While the results of the preceding chapter provide a satisfactory treatment of type inference with qualified types, we have not yet made any attempt to discuss the semantics or evaluation of overloaded terms. For example, given a generic equality operator (==) of type ∀a.Eq a ⇒ a → a → Bool and integer valued expressions E and F, we can determine that the expression E == F has type Bool in any environment which satisfies Eq Int. However, this information is not sufficient to determine the value of E == F; this is only possible if we are also provided with the value of the equality operator which makes Int an instance of Eq.

Our aim in the next two chapters is to present a general approach to the semantics and implementation of objects with qualified types based on the concept of evidence. The essential idea is that an object of type π ⇒ σ can only be used if we are also supplied with suitable evidence that the predicate π does indeed hold. In this chapter we concentrate on the role of evidence for the systems of predicates described in Chapter 2 and then, in the following chapter, extend the results of Chapter 3 to give a semantics for OML.

As an introduction, Section 4.1 describes some simple techniques used in the implementation of particular forms of overloading and shows why these methods are unsuitable for the more general systems considered in this thesis.