Proof in VDM: A Practitioner’s Guide

Formal specifications were first used in the description of program­ ming languages because of the central role that languages and their compilers play in causing a machine to perform the computations required by a programmer. In a relatively short time, specification notations have found their place in industry and are used for the description of a wide variety of software and hardware systems. A formal method - like VDM - must offer a mathematically-based specification language. On this language rests the other key element of the formal method: the ability to reason about a specification. Proofs can be empioyed in reasoning about the potential behaviour of a system and in the process of showing that the design satisfies the specification. The existence of a formal specification is a prerequisite for the use of proofs; but this prerequisite is not in itself sufficient. Both proofs and programs are large formal texts. Would-be proofs may therefore contain errors in the same way as code. During the difficult but inevitable process of revising specifications and devel­ opments, ensuring consistency is a major challenge. It is therefore evident that another requirement - for the successful use of proof techniques in the development of systems from formal descriptions - is the availability of software tools which support the manipu­ lation of large bodies of formulae and help the user in the design of the proofs themselves.

1 Introduction.- 1.1 Background.- 1.2 How proofs arise in practice: an introductory example.- 1.3 A logical framework for proofs.- 1.4 Summary.- I A Logical Basis for Proof in VDM.- 2 Propositional LPF.- 3 Predicate LPF with Equality.- 4 Basic Type Constructors.- 5 Numbers.- 6 Finite Sets.- 7 Finite Maps.- 8 Finite Sequences.- 9 Booleans.- II Proof in Practice.- 10 Proofs From Specifications.- 11 Verifying Reifications.- 12 A Case Study in Air-Traffic Control.- 13 Advanced Topics.- III Directory of Theorems.- 14 Directory of Theorems.- Index of Symbols.- Index of Rules.