Coercive Subtyping in Type Theory

We propose and study coercive subtyping, a formal extension with subtyping of dependent type theories such as Martin-LL of's type theory NPS90] and the type theory UTT Luo94]. In this approach, subtyping with speciied implicit coercions is treated as a feature at the level of the logical framework; in particular, subsumption and coercion are combined in such a way that the meaning of an object being in a supertype is given by coercive deenition rules for the deenitional equality. It is shown that this provides a conceptually simple and uniform framework to understand subtyping and coercion relations in type theories with sophisticated type structures such as inductive types and universes. The use of coercive subtyping in formal development and in reasoning about subsets of objects is discussed in the context of computer-assisted formal reasoning.