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HIGHER ORDER
LOGIC THEOREM
PROVING A ND ITS
APPLICATIONS
Proceedings of the IFIP TC10/WG10.2 International Workshop on
Higher Order Logic Theorem Proving and its Applications - HOL '92
organized by CHEOPS ESPRIT BRA 3215
sponsored by IMEC and the Commission of the European Communities
Leuven, Belgium, 21-24 September 1992
Edited by
LUCJ.M.CLAESEN
Interuniversity Micro-Electronics Center and
Katholieke Universiteit Leuven
Leuven, Belgium
MICHAEL J.C. GORDON
Computer Laboratory
University of Cambridge
Cambridge, England
N-H
WM
m
1993
NORTH-HOLLAND
AMSTERDAM · LONDON · NEW YORK · TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V.
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Keywords are chosen from the ACM Computing Reviews Classification System, ©1991, with permission.
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ISBN: 0 444 89880 8
ISSN: 0926-5473
© 1993 IFIP. All rights reserved.
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pp. 117-132: Copyright not transferred
This book is printed on acid-free paper.
Printed in The Netherlands
ν
Preface
This book contains all the papers submitted to the workshop entitled Higher Order Logic
Theorem Proving and its Applications. These range from reports on student projects to
original research contributions.
The workshop was organized in IMEC, Leuven, Belgium, from 21 to 24 September 1992
by the CHEOPS ESPRIT Basic Research Action in cooperation with IFIP working group
10.2.
The meeting was the fifth in a series of annual workshops dedicated to the topic of higher
order logic theorem proving, its usage in the HOL system and its applications. Previous
workshops have taken place in Cambridge UK, Aarhus Denmark and Davis California.
The HOL system is a higher order logic theorem proving system implemented at Edin-
burgh University, Cambridge University and INRIA. It has found many applications, from
the verification of hardware designs at all levels to the vérification of programs and com-
munication protocols. Contributions and experiences with other systems based on higher
order logic, namely Nuprl and LAMBDA, are also presented here.
The papers in this book cover the following topics:
• Mathematical Logic.
• Induction.
• General Modelling and Proofs.
• Formalizing and Modelling of Automata.
• Program Verification.
• Hardware Description Language Semantics.
• Hardware Verification Methodologies.
• Simulation in Higher Order Logic.
• Extended uses of Higher Order Logic.
vi
The workshop was attended by 60 participants (Europe: 45, North America: 15)
Tutorials were presented by Roger B. Jones on Proof Power and by Carl Seger and Jeff
Joyce on Hybrid Formal Verification Techniques.
In addition to regular presentations on the practical application of higher order logic and
the HOL system, the workshop included a HOL users clinic organized by Tom Melham
for in-depth discussion of issues concerning the use of the HOL system.
A special LAMBDA users meeting session was organized by Holger Bush.
The next workshop in the series will be organized by Jeff Joyce and Carl Seger at the
University of Britisch Columbia in Vancouver Canada.
We are grateful to the Commission of the European Communities for sponsoring this
workshop by providing grants for students to participate.
Luc Claesen, Michael Gordon
November 1992
vii
Conference organization.
Workshop Chair:
Luc Claesen
Interuniversity Micro Electronics Center
h Katholieke Universiteit Leuven
Kapeldreef 75,
B-3001 Leuven (Belgium)
e-mail: [email protected]
Workshop Co-Chair:
Michael Gordon
University of Cambridge
Computer Laboratory
Pembroke Street
Cambridge CB2 3QG (U.K.)
email: [email protected]
Program Committee
Myla Archer (University of California, Davis, USA)
Graham Birtwistle (University of Calgary, CA)
Holger Bush (Siemens AG, D)
Albert Camilleri (Hewlett-Packard, UK)
Shui-Kai Chin (Syracuse University, USA)
Luc Claesen (IMEC / Kath. Univ. Leuven, B)
Simon Finn (Abstract Hardware Ltd., UK)
Michael Gordon (University of Cambridge, UK)
Elsa L. Gunter (AT&T Bell Labs, USA)
John Herbert (SRI International, UK)
Roger B. Jones (ICL, UK)
Jeff Joyce (University of British Columbia, CA)
Ton Kalker (Philips, NL)
Matt Kaufmann (Computational Logic Inc., USA)
Kurt Keutzer (Synopsys, USA)
Ramayya Kumar (Univ. of Karlsruhe, D)
viii
Miriam Leeser (Cornell University, USA)
Tim Leonard (Digital Equipement Corp., USA)
Paul Loewenstein (Sun Microsystems, USA)
Tom Melham (University of Cambridge, UK)
Carl Seger (University of British Columbia, CA)
David Shepherd (Inmos Ltd., UK)
Gerd Venzl (Siemens AG, D)
Phillip J. Windley (University of Idaho, USA)
Local Organizing Committee.
Catia Angelo Marcondes, Luc Claesen, Geert De Pril, Peter De Vijt, Mark Genoe, Peter
Johannes, Wim Ploegaerts, Hans Samsom, Robert Severyns, Annemie Stas, John Tytgat,
Jan Vandenbergh, Diederik Verkest, Eric Verlind
Reviewers.
J. Alves-Foss, C. Angelo Marcondes, M. Archer, G. Birtwistle, H. Bush, A. Camilleri,
S-K. Chin, L. Claesen, S. Finn, M. Gordon, E.L. Gunter, J. Grundy, J. Herbert, R.B.
Jones, J. Joyce, A. Kalker, M. Kaufmann, Κ. Keutzer, R. Kumar, M. Leeser, T. Leonard,
P. Loewenstein, T. Melham, W. Ploegaerts, C. Seger, D. Shepherd, G. Venzl, D. Verkest,
Ph.J. Windley
xiii
Higher Order Logic Theorem Proving and its Applications (A-20)
L.J.M. Claesen and M.J.C. Gordon (Editors)
Elsevier Science Publishers B.V. (North-Holland) 3
© 1993 IFIP. All rights reserved.
The HOL Logic Extended with Quantification over
Type Variables
Thomas F. Melham
University of Cambridge Computer Laboratory, New Museums Site, Pembroke Street,
Cambridge, CB2 3QG, England.
Abstract
This paper discusses a proposal to extend the primitive basis of the HOL logic with a very
simple form of quantification over types. It is shown how certain practical problems with
using the definitional mechanisms of HOL would be solved by the additional expressive
power gained by making this extension.
Keyword Codes: F.4.1; 1.2.3
Keywords: Mathematical Logic; Deduction and Theorem Proving.
1 Introduction
The version of higher order logic mechanized by the HOL system is essentially Church's
formulation of simple type theory [2] extended with explicit rules of definition and with
object-language polymorphism of the kind developed by Milner for the LCF logic PPA [4].
In this paper a further extension to the logic is proposed—namely, the addition of a
very limited form of object-language quantification over types. The motivation for this
extension comes from a particular technical problem that arises when using the definitional
mechanisms provided by the HOL logic. This paper explains this problem and shows how
the proposed extension solves it.
It is assumed that the reader is familiar with the details of the HOL logic given in the
DESCRIPTION volume of the HOL documentation [5].
2 Types and polymorphism in HOL
Type expressions in the HOL logic have the following syntax:
σ ::= c | a | σ σ | (σι,...,σ )ορ
λ 2 η
where σ, σι, ..., σ range over types, c ranges over type constants, a ranges over type
η
variables, and op ranges over n-ary type operators (for η > 1). It is the inclusion of
4
type variables in this syntax of types that makes the HOL logic polymorphic. Typing of
terms takes place within the context of an assignment of generic types to constants, and
a constant is well-typed at any substitution instance of its generic type.
Theorems that contain polymorphic types are also true for any substitution instance
of them, so there is a limited form of implicit universal quantification over types in the
HOL logic. More precisely, there is an implicit universal quantification over type variables
at the level of sequents. A sequent Γ h Ρ means that Ρ is provable by natural deduction
from the hypotheses Γ. All occurrences of a type variable in such a sequent are identified
by the semantics, and there is an implicit universal quantification over the value of this
variable whose scope is the entire sequent. This is reflected in the primitive rule of type
instantiation shown below.
, ..., α not in Γ.
Γ h Ρ[σι,...,σ / αϊ,..., η
η
where Ρ [σι,..., σ / αϊ,..., α ] means the result of simultaneously substituting the type
η η
σ{ for the type variable a; for 1 < i < η at every occurrence of a; in the term P. The
side condition on this rule is a consequence of the fact that if a type variable occurs in
both the hypotheses and the conclusion of a sequent, then both occurrences are assumed
to denote the same set. In fact, from this primitive rule one can derive the more general
instantiation rule
ΓΚΡ
Γ[σι,..., σ / α ,..., α ] h Ρ [σι,..., σ / a ..., α„]
η χ η η u
which allows one to substitute types for type variables throughout an entire sequent.
There is an additional side condition on both of the rules shown above, namely the
condition that no two distinct term variables in Ρ (and, for the second rule, Γ) may
become identified as a result of doing the substitution. The HOL system deals with this
side condition by renaming variables as required.
3 Motivation
The proposal of this paper is to extend the very simple form of polymorphism described
above with limited object-language quantification over types. In particular, we wish to
add primitive terms to the HOL logic of the forms Va. Ρ and 3α. P, where α is a type
variable and Ρ is a boolean term. Informally, the intended interpretation is that Ρ[σ/α] is
true for all types σ and for some type σ, respectively. Note that we are not proposing an
extension to the type language of HOL—the quantifications Va. Ρ and 3α. Ρ are new term
constructs, and not type constructs of the kind found (for example) in Girard's system
F [3]. The extended logic proposed here resembles system Q, a transfinite type theory
due to Andrews [1]. It is, however, still much weaker than Andrews' system.
The motivation for this extension originally arose in connection with work on new
derived rules of definition for HOL—particularly in work on derived rules for defining
abstract data types. The following sections explain this motivational background. Readers
primarily interested in the semantics and other details of the proposed extension may wish
to skip to section 4.
