Two-dimensional proof-relevant parametricity

Relational parametricity is a fundamental concept within theoretical computer science and the foundations of programming languages, introduced by John Reynolds [6]. His fundamental insight was that types can be interpreted not just as functors on the category of sets, but also as equality preserving functors on the category of relations. This gives rise to a model where polymorphic functions are uniform in a suitable sense; this can be used to establish e.g. representation independence, equivalences between programs, or deriving useful theorems about programs from their type alone [7]. The relations Reynolds considered were proof-irrelevant, which from a type theoretic perspective is a little limited. As a result, one might like to extend his work to deal with proof-relevant, i.e. set-valued relations. However naive attempts to do this fail: the fundamental property of equality preservation cannot be established. Our insight is that just as one uses parametricity to restrict definable elements of a type, one can use parametricity to restrict definable proofs to ensure equality preservation in the proof-relevant setting. Proof-relevant logical relations also appear in Benton, Hofmann and Nigam’s recent work [1], but for unary relations, and in a setting without ∀-types. Hence they do not need to consider equality preservation. In our work, we consider proof-relevant binary relations. A proof-relevant relation R over two sets A and B consists of a map R : A × B → Set which, for every two elements (a, b) ∈ A×B, associates a set of proofs that they are related. On top of this, we have a new proof-irrelevant layer: a 2-dimensional relation Q is defined over four sets and four binary proof-relevant relations. It can be thought of as over a square, where the vertices are sets and the edges are the binary proof-relevant relations: