The connection between two ways of reasoning about partial functions

Abstract

Undefined terms involving the application of partial functions and operators are common in program specifications and in discharging proof obligations that arise in design. One way of reasoning about partial functions with classical First-order Predicate Calculus (FoPC) is to use a non-strict equality notion so as to insulate logical operators from undefined operands. An alternative approach is to work only with strict (weak) equality but use an alternative Logic of Partial Functions (LPF)—a logic in which the “Law of the Excluded Middle” does not hold. This paper explores the relationships between the theorems that can be proved in the two approaches. The main result is that theorems in LPF using weak equality can be straightforwardly translated into ones that are true in FoPC; translation in the other direction results, in general, in more complicated expressions but in many cases these can be readily simplified. Such results are important if the laudable move towards interworking of formal methods tools is to be sound.