Observational equality: now for good

Abstract

Building on the recent extension of dependent type theory with a universe of definitionally proof-irrelevant types, we introduce TTobs, a new type theory based on the setoidal interpretation of dependent type theory. TTobs equips every type with an identity relation that satisfies function extensionality, propositional extensionality, and definitional uniqueness of identity proofs (UIP). Compared to other existing proposals to enrich dependent type theory with these principles, our theory features a notion of reduction that is normalizing and provides an algorithmic canonicity result, which we formally prove in Agda using the logical relation framework of Abel et al. Our paper thoroughly develops the meta-theoretical properties of TTobs, such as the decidability of the conversion and of the type checking, as well as consistency. We also explain how to extend our theory with quotient types, and we introduce a setoidal version of Swan's Id types that turn it into a proper extension of MLTT with inductive equality.