S-C. Mu and R. S. Bird. In *Science of Computer Programming Vol. 51 Special Issue for Mathematics of Program Construction 2002*, pp. 87-116, 2003.

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This paper is devoted to the proof, applications, and generalisation of a theorem, due to Bird and de Moor, that

gave conditions under which a total function can be expressed as a relational fold. The theorem is illustrated with three problems, all dealing with constructing trees with various properties. It is then generalised to give conditions under which the inverse of a partial function can be expressed as a relational hylomorphism. Its proof makes use of Doornbos and Backhouse’s theory on well-foundedness and reductivity. Possible applications of the generalised theorem is discussed.

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