  In recent work we have shown how it is possible to define very precise type
systems for object-oriented languages by abstractly compiling a program into a
Horn formula f. Then type inference amounts to resolving a certain goal w.r.t.
the coinductive (that is, the greatest) Herbrand model of f.
  Type systems defined in this way are idealized, since in the most interesting
instantiations both the terms of the coinductive Herbrand universe and goal
derivations cannot be finitely represented. However, sound and quite expressive
approximations can be implemented by considering only regular terms and
derivations. In doing so, it is essential to introduce a proper subtyping
relation formalizing the notion of approximation between types.
  In this paper we study a subtyping relation on coinductive terms built on
union and object type constructors. We define an interpretation of types as set
of values induced by a quite intuitive relation of membership of values to
types, and prove that the definition of subtyping is sound w.r.t. subset
inclusion between type interpretations. The proof of soundness has allowed us
to simplify the notion of contractive derivation and to discover that the
previously given definition of subtyping did not cover all possible
representations of the empty type.
