Application of monadic substitution to recursive type containment

In this paper, we present a computer-checked, constructive soundness and completeness result for prototypic recursive type containment with respect to containment of nonwellfounded (finite or infinite) trees. The central role is played by formalisation of substitution of recursive types as a monad, with a traverse function implementing a strategy for potentially infinite recursive unfolding. In the rigorous setting of constructive type theory, the very definition of subtyping should be scrutinised to allow a prover to accept it. As a solution, we adapt the method of weak bisimilarity by mixed induction-coinduction recently introduced to Coq by Nakata and Uustalu. However, our setting is different in that we work with a notion of weak similarity that corresponds to subtyping. We accomplish the task of representing infinitary subtyping in the Calculus of (Co-)Inductive Constructions. Our technique does not require extensions of the Calculus and therefore can be ported to other dependently typed languages.