Proposition 3.3 (soundness of Non-structural Subtyping)

9 union types and existential types as well as intersection types; a similar observation is in 12] where, rather than interpreting types as sets of terms in SN, types are interpreted as sets of values (with respect to a speciied evaluation relation) of terms in SN. In 12] it is shown how this approach yields clean normalization proofs for many powerful extensions of system F 2 , including several extensions with subtyping. Important examples are system F-bounded and Ghelli's system F ; system F was rst shown to be strongly normalizing in Ghelli's thesis 5], which uses saturated sets. Termination of system F-bounded is also proven in 6] using similar methods. Recent generalizations of techniques for proving strong normalization can be found in 8], 16], 3]. Comparing the typing power of the non-structural subtyping system to other systems, it is noted in 24] that system F 2 and the system (!; ?; >; B) are incomparable with respect to typing power, since they both type some terms that have no type in the other system; this is observed to hold also for F 2 in comparison with Thatte's partial typing. The system (!; ?; >; B) is strictly more powerful than partial typing 24]. It is also worth noting that a bottom type (a subtype of every type) is not deenable in system F , since, as is shown in 5], there are types with no common lower bound in system F. Palsberg 18] shows that every lambda term in normal form has a partial type in Thatte's system. Type inference algorithms for partial types and non-structural subtyping can be found in 9], 17]. Acknowledgement Thanks to an anonymous referee for helpful comments. 8 2 The last part of our proof is to extend the proof of Proposition 2.4 with one new case of the induction, covering the subtyping rule via Lemma 3.2. proof The proof is by induction on the derivation of ? ` M :. We consider only the new case where we use the subtyping rule 2 Finally, we note that the extended version of Theorem 2.5 goes through as usual, using Proposition 3.3 in the standard way. This completes the proof of strong normalization for non-structural subtyping. 4 Other systems It is not diicult to see that appropriate analogues of Lemma 3.2 will hold for many other subtyping systems than the one considered here. So long …