Recursive subtyping revealed

Algorithms for checking subtyping between recursive types lie at the core of many programming language implementations. But the fundamental theory of these algorithms and how they relate to simpler declarative speci cations is not widely understood, due in part to the di culty of the available introductions to the area. This tutorial paper o ers an \end-to-end" introduction to recursive types and subtyping algorithms, from basic theory to e cient implementation, set in the unifying mathematical framework of coinduction.