how to represent Keywords-model-driven engineering, model transformations, models and metamodels in the CIC. This leads to the correctness, constructive type theory consideration of entities (e.g. classes) that mutually reference each other (e.g. through associations) and their possible I. INTRODUCTION representation by means of co-inductive or, alternatively, This paper reports ongoing work on the application of a mutually inductive types, which is the approach that we type-theoretic framework to the formal treatment of model eventually favor. This is done in section III below, after transformations in the context of Model-Driven Engineering having provided a brief outlook on the CIC in Section II. (MDE) [1]. Model transformations take as input models We have carried out our method on the well-known Class conforming to a given source metamodel and produce output to Relational model transformation [5], which we have fully models conforming to a given target metamodel. That is, developed in Coq. The corresponding code is available at [6]. metamodels specify collections of models and can thus Finally, in section IV, besides exposing some conclusions, be equated to model languages. In turn, they conform to we discuss several ways in which the type-theoretic frame- metametamodels, which can be understood as specifying work can be used either for the development of provably model languages and usually conform just to themselves. correct transformations or for the verification of pre-existing We explore the idea of using the Calculus of Inductive ones, which leads to the formulation of further work. Constructions (CIC) [2], [3] as a technical space for repre- The idea of using type theory in the context of MDE has senting and dealing with the schema above and obtaining been formulated before by Poernomo in [7], [8]. Poernomo provably correct model transformations. The choice of the formulates a type theory of his own –a variant of Martin- CIC is dictated by its very considerable expressive power Lof’s¨ constructive type theory– and outlines the methods as well as by the fact that it is supported by tools of for representing MOF [10] models as types as well as industrial strength, e.g. the Coq proof assistant [3], [4]. specifications of transformations as 89 formulæ. However, More specifically, the CIC is a type theory with dependent he requires the use of co-inductive types to treat cyclic types which allows to write logical formulæ of higher order references within metamodels, which we here avoid. An- about objects of inductive and functional types as well as other difference from Poernomo’s work is that we have fully about potentially infinite structures of co-inductive types. implemented a significant case study on machine, thanks to Within this framework, metamodels are represented as types, the choice of the CIC and the Coq tool. whilst model transformation can be specified for example Other works that propose type systems of their own by means of logical formulæ of 89 form, i.e. assertions that for object models are [11], [12], but they are oriented for every input model conforming to a given metamodel towards defining executable transformation languages and not towards reasoning about models and transformations. In inductive types, by giving the corresponding proof construc- contrast, [13], [14] explore the use of Maude (an executable tors. The type of propositions is called Prop. The logical rewriting logic language) as a formal notation for describing constants are defined accordingly, which means in particu- models and metamodels, and proving certain properties lar that each universally quantified proposition 8x:A:P (x) about them. In these works references are represented by is a type of functions taking objects a of type A into means of object identifiers, thus introducing a level of proofs of P (a), whereas existentially quantified propositions indirection. Our representation, which consists of the mutual 9x:A:P (x) are types of pairs formed by objects a of type definition of inductive types, allows for the use of induction A and proofs of P (a). In view of the latter, a formula of 89 and recursion for the purpose of navigation. Also, an im- form, i.e. 8x:AjP (x) 9y:BjQ(x; y) is proven by a function portant difference with these works is that our framework that maps input objects of type A satisfying precondition P is oriented towards the specification and formal verification to output of type B which stands in the relation Q with the of model transformations, whereas this is not considered in input, together with the proof that this latter condition holds. the former works. Then such formulæ constitute a natural way of expressing program specifications. From each proof of one such formula II. THE CALCULUS OF INDUCTIVE CONSTRUCTIONS it is possible to mechanically extract a function in the The Calculus of Inductive Constructions is a type theory, sense of ordinary functional programming languages which i.e. in brief, a higher order logic in which the individuals operates only on the input and output data and leaves the are classified into a hierarchy of types. The types work proof of the condition Q aside. The resulting program, say very much as in strongly typed functional programming f, remains provably correct with respect to the original languages which means that, to begin with, there are basic specification, i.e. for all a:A satisfying P , Q(a; f(a)) holds. elementary types, recursive types defined by induction like lists and trees (called inductive types) and function types. III. FORMALIZATION OF METAMODELS AND MODELS An example of inductive type is given by the following This section introduces the method used to describe meta- definition of the lists of elements of (parametric) type A, models and models within the CIC. Although the approach which we give in Coq notation (data types are called “Sets” does not tie itself to a specific metamodel specification in CIC): language, we use, for the sake of presentation, a simplified Inductive list : Set := version of the MOF [10] equivalent to the Kernel MetaMeta- | nil : list Model (KM3) [15]. As to the CIC, we use the syntax of the | cons : A -> list -> list. Coq proof assistant [4]. The type is defined by its constructors, in this case We use the well-known Class to Relational model trans- nil: list A and cons : A -> list A -> list A formation [5] as an example, actually in the simplified and it is understood that its elements are obtained as finite version found in the QVT Specification [16]. The transfor- combinations of the constructors. Well-founded recursion for mation describes how persistent classes of a simple UML these types is available via the Fixpoint operator. class diagram are mapped to tables of a RDBMS model. A (dependent) record type is a non-recursive inductive Here we shall focus on the representation of metamodels type with a single constructor and projection functions for and their models rather than on the transformation itself. each field of the type. Nevertheless, one version of the complete transformation When the requirement of finiteness is removed we obtain developed following our approach can be found at [6]. the possibility of defining infinite structures, as in In Figure 1 we show a metamodel of UML class diagrams. Classes can contain one or more attributes and can belong CoInductive Stream : Set := to a class hierarchy. Each attribute has a type that can Cons : A -> Stream -> Stream. be another class or a primitive datatype (string, boolean, The possibly infinite elements of coinductive types are integer, etc.).
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • An Introduction to Lean
    An Introduction to Lean Jeremy Avigad Leonardo de Moura Gabriel Ebner and Sebastian Ullrich Version 1fc176a, updated at 2017-01-09 14:16:26 -0500 2 Contents Contents 3 1 Overview 5 1.1 Perspectives on Lean ............................... 5 1.2 Where To Go From Here ............................. 12 2 Defining Objects in Lean 13 2.1 Some Basic Types ................................ 14 2.2 Defining Functions ................................ 17 2.3 Defining New Types ............................... 20 2.4 Records and Structures ............................. 22 2.5 Nonconstructive Definitions ........................... 25 3 Programming in Lean 27 3.1 Evaluating Expressions .............................. 28 3.2 Recursive Definitions ............................... 30 3.3 Inhabited Types, Subtypes, and Option Types . 32 3.4 Monads ...................................... 34 3.5 Input and Output ................................ 35 3.6 An Example: Abstract Syntax ......................... 36 4 Theorem Proving in Lean 38 4.1 Assertions in Dependent Type Theory ..................... 38 4.2 Propositions as Types .............................. 39 4.3 Induction and Calculation ............................ 42 4.4 Axioms ...................................... 45 5 Using Automation in Lean 46 6 Metaprogramming in Lean 47 3 CONTENTS 4 Bibliography 48 1 Overview This introduction offers a tour of Lean and its features, with a number of examples foryou to play around with and explore. If you are reading this in our online tutorial system, you can run examples like the one below by clicking the button that says “try it yourself.” #check "hello world!" The response from Lean appears in the small window underneath the editor text, and also in popup windows that you can read when you hover over the indicators in the left margin. Alternatively, if you have installed Lean and have it running in a stand-alone editor, you can copy and paste examples and try them there.
    [Show full text]
  • 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).
    [Show full text]
  • 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) ....................................................................................................................................................................................
    [Show full text]
  • 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.
    [Show full text]