As mentioned before, by inserting a constraint into a program, we shrink the domain for which it is defined. To reverse this process, we need to be able to speak of, and make use of, a set of subprograms. To this end, a new programming construct called a program set is now introduced. The meaning of a program set, or a set of programs, is identical to the conventional notion of a set of other objects. As usual, a set of n programs is denoted by {P1, P2, …, Pn}. When used as a programming construct, it describes the computation prescribed by its elements. Formally, the semantics of such a set is defined in Axiom 4.1.

Axiom 4.1

wp({P1, P2, …, Pn}, R) ≡ wp(P1, R) ⋁ wp(P2, R) ⋁ … ⋁ wp(Pn, R).

The choice of this particular semantics is explained in detail at the end of this chapter. A program set so defined has all properties commonly found in an ordinary set. For instance, because the logical operation of disjunction is commutative, a direct consequence of Axiom 4.1 is Corollary 4.2.

Corollary 4.2

The ordering of elements in a program set is immaterial, i.e.,

{P1, P2} ⇔ {P2, P1}.

Furthermore, because every proposition is an idempotent under the operation of disjunction, we have Corollary 4.3.

Corollary 4.3

P ⇔ {P} ⇔ {P, P} for any program P.

In words, a set is unchanged by listing any of its elements more than once.