Type theory in type theory using quotient inductive types

Abstract

We present an internal formalisation of a type heory with dependent types in Type Theory using a special case of higher inductive types from Homotopy Type Theory which we call quotient inductive types (QITs). Our formalisation of type theory avoids referring to preterms or a typability relation but defines directly well typed objects by an inductive definition. We use the elimination principle to define the set-theoretic and logical predicate interpretation. The work has been formalized using the Agda system extended with QITs using postulates.