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Intelligent Computer Mathematics International Conference, CICM 2015 Washington, DC, USA, July 13–17, 2015 Proceedings Manfred Kerber · Jacques Carette Cezary Kaliszyk · Florian Rabe Volker Sorge (Eds.) Intelligent LNAI 9150 Computer Mathematics International Conference, CICM 2015 Washington, DC, USA, July 13–17, 2015 Proceedings 123 Lecture Notes in Artificial Intelligence 9150 Subseries of Lecture Notes in Computer Science LNAI Series Editors Randy Goebel University of Alberta, Edmonton, Canada Yuzuru Tanaka Hokkaido University, Sapporo, Japan Wolfgang Wahlster DFKI and Saarland University, Saarbrücken, Germany LNAI Founding Series Editor Joerg Siekmann DFKI and Saarland University, Saarbrücken, Germany Manfred Kerber • Jacques Carette Cezary Kaliszyk • Florian Rabe Volker Sorge (Eds.) Intelligent Computer Mathematics International Conference, CICM 2015 Washington, DC, USA, July 13–17, 2015 Proceedings 123 Editors Manfred Kerber Florian Rabe University of Birmingham Jacobs University Bremen Birmingham Bremen UK Germany Jacques Carette Volker Sorge McMaster University University of Birmingham Hamilton, ON Birmingham Canada UK Cezary Kaliszyk University of Innsbruck Innsbruck Austria ISSN 0302-9743 ISSN 1611-3349 (electronic) Lecture Notes in Artificial Intelligence ISBN 978-3-319-20614-1 ISBN 978-3-319-20615-8 (eBook) DOI 10.1007/978-3-319-20615-8 Library of Congress Control Number: 2015942531 LNCS Sublibrary: SL7 – Artificial Intelligence Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Contents Invited Talks Mining the Archive of Formal Proofs . 3 Jasmin Christian Blanchette, Maximilian Haslbeck, Daniel Matichuk, and Tobias Nipkow Math Search for the Masses: Multimodal Search Interfaces and Appearance-Based Retrieval . 18 Richard Zanibbi and Awelemdy Orakwue Calculemus Towards Formal Fault Tree Analysis Using Theorem Proving. 39 Waqar Ahmed and Osman Hasan Optimizing a Certified Proof Checker for a Large-Scale Computer-Generated Proof . 55 Luís Cruz-Filipe and Peter Schneider-Kamp A First Class Boolean Sort in First-Order Theorem Proving and TPTP . 71 Evgenii Kotelnikov, Laura Kovács, and Andrei Voronkov Type Inference for ZFH. 87 Steven Obua, Jacques Fleuriot, Phil Scott, and David Aspinall Generic Literals . 102 Florian Rabe Ranking/Unranking of Lambda Terms with Compressed de Bruijn Indices . 118 Paul Tarau Digital Mathematics Libraries A Flexiformal Model of Knowledge Dissemination and Aggregation in Mathematics . 137 Mihnea Iancu and Michael Kohlhase Mathematical Knowledge Management Structure Formation in Large Theories . 155 Serge Autexier and Dieter Hutter XX Contents Formal Logic Definitions for Interchange Languages . 171 Fulya Horozal and Florian Rabe Math Literate Knowledge Management via Induced Material . 187 Mihnea Iancu and Michael Kohlhase Strategies for Parallel Markup. 203 Bruce R. Miller Readable Formalization of Euler’s Partition Theorem in Mizar . 211 Karol Pąk Automating Change of Representation for Proofs in Discrete Mathematics . 227 Daniel Raggi, Alan Bundy, Gudmund Grov, and Alison Pease Performance Evaluation and Optimization of Math-Similarity Search. 243 Qun Zhang and Abdou Youssef Projects and Surveys Mizar: State-of-the-Art and Beyond. 261 Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak,̧ and Josef Urban Growing the Digital Repository of Mathematical Formulae with Generic LTA E X Sources . 280 Howard S. Cohl, Moritz Schubotz, Marjorie A. McClain, Bonita V. Saunders, Cherry Y. Zou, Azeem S. Mohammed, and Alex A. Danoff Formalizing Physics: Automation, Presentation and Foundation Issues . 288 Cezary Kaliszyk, Josef Urban, Umair Siddique, Sanaz Khan-Afshar, Cvetan Dunchev, and Sofiène Tahar A Survey on Retrieval of Mathematical Knowledge. 296 Ferruccio Guidi and Claudio Sacerdoti Coen Towards the Formalization of Fractional Calculus in Higher-Order Logic. 316 Umair Siddique, Osman Hasan, and Sofiène Tahar LeoPARD — A Generic Platform for the Implementation of Higher-Order Reasoners. 325 Max Wisniewski, Alexander Steen, and Christoph Benzmüller Contents XXI Systems and Data TIP: Tons of Inductive Problems. 333 Koen Claessen, Moa Johansson, Dan Rosén, and Nicholas Smallbone Semantic Enrichment of Mathematics via ‘tooltips’ ................... 338 Ross Moore Documentation Generator Focusing on Symbols for the HTML-ized Mizar Library . 343 Kazuhisa Nakasho and Yasunari Shidama Tools for MML Environment Analysis . 348 Adam Naumowicz Enabling Symbolic and Numerical Computations in HOL Light . 353 Ons Seddiki, Cvetan Dunchev, Sanaz Khan-Afshar, and Sofiène Tahar Author Index ............................................ 359 Mizar: State-of-the-Art and Beyond Grzegorz Bancerek1, Czeslaw Byli´nski2,AdamGrabowski3, B Artur Kornilowicz3, Roman Matuszewski4, Adam Naumowicz3( ), Karol P¸ak3,andJosefUrban5 1 Association of Mizar Users, Bialystok, Poland [email protected] 2 Section of Computer Systems and Teleinformatic Networks, University of Bialystok, Bialystok, Poland [email protected] 3 Institute of Informatics, University of Bialystok, Bialystok, Poland {adam,arturk,adamn,karol}@mizar.org 4 Department of Applied Linguistics, Faculty of Philology, University of Bialystok, Bialystok, Poland [email protected] 5 Radboud University, Nijmegen, The Netherlands [email protected] Abstract. Mizar is one of the pioneering systems for mathematics formalization, which still has an active user community. The project has been in constant development since 1973, when Andrzej Trybulec designed the fundamentals of a language capable of rigorously encod- ing mathematical knowledge in a computerized environment which guarantees its full logical correctness. Since then, the system with its feature-rich language devised to approximate mathematics writing has influenced other formalization projects and has given rise to a number of Mizar modes implemented on top of other systems. However, the infor- mation about the system as a whole is not readily available to developers of other systems. Various papers describing Mizar features have been rather incremental and focused only on particular newly implemented Mizar aspects. The objective of the current paper is to give a survey of the most important Mizar features that distinguish it from other popu- lar proof checkers. We also go a step further and describe most important current trends and lines of development that go beyond the state-of-the- art system. 1 Introduction The Mizar [21,38] project is a long-term effort originally aimed at developing a computer environment to support mathematicians in preparing papers. Around 1973, Andrzej Trybulec, the leader of the project, has designed a language for writing formal mathematics. The implemented processor was intended to check written texts for logical consistency and correctness. For fifteen years numerous implementations of the system were developed in order to choose suitable under- lying logic and expressive power of the language (PC – propositional calculus, c Springer International Publishing Switzerland 2015 M. Kerber et al. (Eds.): CICM 2015, LNAI 9150, pp. 261–279, 2015. DOI: 10.1007/978-3-319-20615-8 17 262 G. Bancerek et al. MSE – multi-sorted with equality, QC – quantifier calculus, etc.). An impor- tant issue was also to find the proper technology of automated cross-referencing between papers. The history of the first 30 years of Mizar development has been described in [32]. All these experiments resulted in choosing the principles of the current Mizar. The logical structure of the language is based on a variant of the classical first-order logic natural deduction system proposed by Ja´skowski [27]. The texts written in the language (called Mizar articles) are organized into adatabase–theMizar Mathematical Library, MML. The Tarski-Grothendieck set theory, which is basically ZFC set theory with the axiom of infinity replaced by Tarski’s axiom of existence of arbitrarily large, strongly inaccessible cardinals forms the basis of doing mathematics in contemporary Mizar. Today there are a number of other projects committed to address prob- lems related to computerized theorem proving developed at various research centers around the world. Most significantly: the HOL Light theorem prover1, Isabelle2, the Coq Proof Assistant3, Metamath4, ProofPower5, Nqthm/ACL26, the PVS Specification and Verification
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