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Induction and Recursion on Datatypes
, 1995
"... 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 rel ..."
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Cited by 16 (7 self)
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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 wellfoundedness, 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...
Category Theory as Coherently Constructive Lattice Theory: An Illustration
, 1994
"... Dijkstra and Scholten have formulated a theorem stating that all disjunctivity properties of a predicate transformer are preserved by the construction of least prefix points. An alternative proof of their theorem is presented based on two fundamental fixed point theorems, the abstraction theorem ..."
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Cited by 1 (0 self)
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Dijkstra and Scholten have formulated a theorem stating that all disjunctivity properties of a predicate transformer are preserved by the construction of least prefix points. An alternative proof of their theorem is presented based on two fundamental fixed point theorems, the abstraction theorem and the fusion theorem, and the fact that suprema in a lattice are defined by a Galois connection. The abstraction theorem seems to be new; the fusion theorem is known but its importance does not seem to be fully recognised. The abstraction theorem, the fusion theorem, and Dijkstra and Scholten's theorem are then generalised to the context of category theory and shown to be valid. None of the theorems in this context seems to be known, although specific instances of Dijkstra and Scholten's theorem are known. The main point of the paper is to discuss the process of drawing inspiration from lattice theory to formulate theorems in category theory (first advocated by Lambek in 1968). W...