Computational Higher Type Theory III: Univalent Universes and Exact Equality

This is the third in a series of papers extending Martin-Löf’s meaning explanations of dependent type theory to a Cartesian cubical realizability framework that accounts for higherdimensional types. We extend this framework to include a cumulative hierarchy of univalent Kan universes of Kan types; exact equality and other pretypes lacking Kan structure; and a cumulative hierarchy of pretype universes. As in Parts I and II, the main result is a canonicity theorem stating that closed terms of boolean type evaluate to either true or false. This establishes the computational interpretation of Cartesian cubical higher type theory based on cubical programs equipped with a deterministic operational semantics.