Coercive Subtyping via Mappings of Reduction Behaviour

Abstract

This paper reports preliminary work on a novel approach to Coercive Subtyping that is based on relationships between reduction behaviour of source and target types in coerced terms. Coercive Subtyping is a superset of record-based subtyping, allowing so-called coercion functions to carry the subtyping. This allows many novel and powerful forms of subtyping and abbreviation, with applications including interfaces to theorem provers and programming with dependent type systems. However, the use of coercion functions introduces non-trivial overheads, and requires difficult proof of properties such as coherence in order to guarantee sensible results. These points restrict the practicality of coercive subtyping. We begin from the idea that coercing a value v from type U to a type T intuitively means that we wish to compute with v as if it was a value in T, not that v must be converted into a value in T. Instead, we explore how to compute on U in terms of computation on T, and develop a framework for mapping computations on some T to computations on some U via a simple extension of the elimination rule of T. By exposing how computations on different types are related, we gain insight on and make progress with several aspects of coercive subtyping, including (a) distinguishing classes of coercion and finding reasons to deprecate use of some classes; (b) alternative techniques for proving key properties of coercions; (c) greater efficiency from implementations of coercions.