ELIOS-OBJ Theorem Proving in a Specification Language

1 I n t r o d u c t i o n OBJ3 is a programming language based on equational logic: programs are given in terms of abst ract da ta types and their semantics relies on order sorted algebras, which enables inclusion of types. The problem approached here is the correctness of axiom sets, in the following sense. The operational semantics of OBJ3 is rewriting, which means tha t when a program is executed on a given value, the set of axioms is interpreted and used as a set of rewrite rules that reduces the value to its normal form. We have to establish whether computations are correct with respect to validity in initial models, whether results are unique and last but not least whether computat ion always terminates. The completion process of a rewrite rule set is able to ensure the previous requirements. Start ing from any axiom set, it provides, when it succeeds (this is a semi-decidable problem), an equivalent set of rules with the same deduction power, confluent (the result of rewriting an expression does not depend of the way the rules are applied: it is unambiguous), and terminating (there is no infinite rewrite chain). Hence, it can be seen as an automat ic prover of program correctness. We intend here to design and implement an integrated programming environment, named ELIOS-OBJ, allowing programming and proving in the same context. Our goal here has three aspects: to provide the user with a tool for proving correctness of specifications in the context of OBJ3, namely with an order-sorted semantics; to propose an implementat ion of order-sorted completion described and proved in [6]; and to point out some problems arising in integrating theorem proving aspects (completion here) with programming aspects (the OBJ3 language).