Theories, Implementations, and Transformations

The purpose of this paper is to try to put theory presentation and structuring in the simplest possible logical setting in order to improve our understanding of it. We look at how theories can be combined, and compared for strength. We look at theory refinement and implementation, and what constitutes proof of correctness. Our examples come from both the functional style and imperative (state-changing) style of theory. Finally, we explore how one implementation can be transformed to another.