Theory and implementation of coercive subtyping

Coercive subtyping is a useful and powerful framework of subtyping for type theories. In this thesis, we point out the problem in the old formulation of coercive subtyping in [Luo99], give a new and adequate formulation T [C], the system that extends a type theory T with coercive subtyping based on a set C of basic subtyping judgements, and show that coercive subtyping is a conservative extension and, in a more general sense, a definitional extension. We introduce an intermediate system, the star-calculus T [C]∗, in which the positions that require coercion insertions are marked, and show that T [C]∗ is a conservative extension of T and that T [C]∗ is equivalent to T [C]. Further more, in order to capture all the properties of the coercive subtyping framework, we introduce another intermediate system T [C]0K which does not contain the coercion application rules. We show that T [C]∗ is actually a definitional extension of T [C]0K , which is a conservative extension of T . This makes clear what we mean by coercive subtyping being a conservative extension and amends a technical problem that has led to a gap in the earlier conservativity proof. Another part of the work in this thesis concerns the implementation of coercive subtyping in the proof assistant Plastic. Coercive subtyping was implemented in Plastic by Paul Callaghan [CL01]. We have done some improvement based on that work, fixed some problems of Plastic, and implemented a new kind of data type called dot-types, which are special data types useful in formal semantics to describe interesting linguistic phenomena such as copredication, in Plastic.