A Calculus for Overload Functions with Subtyping

We present a simple extension of typed-calculus where functions can be overloaded by putting diierent \branches of code" together. When the function is applied, the branch to execute is chosen according to a particular selection rule which depends on the type of the argument. The crucial feature of the present approach is that the branch selection depends on the \run-time type" of the argument , which may diier from its compile-time type, because of the existence of a subtyping relation among types. Hence overloading cannot be eliminated by a static analysis of code, but is an essential feature to be dealt with during computation. We obtain in this way a type-dependent calculus, which diiers from the various-calculi where types do not play any r^ ole during computation. We prove Connuence and a generalized Subject-Reduction theorem for this calculus. We prove Strong Normalization for a \stratiied" subcalculus. The deenition of this calculus is guided by the understanding of object-oriented features and the connections between our calculus and object-orientedness are extensively stressed. We show that this calculus provides a foundation for typed object-oriented languages which solves some of the problems of the standard record-based approach.