Formalising Mathematics in Simple Type Theory
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The Common HOL Platform
The Common HOL Platform Mark Adams Proof Technologies Ltd, UK Radboud University, Nijmegen, The Netherlands The Common HOL project aims to facilitate porting source code and proofs between members of the HOL family of theorem provers. At the heart of the project is the Common HOL Platform, which defines a standard HOL theory and API that aims to be compatible with all HOL systems. So far, HOL Light and hol90 have been adapted for conformance, and HOL Zero was originally developed to conform. In this paper we provide motivation for a platform, give an overview of the Common HOL Platform’s theory and API components, and show how to adapt legacy systems. We also report on the platform’s successful application in the hand-translation of a few thousand lines of source code from HOL Light to HOL Zero. 1 Introduction The HOL family of theorem provers started in the 1980s with HOL88 [5], and has since grown to include many systems, most prominently HOL4 [16], HOL Light [8], ProofPower HOL [3] and Is- abelle/HOL [12]. These four main systems have developed their own advanced proof facilities and extensive theory libraries, and have been successfully employed in major projects in the verification of critical hardware and software [1, 11] and the formalisation of mathematics [7]. It would clearly be of benefit if these systems could “talk” to each other, specifically if theory, proofs and source code could be exchanged in a relatively seamless manner. This would reduce the considerable duplication of effort otherwise required for one system to benefit from the major projects and advanced capabilities developed on another. -
Enhancement of Properties in Mizar
Enhancement of properties in Mizar Artur Korniªowicz Institute of Computer Science, University of Bialystok, Bialystok, Poland ABSTRACT A ``property'' in the Mizar proof-assistant is a construction that can be used to register chosen features of predicates (e.g., ``reflexivity'', ``symmetry''), operations (e.g., ``involutiveness'', ``commutativity'') and types (e.g., ``sethoodness'') declared at the definition stage. The current implementation of Mizar allows using properties for notions with a specific number of visible arguments (e.g., reflexivity for a predicate with two visible arguments and involutiveness for an operation with just one visible argument). In this paper we investigate a more general approach to overcome these limitations. We propose an extension of the Mizar language and a corresponding enhancement of the Mizar proof-checker which allow declaring properties of notions of arbitrary arity with respect to explicitly indicated arguments. Moreover, we introduce a new property—the ``fixedpoint-free'' property of unary operations—meaning that the result of applying the operation to its argument always differs from the argument. Results of tests conducted on the Mizar Mathematical Library are presented. Subjects Digital Libraries, Theory and Formal Methods, Programming Languages Keywords Formal verification, Mizar proof-assistant, Mizar Mathematical Library INTRODUCTION Classical mathematical papers consist of sequences of definitions and justified facts classified into several categories like: theorems, lemmas, corollaries, and so on, often interspersed with some examples and descriptions. Submitted 25 June 2020 Mathematical documents prepared using various proof-assistants (e.g., Isabelle (Isabelle, Accepted 29 October 2020 2020), HOL Light (HOL Light, 2020), Coq (Coq, 2020), Metamath (Metamath, 2020), Published 30 November 2020 Lean (Lean, 2020), and Mizar (Mizar, 2020)) can also contain other constructions that are Corresponding author Artur Korniªowicz, processable by dedicated software. -
Interactive Theorem Proving (ITP) Course Web Version
Interactive Theorem Proving (ITP) Course Web Version Thomas Tuerk ([email protected]) Except where otherwise noted, this work is licened under Creative Commons Attribution-ShareAlike 4.0 International License version 3c35cc2 of Mon Nov 11 10:22:43 2019 Contents Part I Preface 3 Part II Introduction 6 Part III HOL 4 History and Architecture 16 Part IV HOL's Logic 24 Part V Basic HOL Usage 38 Part VI Forward Proofs 46 Part VII Backward Proofs 61 Part VIII Basic Tactics 83 Part IX Induction Proofs 110 Part X Basic Definitions 119 Part XI Good Definitions 163 Part XII Deep and Shallow Embeddings 184 Part XIII Rewriting 192 Part XIV Advanced Definition Principles 246 Part XV Maintainable Proofs 267 Part XVI Overview of HOL 4 289 Part XVII Other Interactive Theorem Provers 302 2 / 315 Part I Preface Except where otherwise noted, this work is licened under Creative Commons Attribution-ShareAlike 4.0 International License. Preface these slides originate from a course for advanced master students it was given by the PROSPER group at KTH in Stockholm in 2017 (see https://www.kth.se/social/group/interactive-theorem-) the course focused on how to use HOL 4 students taking the course were expected to I know functional programming, esp. SML I understand predicate logic I have some experience with pen and paper proofs the course consisted of 9 lectures, which each took 90 minutes there were 19 supervised practical sessions, which each took 2 h usually there was 1 lecture and 2 practical sessions each week students were expected to work about 10 h each -
HOL-Testgen Version 1.8 USER GUIDE
HOL-TestGen Version 1.8 USER GUIDE Achim Brucker, Lukas Brügger, Abderrahmane Feliachi, Chantal Keller, Matthias Krieger, Delphine Longuet, Yakoub Nemouchi, Frédéric Tuong, Burkhart Wolff To cite this version: Achim Brucker, Lukas Brügger, Abderrahmane Feliachi, Chantal Keller, Matthias Krieger, et al.. HOL-TestGen Version 1.8 USER GUIDE. [Technical Report] Univeristé Paris-Saclay; LRI - CNRS, University Paris-Sud. 2016. hal-01765526 HAL Id: hal-01765526 https://hal.inria.fr/hal-01765526 Submitted on 12 Apr 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. R L R I A P P O R HOL-TestGen Version 1.8 USER GUIDE T BRUCKER A D / BRUGGER L / FELIACHI A / KELLER C / KRIEGER M P / LONGUET D / NEMOUCHI Y / TUONG F / WOLFF B D Unité Mixte de Recherche 8623 E CNRS-Université Paris Sud-LRI 04/2016 R Rapport de Recherche N° 1586 E C H E R C CNRS – Université de Paris Sud H Centre d’Orsay LABORATOIRE DE RECHERCHE EN INFORMATIQUE E Bâtiment 650 91405 ORSAY Cedex (France) HOL-TestGen 1.8.0 User Guide http://www.brucker.ch/projects/hol-testgen/ Achim D. Brucker Lukas Brügger Abderrahmane Feliachi Chantal Keller Matthias P. -
Translating Scala Programs to Isabelle/HOL (System
Translating Scala Programs to Isabelle/HOL System Description Lars Hupel1 and Viktor Kuncak2 1 Technische Universität München 2 École Polytechnique Fédérale de Lausanne (EPFL) Abstract. We present a trustworthy connection between the Leon verification system and the Isabelle proof assistant. Leon is a system for verifying functional Scala programs. It uses a variety of automated theorem provers (ATPs) to check verification conditions (VCs) stemming from the input program. Isabelle, on the other hand, is an interactive theorem prover used to verify mathematical spec- ifications using its own input language Isabelle/Isar. Users specify (inductive) definitions and write proofs about them manually, albeit with the help of semi- automated tactics. The integration of these two systems allows us to exploit Is- abelle’s rich standard library and give greater confidence guarantees in the cor- rectness of analysed programs. Keywords: Isabelle, HOL, Scala, Leon, compiler 1 Introduction This system description presents a new tool that aims to connect two important worlds: the world of interactive proof assistant users who create a body of verified theorems, and the world of professional programmers who increasingly adopt functional programming to develop important applications. The Scala language (www.scala-lang.org) en- joys a prominent role today for its adoption in industry, a trend most recently driven by the Apache Spark data analysis framework (to which, e.g., IBM committed 3500 researchers recently [16]). We hope to introduce some of the many Scala users to formal methods by providing tools they can use directly on Scala code. Leon sys- tem (http://leon.epfl.ch) is a verification and synthesis system for a subset of Scala [2, 10]. -
Arxiv:2005.03089V1 [Cs.LO] 5 May 2020 Abstract ??? Accepted: / ??? Received: Optrsine a Erlangen-N¨Urnberg E.G
Noname manuscript No. (will be inserted by the editor) Experiences from Exporting Major Proof Assistant Libraries Michael Kohlhase · Florian Rabe Received: ??? / Accepted: ??? Abstract The interoperability of proof assistants and the integration of their li- braries is a highly valued but elusive goal in the field of theorem proving. As a preparatory step, in previous work, we translated the libraries of multiple proof assistants, specifically the ones of Coq, HOL Light, IMPS, Isabelle, Mizar, and PVS into a universal format: OMDoc/MMT. Each translation presented tremendous theoretical, technical, and social chal- lenges, some universal and some system-specific, some solvable and some still open. In this paper, we survey these challenges and compare and evaluate the solutions we chose. We believe similar library translations will be an essential part of any future system interoperability solution and our experiences will prove valuable to others undertaking such efforts. 1 Introduction Motivation The hugely influential QED manifesto [2] of 1994 urged the automated reasoning community to work towards a universal, computer-based database of all mathematical knowledge, strictly formalized in logic and supported by proofs that can be checked mechanically. The QED database was intended as a communal re- source that would guide research and allow the evaluation of automated reasoning tools and systems. This database was never realized, but the interoperability of proof assistants and the integration of their libraries has remained a highly valued but elusive goal. This is despite the large and growing need for more integrated and easily reusable libraries of formalizations. For example, the Flyspeck project [20] build a formal proof of the Kepler conjecture in HOL Light. -
A Comparison of the Mathematical Proof Languages Mizar and Isar
A comparison of the mathematical proof languages Mizar and Isar Markus Wenzel ([email protected]) ∗ Institut f¨urInformatik, Technische Universit¨atM¨unchen,Germany Freek Wiedijk ([email protected]) y Department of Computer Science, University of Nijmegen, the Netherlands Abstract. The mathematical proof checker Mizar by Andrzej Trybulec uses a proof input language that is much more readable than the input languages of most other proof assistants. This system also differs in many other respects from most current systems. John Harrison has shown that one can have a Mizar mode on top of a tactical prover, allowing one to combine a mathematical proof language with other styles of proof checking. Currently the only fully developed Mizar mode in this style is the Isar proof language for the Isabelle theorem prover. In fact the Isar language has become the official input language to the Isabelle system, even though many users still use its low-level tactical part only. In this paper we compare Mizar and Isar. A small example, Euclid's proof of the existence of infinitely many primes, is shown in both systems. We also include slightly higher-level views of formal proof sketches. Moreover a list of differences between Mizar and Isar is presented, highlighting the strengths of both systems from the perspective of end-users. Finally, we point out some key differences of the internal mechanisms of structured proof processing in either system. Keywords: Formalized mathematics, structured proof languages, proof sketches. 1. Introduction 1.1. Overview We compare the proof languages of two systems for formalizing math- ematics in the computer: the Mizar system by Andrzej Trybulec from Bia lystok,Poland (Rudnicki, 1992; Trybulec, 1993; Muzalewski, 1993; Wiedijk, 1999), and the Isar interface by Markus Wenzel from Munich, Germany (Wenzel, 1999; Wenzel, 2002a; Wenzel, 2002b) to the Isabelle system by Larry Paulson and Tobias Nipkow (Paulson, 1994; Nip- kow et al., 2002). -
Importing HOL Light Into Coq Chantal Keller, Benjamin Werner
Importing HOL Light into Coq Chantal Keller, Benjamin Werner To cite this version: Chantal Keller, Benjamin Werner. Importing HOL Light into Coq. ITP - Interactive Theorem Proving, First International Conference - 2010, Jul 2010, Edimbourg, United Kingdom. pp.307-322. inria-00520604 HAL Id: inria-00520604 https://hal.inria.fr/inria-00520604 Submitted on 23 Sep 2010 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. Importing HOL Light into Coq Chantal Keller1,2 and Benjamin Werner2 1 ENS Lyon 2 INRIA Saclay–ˆIle-de-France at Ecole´ polytechnique Laboratoire d’informatique (LIX) 91128 Palaiseau Cedex France [email protected] [email protected] Abstract. We present a new scheme to translate mathematical devel- opments from HOL Light to Coq, where they can be re-used and re- checked. By relying on a carefully chosen embedding of Higher-Order Logic into Type Theory, we try to avoid some pitfalls of inter-operation between proof systems. In particular, our translation keeps the mathe- matical statements intelligible. This translation has been implemented and allows the importation of the HOL Light basic library into Coq. 1 Introduction 1.1 The curse of Babel? Proof-systems are software dedicated to the development of mechanically checked formal mathematics. -
Towards a Mechanized Metatheory of Standard ML ∗
Towards a Mechanized Metatheory of Standard ML ∗ Daniel K. Lee Karl Crary Robert Harper Carnegie Mellon University {dklee,crary,rwh}@cs.cmu.edu Abstract small case-studies that expose pitfalls. But even using these best We present an internal language with equivalent expressive power practices, errors and inconsistencies arise that are not easily dis- to Standard ML, and discuss its formalization in LF and the covered. Moreover, as languages evolve, so must the metatheory machine-checked verification of its type safety in Twelf. The in- that supports it, introducing further opportunities for error. ternal language is intended to serve as the target of elaboration in A promising approach to reducing error is to use mechanized an elaborative semantics for Standard ML in the style of Harper and verification tools to ease the burden of proving properties of Stone. Therefore, it includes all the programming mechanisms nec- language definitions. Ideally, a language definition would come essary to implement Standard ML, including translucent modules, equipped with a body of metatheory that is mechanically checked abstraction, polymorphism, higher kinds, references, exceptions, against the definition and that can be extended as need and interest recursive types, and recursive functions. Our successful formaliza- demands. With the development of powerful tools such as mechani- tion of the proof involved a careful interplay between the precise cal theorem provers and logical frameworks, it is becoming feasible formulations of the various mechanisms, and required the invention to put this idea into practice. For example, Klein and Nipkow [17] of new representation and proof techniques of general interest. have recently used the Isabelle theorem prover [23] to formalize a large part of the Java programming language and to prove type Categories and Subject Descriptors D.3.1 [Programming Lan- safety for it. -
Two Lectures on Constructive Type Theory
Two Lectures on Constructive Type Theory Robert L. Constable July 20, 2015 Abstract Main Goal: One goal of these two lectures is to explain how important ideas and problems from computer science and mathematics can be expressed well in constructive type theory and how proof assistants for type theory help us solve them. Another goal is to note examples of abstract mathematical ideas currently not expressed well enough in type theory. The two lec- tures will address the following three specific questions related to this goal. Three Questions: One, what are the most important foundational ideas in computer science and mathematics that are expressed well in constructive type theory, and what concepts are more difficult to express? Two, how can proof assistants for type theory have a large impact on research and education, specifically in computer science, mathematics, and beyond? Three, what key ideas from type theory are students missing if they know only one of the modern type theories? The lectures are intended to complement the hands-on Nuprl tutorials by Dr. Mark Bickford that will introduce new topics as well as address these questions. The lectures refer to recent educational material posted on the PRL project web page, www.nuprl.org, especially the on-line article Logical Investigations, July 2014 on the front page of the web cite. Background: Decades of experience using proof assistants for constructive type theories, such as Agda, Coq, and Nuprl, have brought type theory ever deeper into computer science research and education. New type theories implemented by proof assistants are being developed. More- over, mathematicians are showing interest in enriching these theories to make them a more suitable foundation for mathematics. -
Ott: Effective Tool Support for the Working Semanticist
JFP 20 (1): 71–122, 2010. c Cambridge University Press 2010 71 doi:10.1017/S0956796809990293 Ott: Effective tool support for the working semanticist PETER SEWELL1, FRANCESCO ZAPPA NARDELLI2, SCOTT OWENS1, GILLES PESKINE1, THOMAS RIDGE1, SUSMIT SARKAR1 andROK STRNISAˇ 1 1University of Cambridge, Computer Laboratory, William Gates Building, 15 JJ Thomson Avenue, Cambridge CB3 0FD, United Kingdom and 2INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France (e-mail: [email protected]) Abstract Semantic definitions of full-scale programming languages are rarely given, despite the many potential benefits. Partly this is because the available metalanguages for expressing semantics – usually either LATEX for informal mathematics or the formal mathematics of a proof assistant – make it much harder than necessary to work with large definitions. We present a metalanguage specifically designed for this problem, and a tool, Ott, that sanity-checks such definitions and compiles them into proof assistant code for Coq, HOL, and Isabelle/HOL, together with LATEX code for production-quality typesetting, and OCaml boilerplate. The main innovations are (1) metalanguage design to make definitions concise, and easy to read and edit; (2) an expressive but intuitive metalanguage for specifying binding structures; and (3) compilation to proof assistant code. This has been tested in substantial case studies, including modular specifications of calculi from the TAPL text, a Lightweight Java with Java JSR 277/294 module system proposals, and a large fragment of OCaml (OCamllight, 310 rules), with mechanised proofs of various soundness results. Our aim with this work is to enable a phase change: making it feasible to work routinely, without heroic effort, with rigorous semantic definitions of realistic languages. -
Formalization of Logic in the Isabelle Proof Assistant
PhD thesis Formalization of Logic in the Isabelle Proof Assistant Anders Schlichtkrull Technical University of Denmark September 14, 2018 PHD-2018-493 Abstract Isabelle is a proof assistant, i.e. a computer program that helps its user to define concepts in mathematics and computer science as well as to prove properties about them. This process is called formalization. Proof assistants aid their users by ensuring that proofs are constructed correctly and by conducting parts of the proofs automatically. A logical calculus is a set of rules and axioms that can be applied to construct theorems of the calculus. Logical calculi are employed in e.g. tools for formal verification of computer programs. Two important properties of logical calculi are soundness and completeness, since they state, respectively, that all theorems of a given calculus are valid, and that all valid statements are theorems of the calculus. Validity is defined by a semantics, which gives meaning to formulas. This thesis describes formalizations in Isabelle of several logics as well as tools built upon these. Specifically this thesis explains and discusses the following contributions of my PhD project: A formalization of the resolution calculus for first-order logic, Herbrand’s theorem and the • soundness and completeness of the calculus. A formalization of the ordered resolution calculus for first-order logic, an abstract prover • based on it and the prover’s soundness and completeness. A verified automatic theorem prover for first-order logic. The prover is a refinement of the • above formalization of an abstract prover. This explicitly shows that the abstract notion of a prover can describe concrete computer programs.