Definitional Extension in Type Theory

When we extend a type system, the relation between the original system and its extension is an important issue we want to know. Conservative extension is a traditional relation we study with. But in some cases, like coercive subtyping, it is not strong enough to capture all the properties, more powerful relation between the systems is required. We bring the idea definitional extension from mathematical logic into type theory. In this paper, we study the notion of definitional extension for type theories and explicate its use, both informally and formally, in the context of coercive subtyping