. In this paper we present an approach to prove the equality between terms in a goaldirected way developed in the field of inductive theorem proving. The two terms to be equated are syntactically split into expressions which are common to both and those which occur only in one term. According to the computed differences we apply appropriate equations to the terms in order to reduce the differences in a goal-directed way. Although this approach was developed for purposes of inductive theorem proving - we use this technique to manipulate the conclusion of an induction step to enable the use of the hypothesis - it is a powerful method for the control of equational reasoning in general. 1. Introduction The automation of equational reasoning is one of the most important obstacles in the field of automating deductions. Even small equational problems result in a huge search space, and finding a proof often fails due to the combinatorial explosion. Proving (conditional) equations by inductio...

Defining data types as initial algebras, or dually as final co-algebras, is beneficial, if not indispensible, for an algebraic calculus for program construction, in view of the nice equational properties that then become available. It is not hard to render finite lists as an initial algebra and, dually, infinite lists as a final co-algebra. However, this would mean that there are two distinct data types for lists, and then a program that is applicable to both finite and infinite lists is not possible, and arbitrary recursive definitions are not allowed. We prove the existence of algebras that are both initial in one category of algebras and final in the closely related category of co-algebras, and for which arbitrary (continuous) fixed point definitions ("recursion") do have a solution. Thus there is a single data type that comprises both the finite and the infinite lists. The price to be paid, however, is that partiality (of functions and values) is unavoidable.

Introduction From the early days of computing, many individuals have recognized that algebras provide interesting mathematical models for at least some aspects of programs. In mathematics, an algebra consists of a set (called the carrier of the algebra), together with a finite set of total functions that have the carrier set as their common codomain. The algebras we learn in school, however, are usually those derived from number theory and programs are more diverse, if not richer, than operations on numbers. A somewhat more abstract notion, called signature algebras, has been used for some time to to model abstract data types [GTW78]. A signature defines a set of typed operator symbols without specifying functions that would be the actual operators. Thus a signature defines a class of algebras, namely the algebras whose operators conform to the typing constraints imposed by the signature. Signature algebras have been helpful in understanding the issues involved in abstract dat

In this paper we study final algebra semantics for constructive equational systems. A class of models of a constructive system is described, and proven to haveafinal algebra. Then wedevelop a method for proof by consistency with respect to the final model. Finally weshowthatthemethod contains the proof methods of Musser [11], Goguen [2], and Huet and Hullot [5] as special cases.

Z is a formal specification language that is extensively used in both academia and industry. Several tools have been developed for reasoning about Z specifications, but they all lack substantial facilities for inductive reasoning. We propose the development of such a reasoning assistant for Z. Utilising the concept of "ordered covering", our proposed approach will combine aspects of both the classical and rewrite-based approaches to proof by induction. In particular, it will include the classical induction "on variables " and use of heuristics, and the rewrite-based induction "on subterms " and unification for generation of induction cases. In this way, we will not only be developing an inductive reasoner for Z but will be furthering our understanding of the automation of proof by induction. To ensure that our reasoning principles and strategies are appropriate for Z, we propose to implement them in CADiZ, an interactive reasoning tool for Z developed at the University of York. We will evaluate the utility of the resulting system by attempting to construct a broad range of proofs, including some of practical importance to computer security, safety-critical systems and compiler correctness.

This thesis is primarily concerned with the algebraic semantics of non-terminating term rewriting systems. The usual semantics for rewrite system is based in interpreting rewrite rules as equations and rewriting as a particular case of equational reasoning. The termination of a rewrite system ensures that every term has a value (normal form). But, in general we cannot guarantee this. The research that has been done on non-terminating rewrite systems is centered on seeking semantics for these systems where the usual properties of confluent systems (like uniqueness of normal forms) still hold. These approaches extend the original set of terms (with infinite terms) in such a way that every term has a value. We propose a new semantics for rewrite systems based on interpreting rewrite rules as inequations between terms in an ordered algebra. We show that a variant of equational logic -- inequational logic -- is an institution and we further prove that rewriting is a sound and complete proof...