A Practical Approach to Partiality - A Proof Based Approach

Partial functions are frequently used when specifying and reasoning about computer programs. Using partial functions entails reasoning about potentially ill-defined expressions. In this paper we show how to formally reason about partial functions without abandoning the well understood domain of classical two-valued predicate calculus. In order to achieve this, we extend standard predicate calculus with the notion of well-definedness which is currently used to filter out potentially ill-defined statements from proofs. The main contribution of this paper is to show how the standard predicate calculus can be extended with a new set of derived proof rules that can be used to preserve well-definedness in order to make proofs involving partial functions less tedious to perform.