Termination of system F-bounded (Note)

System F-bounded is a second order lambda calculus with subtyping. It extends system F≤ (see [Cardelli Wegner 85, Curien Ghelli 92, Ghelli 90, Cardelli et al. 91]) in that bounds of typed quantification may contain the bounded variable. In object-oriented terms, this feature allows one to write functions which accept parameters belonging to all the classes which inherit from one class. F-bounded quantification was introduced in [Canning et al. 89] and is currently included in many proposals for strongly typed object-oriented languages (e.g., [Bruce 93, Mitchell 90, Katiyar et al. 94]). A typed l-calculus is strongly normalizing (or terminating) when no infinite reduction chain starts from a typed term of that calculus. Termination is related to the possibility of solving some recursive type equations. For example, in a system with subtyping, if the disequation system {a ≤ a¤b, a¤b ≤ g, a¤b ≤ g¤b'}2 has a solution, then the non-terminating term (lx:a.x(x))(lx:a.x(x)):b ' is well typed. Some recursive disequations (e.g. a ≤ a¤b) can already be solved in F≤ by exploiting the Top type (e.g. a=Top¤b). Fbounded quantification introduces a new class of solutions, involving type variables (e.g. a=t when t ≤ t¤b). This raises the question of whether the addition of F-bounded quantification may allow non-terminating terms to be written. In this note we prove that such terms cannot, in fact, be written. Our proof is based on the computability method [Girard 72], and our approach is similar to the one used in [Mitchell 86] to prove termination for F, and, more closely, to the proof of termination of F≤ given in [Ghelli 90]. We first show that the type erasure of each well-typed F-bounded term is strongly normalizing. To this aim, we define an interpretation of F-bounded types such that each type is interpreted with a superset of the type erasures of all F-bounded terms with that type, and this superset is small enough to be contained in the set SN of strongly normalizing l-terms: G ∫ a: T „ typeErasure(a) Ï [G ∫ T] and [G ∫ T] Î SN. From this, we derive strong normalization for system F-bounded. System F-bounded is introduced in Section 2. Strong normalization is proved in Section 3.