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Partial functions in induction theorem proving
 THE PROCEEDINGS OF CADE15 WORKSHOP ON MECHANISATION OF PARTIAL FUNCTIONS
, 1998
"... We present an approach for automated induction proofs with partial functions. Most wellknown techniques developed for (explicit) induction theorem proving are unsound when dealing with partial functions. But surprisingly, by slightly restricting the application of these techniques, it is possible t ..."
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We present an approach for automated induction proofs with partial functions. Most wellknown techniques developed for (explicit) induction theorem proving are unsound when dealing with partial functions. But surprisingly, by slightly restricting the application of these techniques, it is possible to develop a calculus for automated induction proofs with partial functions. In particular, under certain conditions one may even generate induction schemes from the recursions of nonterminating algorithms. The need for such induction schemes and the power of our approach have been demonstrated on a large collection of nontrivial theorems (including Knuth and Bendix' critical pair lemma). In this way, existing induction theorem provers can be directly extended to partial functions without changing their logical framework.
Partial Functions in the Z Specification Language
"... . One of the principal benefits of the specification language Z to the formalmethods community is its simple type system, which allows the early detection of meaningless expressions. However, partly as a consequence of this simple approach to types, partial functions have surprising and, in som ..."
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. One of the principal benefits of the specification language Z to the formalmethods community is its simple type system, which allows the early detection of meaningless expressions. However, partly as a consequence of this simple approach to types, partial functions have surprising and, in some cases, problematic properties. We outline a semantics for partial functions that resolves these difficulties. We proceed to utilise our semantics as a basis for reasoning about the inductive properties of partial functions. Our semantics allows us to construct straightforward "closure axioms" for partiallydefined functions that specify when the functions are not defined, thus allowing verification of the properties of the functions over their defined cases via structural induction. Our semantics also provides justification, under certain conditions, for an alternative approach to verification that utilises a speciallydesigned rule of induction. We sketch and compare these two ap...