Abstract

this paper we introduce a notion of induction over an arbitrary datatype and go on to show how the notion is used to establish unicity of a certain (broad) class of equations. Our overall goal is to develop a calculational theory of mathematical induction. That is we want to be able to calculate relations on which inductive arguments may be based using laws that relate admitting induction to the mechanisms for constructing datatypes. We also want to incorporate such calculations into a methodology for calculating inductive hypotheses rather than leaving their creation to inspired guesswork. This is a bold aim, in view of the vast amount of knowledge and experience that already exists on proof by induction, but recent advances in the role played by Galois connections in the calculus of relations have led us to speculate that significant progress can be made in the short term. The theory developed in this paper is general and not specific to any particular datatype. We define a notion of F -reductivity (so called in order to avoid confusion with existing notions of inductivity), where F stands for a "relator", and show that F -reductive relations always exist, whatever the value of F . We also give laws for constructing reductive relations from existing reductive relations. We conclude the paper by introducing the dual notion of F -inductivity and briefly contrast it with F -reductivity. The organisation of this note is as follows. In section 2 we give a very brief introduction to the relational calculus. In section 3 the notion of reductivity is defined. This notion is a generalisation of well-foundedness, or inductivity. Then in section 4 we define a class of equations and prove that an equation from that class has a unique solution if one of its components enjoys a red...

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