This paper introduces a coalgebraic foundation for coinductive types, interpreted as sets of values and extended with set theoretic union. We give a sound and complete characterization of semantic sub-typing in terms of inclusion of maximal traces. Further, we provide a technique for reducing subtyping to inclusion between sets of finite traces, based on approximation. We obtain inclusion of tree languages as a sound and complete method to show semantic subtyping of recursive types with basic types, product and union, interpreted coinductively.
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