Internal and Observational Parametricity for Cubical Agda

Two approaches exist to incorporate parametricity into proof assistants based on dependent type theory. On the one hand, parametricity translations conveniently compute parametricity statements and their proofs solely based on individual well-typed polymorphic programs. But they do not offer internal parametricity: formal proofs that any polymorphic program of a certain type satisfies its parametricity statement. On the other hand, internally parametric type theories augment plain type theory with additional primitives out of which internal parametricity can be derived. But those type theories lack mature proof assistant implementations and deriving parametricity in them involves low-level intractable proofs. In this paper, we contribute Agda --bridges: the first practical internally parametric proof assistant. We provide the first mechanized proofs of crucial theorems for internal parametricity, like the relativity theorem. We identify a high-level sufficient condition for proving internal parametricity which we call the structure relatedness principle (SRP) by analogy with the structure identity principle (SIP) of HoTT/UF. We state and prove a general parametricity theorem for types that satisfy the SRP. Our parametricity theorem lets us obtain one-liner proofs of standard internal free theorems. We observe that the SRP is harder to prove than the SIP and provide in Agda --bridges a shallowly embedded type theory to compose types that satisfy the SRP. This type theory is an observational type theory of logical relations and our parametricity theorem ought to be one of its inference rules.