A Framework for Internalizing Relations into Type Theory

This paper introduces the concept of internalization structure, which can be used to incorporate certain relations into F , a variant of system F, while maintaining termination of the new system. We will call this process of incorporation internalization, F the base system and the new system after the incorporation the internalized system. We first specify the syntax, and then the semantics of F via the Tait-Girard reducibility method. We then define internalization structure. We show that we can obtain a terminating internalized system from an internalization structure. Finally, as motivating examples, we demonstrate how our framework can be applied to internalize subtyping, full-beta term equality and term-type inhabitation relations.