Quotient Inductive-Inductive Types
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Pragmatic Quotient Types in Coq Cyril Cohen
Pragmatic Quotient Types in Coq Cyril Cohen To cite this version: Cyril Cohen. Pragmatic Quotient Types in Coq. International Conference on Interactive Theorem Proving, Jul 2013, Rennes, France. pp.16. hal-01966714 HAL Id: hal-01966714 https://hal.inria.fr/hal-01966714 Submitted on 29 Dec 2018 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. Pragmatic Quotient Types in Coq Cyril Cohen Department of Computer Science and Engineering University of Gothenburg [email protected] Abstract. In intensional type theory, it is not always possible to form the quotient of a type by an equivalence relation. However, quotients are extremely useful when formalizing mathematics, especially in algebra. We provide a Coq library with a pragmatic approach in two complemen- tary components. First, we provide a framework to work with quotient types in an axiomatic manner. Second, we program construction mecha- nisms for some specific cases where it is possible to build a quotient type. This library was helpful in implementing the types of rational fractions, multivariate polynomials, field extensions and real algebraic numbers. Keywords: Quotient types, Formalization of mathematics, Coq Introduction In set-based mathematics, given some base set S and an equivalence ≡, one may see the quotient (S/ ≡) as the partition {π(x) | x ∈ S} of S into the sets π(x) ˆ= {y ∈ S | x ≡ y}, which are called equivalence classes. -
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. -
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. -
Arithmetic in MIPS
6 Arithmetic in MIPS Objectives After completing this lab you will: • know how to do integer arithmetic in MIPS • know how to do floating point arithmetic in MIPS • know about conversion from integer to floating point and from floating point to integer. Introduction The definition of the R2000 architecture includes all integer arithmetic within the actual CPU. Floating point arithmetic is done by one of the four possible coprocessors, namely coprocessor number 1. Integer arithmetic Addition and subtraction are performed on 32 bit numbers held in the general purpose registers (32 bit each). The result is a 32 bit number itself. Two kinds of instructions are included in the instruction set to do integer addition and subtraction: • instructions for signed arithmetic: the 32 bit numbers are considered to be represented in 2’s comple- ment. The execution of the instruction (addition or subtraction) may generate an overflow. • instructions for unsigned arithmetic: the 32 bit numbers are considered to be in standard binary repre- sentation. Executing the instruction will never generate an overflow error even if there is an actual overflow (the result cannot be represented with 32 bits). Multiplication and division may generate results that are larger than 32 bits. The architecture provides two special 32 bit registers that are the destination for multiplication and division instructions. These registers are called hi and lo as to indicate that they hold the higher 32 bits of the result and the lower 32 bits respectively. Special instructions are also provided to move data from these registers into the general purpose ones ($0 to $31). -
Quotient Types: a Modular Approach
Quotient Types: A Modular Approach Aleksey Nogin Department of Computer Science Cornell University, Ithaca, NY 14853 [email protected] Abstract. In this paper we introduce a new approach to axiomatizing quotient types in type theory. We suggest replacing the existing mono- lithic rule set by a modular set of rules for a specially chosen set of primitive operations. This modular formalization of quotient types turns out to be much easier to use and free of many limitations of the tradi- tional monolithic formalization. To illustrate the advantages of the new approach, we show how the type of collections (that is known to be very hard to formalize using traditional quotient types) can be naturally for- malized using the new primitives. We also show how modularity allows us to reuse one of the new primitives to simplify and enhance the rules for the set types. 1 Introduction NuPRL type theory differs from most other type theories used in theorem provers in its treatment of equality. In Coq’s Calculus of Constructions, for example, there is a single global equality relation which is not the desired one for many types (e.g. function types). The desired equalities have to be handled explicitly, which is quite burdensome. As in Martin-L¨of type theory [20] (of which NuPRL type theory is an extension), in NuPRL each type comes with its own equality relation (the extensional one in the case of functions), and the typing rules guar- antee that well–typed terms respect these equalities. Semantically, a quotient in NuPRL is trivial to define: it is simply a type with a new equality. -
Mathematics for Computer Scientists (G51MCS) Lecture Notes Autumn 2005
Mathematics for Computer Scientists (G51MCS) Lecture notes Autumn 2005 Thorsten Altenkirch October 26, 2005 Contents 1 1 Types and sets What is a set? Wikipedia 1 says Informally, a set is just a collection of objects considered as a whole. And what is a type? Wikipedia offers different definitions for different subject areas, but if we look up datatype in Computer Science we learn: In computer science, a datatype (often simply type) is a name or label for a set of values and some operations which can be performed on that set of values. Here the meaning of type refers back to the meaning of sets, but it is hard to see a substantial difference. We may conclude that what is called a set in Mathematics is called a type in Computer Science. Here, we will adopt the computer science terminology and use the term type. But remember to translate this to set when talking to a Mathematician. There is a good reason for this choice: I am going to use the type system of the programming language Haskell to illustrate many important constructions on types and to implement mathematical constructions on a computer. Please note that this course is not an introduction to Haskell, this is done in the 2nd semester in G51FUN. If you want to find out more about Haskell now, I refer to Graham Hutton’s excellent, forthcoming book, whose first seven chapters are available from his web page 2 . The book is going to be published soon and is the course text for G51FUN. Another good source for information about Haskell is the Haskell home page 3 While we are going to use Haskell to illustrate mathematical concepts, Haskell is also a very useful language for lots of other things, from designing user interfaces to controlling robots. -
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. -
Quotient Types — a Modular Approach *,**
Quotient Types — a Modular Approach ?;?? Aleksey Nogin Department of Computer Science Cornell University Ithaca, NY 14853 E-mail: [email protected] Web: http://nogin.org/ Abstract. In this paper we introduce a new approach to defining quo- tient types in type theory. We suggest replacing the existing monolithic rule set by a modular set of rules for a specially chosen set of primitive operations. This modular formalization of quotient types turns out to be very powerful and free of many limitations of the traditional monolithic formalization. To illustrate the advantages of the new formalization, we show how the type of collections (that is known to be very hard to formal- ize using traditional quotient types) can be naturally formalized using the new primitives. We also show how modularity allows us to reuse one of the new primitives to simplify and enhance the rules for the set types. 1 Introduction NuPRL type theory differs from most other type theories used in theorem provers in its treatment of equality. In Coq’s Calculus of Constructions, for example, there is a single global equality relation which is not the desired one for many types (e.g. function types). The desired equalities have to be handled explicitly, which is quite burdensome. As in Martin-L¨oftype theory [17] (of which NuPRL type theory is an extension), in NuPRL each type comes with its own equality relation (the extensional one in the case of functions), and the typing rules guar- antee that well–typed terms respect these equalities. Semantically, a quotient in NuPRL is trivial to define: it is simply a type with a new equality. -
Quotient Types
Quotient Types Peter V. Homeier U. S. Department of Defense, [email protected] http://www.cis.upenn.edu/~homeier Abstract. The quotient operation is a standard feature of set theory, where a set is divided into subsets by an equivalence relation; the re- sulting subsets of equivalent elements are called equivalence classes. We reinterpret this idea for Higher Order Logic (HOL), where types are di- vided by an equivalence relation to create new types, called quotient types. We present a tool for the Higher Order Logic theorem prover to mechanically construct quotient types as new types in the HOL logic, and also to de¯ne functions that map between the original type and the quotient type. Several properties of these mapping functions are proven automatically. Tools are also provided to create quotients of aggregate types, in particular lists, pairs, and sums. These require special atten- tion to preserve the aggregate structure across the quotient operation. We demonstrate these tools on an example from the sigma calculus. 1 Quotient Sets Quotient sets are a standard construction of set theory. They have found wide application in many areas of mathematics, including algebra and logic. The following description is abstracted from [3]. A binary relation on S is an equivalence relation if it is reflexive, symmetric, and transitive. » reflexive: x S: x x symmetric: 8x;2 y S: x » y y x transitive: 8x; y;2 z S: x » y ) y »z x z 8 2 » ^ » ) » Let be an equivalence relation. Then the equivalence class of x (modulo )is » » de¯ned as [x] def= y x y . -
Structured Formal Development with Quotient Types in Isabelle/HOL
Structured Formal Development with Quotient Types in Isabelle/HOL Maksym Bortin1 and Christoph Lüth2 1 Universität Bremen, Department of Mathematics and Computer Science [email protected] 2 Deutsches Forschungszentrum für Künstliche Intelligenz, Bremen [email protected] Abstract. General purpose theorem provers provide sophisticated proof methods, but lack some of the advanced structuring mechanisms found in specification languages. This paper builds on previous work extending the theorem prover Isabelle with such mechanisms. A way to build the quotient type over a given base type and an equivalence relation on it, and a generalised notion of folding over quotiented types is given as a formalised high-level step called a design tactic. The core of this paper are four axiomatic theories capturing the design tactic. The applicability is demonstrated by derivations of implementations for finite multisets and finite sets from lists in Isabelle. 1 Introduction Formal development of correct systems requires considerable design and proof effort in order to establish that an implementation meets the required specifi- cation. General purpose theorem provers provide powerful proof methods, but often lack the advanced structuring and design concepts found in specification languages, such as design tactics [20]. A design tactic represents formalised de- velopment knowledge. It is an abstract development pattern proven correct once and for all, saving proof effort when applying it and guiding the development process. If theorem provers can be extended with similar concepts without loss of consistency, the development process can be structured within the prover. This paper is a step in this direction. Building on previous work [2] where an approach to the extension of the theorem prover Isabelle [15] with theory morphisms has been described, the contributions of this paper are the representation of the well-known type quotienting construction and its extension with a generalised notion of folding over the quotiented type as a design tactic. -
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.