Closure Induction in a Z-Like Language

Simply-typed set-theoretic languages such as Z and B are widely used for program and system specifications. The main technique for reasoning about such specifications is induction. However, while partiality is an important concept in these languages, many standard approaches to automating induction proofs rely on the totality of all occurring functions. Reinterpreting the second author's recently proposed induction technique for partial functional programs, we introduce in this paper the new principle of "closure induction" for reasoning about the inductive properties of partial functions in simply-typed set-theoretic languages. In particular, closure induction allows us to prove partial correctness, that is, to prove those instances of conjectures for which designated partial functions are explicitly defined.