Inference of the Definition of the Predicate Transformer wp with Occurrences of the Predicate Domain Based on Denotational Semantics of GCL on ZF Set Theory

Dijkstra recursively defined the predicate transformer wp. Then Gries for each expression Exp of the language, defined domain(Exp), which is a predicate that indicates the states in which Exp is defined. This predicate Gries added it to the recursive formula that defines wp for assignment, and subsequently other authors added it to the rule that recursively defines wp for IF, so that in the bibliography there are several versions of the definition of wp, with and without occurrence of domain. The present work shows an inference of the definition of wp, demonstrating that the occurrence of domain is necessary for wp in assignement, IF and DO. This inference is done through the GCL denotational semantics over the set theory ZF, showing that the classical formulas of Dijkstra to define wp in GCL using domain, are valid if the language of set theory is used to write the assertions.