Identity Inclusion in Relational Type Theory
Where relational semantics for parametric polymorphism often includes a lemma called Identity Extension (discussed in Episode 10, on the paper "Types, Abstraction, and Parametric Polymorphism"), RelTT instead has a refinement of this called Identity Inclusion. Instead of saying that the interpretation of every closed type is the identity relation (Identity Extension), the Identity Inclusion lemma identifies certain types whose relational meaning is included in the identity relation, and certain types which include the identity relation. So there are two subset relations, going in opposite directions. The two classes of types are first, the ones where all quantifiers occur only positively, and second, where they occur only negatively. Using Identity Inclusion, we can derive transitivity for forall-positive types, which is needed to derive induction following the natural generalization of the scheme in Wadler's paper (last episode).