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A Calculational Approach to Mathematical Induction
, 1994
"... Several concise formulations of mathematical induction are presented and proven equivalent. The formulations are expressed in variablefree relation algebra and thus are in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form, when co ..."
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Several concise formulations of mathematical induction are presented and proven equivalent. The formulations are expressed in variablefree relation algebra and thus are in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form, when combined with the explicit use of Galois connections, lends itself very well for use in calculational proofs. Two nontrivial examples are presented. The first is a proof of a Newman's lemma. The second is a calculation of a condition under which the union of two wellfounded relations is wellfounded. In both cases the calculations lead to generalisations of the known results. In the case of the latter example, one lemma generalises three different conditions.
Theory and Applications of Inverting Functions as Folds
"... 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 proper ..."
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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. The proof makes use of Doornbos and Backhouse's theory on wellfoundedness and reductivity. Possible applications of the generalised theorem is then discussed.
Under consideration for publication in J. Functional Programming 1 Algebra of Programming in Agda Dependent Types for Relational Program Derivation
, 2009
"... Relational program derivation is the technique of stepwise refining a relational specification to a program by algebraic rules. The program thus obtained is correct by construction. Meanwhile, dependent type theory is rich enough to express various correctness properties to be verified by the type c ..."
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Relational program derivation is the technique of stepwise refining a relational specification to a program by algebraic rules. The program thus obtained is correct by construction. Meanwhile, dependent type theory is rich enough to express various correctness properties to be verified by the type checker. We have developed a library, AoPA, to encode relational derivations in the dependently typed programming language Agda. A program is coupled with an algebraic derivation whose correctness is guaranteed by the type system. Two nontrivial examples are presented: an optimisation problem, and a derivation of quicksort where wellfounded recursion is used to model terminating hylomorphisms in a language with inductive types. 1