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Thirty Five Years of Automating Mathematics APPLIED LOGIC SERIES VOLUME 28 Managing Editor Dov M. Gabbay, Department oj Computer Science, King's College, London, u.K. Co-Editor Jon Barwiset Editorial Assistant Jane Spurr, Department oj Computer Science, King's College, London, u.K. SCOPE OF THE SERIES Logic is applied in an increasingly wide variety of disciplines, from the traditional subject of philosophy and mathematics to the more recent disciplines of cognitive science, compu ter science, artificial intelligence, and linguistics, leading to new vigor in this ancient subjeci Kluwer, through its Applied Logic Series, seeks to provide a home for outstanding books ani research monographs in applied logic, and in doing so demonstrates the underlying unity ani applicability of logic. The titles published in this series are listed at the end of this volume. Thirty Five Years of Automating Mathematics Edited by FAIROUZ D. KAMAREDDINE Heriot-Watt University, School of Mathematical and Computer Sciences, Mountbatten Building, Riccarton, Edinburgh EH 14 4AS, Scotland, UK SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-6440-0 ISBN 978-94-017-0253-9 (eBook) DOI 10.1007/978-94-017-0253-9 Printed on acid-free paper AII Rights Reserved © 2003 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2003 Softcover reprint ofthe hardcover lst edition 2003 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permis sion from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. CONTENTS Contributors vii Editorial Preface 1 Fairouz Kamareddine A Mathematical Model for Biological Memory and 9 Consciousness N. G. de Bruijn Towards an Interactive Mathematical Proof Mode 25 Henk Barendregt Recent Results in Type Theory and their Relationship to 37 Automath Robert L. Constable Linear Contexts, Sharing Functors: Techniques for 49 Symbolic Computation Gerard Huet De Bruijn's Automath and Pure Type Systems 71 Fairouz Kamareddine, Twan Laan and Rob Nederpelt Hoare Logic with Explicit Contexts 125 Michael Franssen Transitive Closure and the Mechanization of Mathematics 149 Arnon Avron Polymorphic Type-checking for the Ramified Theory of 173 Types of Principia M athematica M. Randall Holmes Termination in ACL2 using Multiset Relations 217 vi J. L. Ruiz-Reina, J. A. Alonso, M. J. Hidalgo and F. J. Martin-Mateos The 7r-Calculus in FM 247 Murdoch J. Gabbay ../2 Proof Development with Omega: The Irrationality of 271 Jorg Siekmann, Christoph Benzmiiller, Armin Fiedler, Andreas Meier, Immanuel Normann and Martin Pollet Index 315 LIST OF CONTRIBUTORS Arnon Avron School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. Email: [email protected] Henk Barendregt Chair of Foundations of Mathematics and Computer Science, Catholic University of Nijmegen, Informatica, PO Box 9010, 6500 GL Nijmegen, The Netherlands. Email: [email protected] N.G. de Bruijn Department of Mathematics and Computing Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email: [email protected] Robert L. Constable Department of Computer Science, 4149 Upson Hall, Cornell University, Ithaca, NY 14853, USA. Email: [email protected] Michael Franssen Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O.Box 513, 5600 MB Eindhoven, the Netherlands. E-mail: [email protected]. Murdoch J. Gabbay University of Cambridge Computer Laboratory William Gates Building, 15 JJ Thomson Avenue Cambridge CB3 OFD, UK Email: [email protected] Randall Holmes Department of Mathematics, Boise State University, 1910 University Drive, Boise, Idaho 83725-1555, USA. Email: [email protected] viii Gerard Huet INRIA, Rocquencourt - BP 105, 78153 Le Chesnay Cedex, France. Email: [email protected] Fairouz Kamareddine School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS. Email: [email protected]. uk Twan Laan Weerdstede 45, 3431 LS Nieuwegein, The Netherlands. Email: [email protected] Rob Nederpelt Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O.Box 513, 5600 MB Eindhoven, the Netherlands. Email: [email protected] Jorg Siekmann, Christoph Benzmiiller, Armin Fiedler, Andreas Meier, Immanuel Normann, and Martin Pollet FR 6.2 Informatik, Universitat des Saarlandes, 66041 Saarbriicken, Germany Email: {siekmann.chris.afiedler.ameier.normann.pollet}@ags.uni-sb.de Jose-Luis Ruiz-Reina, Jose-Antonio Alonso, Marfa-JosE Hidalgo and Francisco Jesus Martfn-Mateos Departamento de Ciencias de la Computacion e Inteligencia Artificial Escuela Tecnica Superior de Ingenieria Informatica, Universidad de Sevilla Avda. Reina Mercedes, sin. 41012 Sevilla, Spain Email:{jruizjalonso.mjoseh.fjesus}@us.es N. G. de Bruijn © Gerard Huet EDITORlAL PREFACE THIRTY FIVE YEARS OF AUTOMATING MATHEMATICS: DEDICATED TO 35 YEARS OF DE BRUIJN'S AUTOMATH N.G. de Bruijn was a well established mathematician before deciding in 1967 at the age of 49 to work on a new direction related to Automating Mathematics. By then, his contributions in mathematics were numerous and extremely influential. His book on advanced asymptotic methods, North Holland 1958, was a classic and was subsequently turned into a book in the well known Dover book series. His work on combinatorics yielded influential notions and theorems of which we mention the de Bruijn-sequences of 1946 and the de Bruijn-Erdos theorem of 1948. De Bruijn's contributions to mathematics also included his work on generalized function theory, analytic number theory, optimal control, quasicrystals, the mathematical analysis of games and much more. In the 1960s de Bruijn became fascinated by the new computer technology and as a result, decided to start the new AUTOMATH project where he could check, with the help of the computer, the correctness of books of mathematics. In each area that de Bruijn approached, he shed a new light and was known for his originality and for making deep intellectual contributions. And when it came to automating mathematics, he again did it his way and introduced the highly influential AUTOMATH. In the past decade he has also been working on theories of the human brain. Through his work on AUTOMATH, de Bruijn started a·revolution in using the computer for verification, and since his AUTOMATH, we have seen more and more proof-checking and theorem-proving systems. Although now Au TO MATH is mainly of historical interest,l its influence remains impressive and its literature [Nederpelt et al., 1994] is indispensable. This is amaz ing considering that only a handful of people really worked on building AUTO MATH whereas these days tens of people are usually involved in any influential theorem prover or proof checker. Even those who do not do proof checking use many of the notions given to us by de Bruijn during his AUTO MATH project. For example: • De Bruijn indices [de Bruijn, 1972] still play an important role in the implementation of programmiug languages and theorem provers. 1 Freek Wiedijk has resurrected AUTOMATH [Wiedijk, 2002] with a new implemen tation (called 'aut') of de Bruijn's Zandleven AUTO MATH checker from the seventies. Wiedijk's implementation describes in some detail the features of aut and was written to restore a damaged version of Jutting's translation [van Benthem-Jutting, 1976] of Landau's book [Landau, 1930]. Wiedijk establishes that aut is quite fast, even when compared to current theorem prover systems (aut can check the translation of a full book in under a second). Fairouz Kamareddifle (ed.), Thirty-Five Years of Automating Mathematics 1-8. © 2003, Kluwer Academic Publishers.