A General Framework for Relational Parametricity

Reynolds' original theory of relational parametricity was intended to capture the observation that polymorphically typed System F programs preserve all relations between inputs. But as Reynolds himself later showed, his theory can only be formulated in a metatheory with an impredicative universe, such as the Calculus of Inductive Constructions. A number of more abstract treatments of relational parametricity have since appeared; however, as we show, none of these seem to express Reynolds' original theory in a satisfactory way. To correct this, we develop an abstract framework for relational parametricity that delivers a model expressing Reynolds' theory in a direct and natural way. This framework is uniform with respect to a choice of meta-theory, which allows us to obtain the well-known PER model of Longo and Moggi as a direct instance in a natural way as well. Underlying the framework is the novel notion of a λ2-fibration with isomorphisms, which relaxes certain strictness requirements on split λ2-fibrations. Our main theorem is a generalization of Seely's classical construction of sound models of System F from split λ2-fibrations: we prove that the canonical interpretation of System F induced by every λ2-fibration with isomorphisms validates System F's entire equational theory on the nose, independently of the parameterizing meta-theory. Moreover, we offer a novel relationally parametric model of System F (after Orsanigo), which is proof-relevant in the sense that witnesses of relatedness are themselves suitably related. We show that, unlike previous frameworks for parametricity, ours recognizes this new model in a natural way. Our framework is thus descriptive, in that it accounts for well-known models, as well as prescriptive, in that it identifies abstract properties that good models of relational parametricity should satisfy and suggests new constructions of such models.