Logic and proof method of recursion

1 What makes a useful calculus? 7 1.0 The r^ ole of proofs and calculi in program development : 9 1.1 Choice of the proof 5 Conclusion 133 xii CONTENTS Chapter 0 Introduction Today there is general agreement that program development is important and diicult. Much research has been done and still is done, in order to make program development reliable and manageable. The main ingredients of program development are a formal calculus, which ensures correctness of development, and a method for its disciplined, goal-directed application. The essence of formal program development is that a program is formally speciied, and a program in executable notation is proved to meet that speciication. It is generally agreed that the executable program should not be guessed and afterwards be proved to meet the speciica-tion. Instead, the program and its proof should be developed hand in hand from the speciication. Of course, thought and insight into the problem domain are needed in the development of a program. As is well-known, program development cannot completely be mechanized. Therefore human guidance is indispensable to program development. The next step towards systematic program development consists in basing decisions rather on the shape of formulae than on insights into the problem domain. Certainly, knowledge of the problem domain cannot completely be replaced by consideration of the shape of formulae. But methods for program development should take the shape of formulae into account as much as possible in order to guide the development process. There is evidence that syntactic considerations can avoid the blind search for ideas to a large extent. Thus, a calculus for program development should be accompanied by a 1 2 CHAPTER 0. INTRODUCTION method for its application. The method should divide the development as clearly as possible into routine steps, and steps that require thought. The latter should be guided as much as possible by the shape of for-mulae. So, more systematic program development could be achieved. Today many calculi exist for program development and proof. But they are rarely accompanied by a method for their application. Undoubtedly, it is diicult to give a method that covers program development in general. But one could concentrate attention on particular development tasks. In this work we will concentrate on a special form of speciications, and nd a development method for them. We have chosen formulae of predicate calculus that contain inequations t v u or equations t …