Abstract
We present two complete systems for subtyping polymorphic types. One system is in the style of natural deduction, while another is a Gentzen style sequent calculus system. We prove several metamathematical 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.
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