Table Of ContentThirty 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.
