A Sequent Calculus for Subtyping Polymorphic Types

We present two complete systems for polymorphic types with sub- typing. One system is in the style of natural deduction, while another is a Gentzen-style sequent calculus system. We prove several meta- mathematical properties for these systems including cut elimination, subject reduction, coherence, and decidability of type reconstruction. Following the approach by J. Mitchell, the sequents are given a simple semantics using logical relations over applicative structures. The systems are complete with respect to this semantics. The logic which emerges from this paper can be seen as a successor to the original Hilbert style system proposed by J. Mitchell in 1988, and to the “half way” sequent calculus of G. Longo, K. Milsted, and S. Soloviev proposed in 1995.