Judgmental subtyping systems with intersection types and modal types

We study how to extend modal type systems based on intuitionistic modal logic S4 or S5 with a subtyping system based on intersection types. In the presence of four type constructors ${!}!rightarrow !{!},,{!}wedge {!},,square {}$ , and $Diamond {}$ , the traditional approach using a binary subtyping relation does not work well because of lack of orthogonality in subtyping rules and presence of a transitivity rule. We adopt the idea from the judgmental formulation of modal logic (Pfenning and Davies in Math Struct Comput Sci 11(4):511–540, 2001) and use subtyping judgments whose definitions express those notions internalized into type constructors directly at the level of judgments. The resultant judgmental subtyping systems admit cut rules similarly to a sequent calculus for intuitionistic logic and play a key role in designing and verifying the relational subtyping systems based on the binary subtyping relation. We use the proof assistant Coq to prove the admissibility of the cut rules and the equivalence between the two kinds of subtyping systems. The lesson from our study is that by using subtyping judgments instead of the binary subtyping relation, we can overcome the limitation usually associated with the syntactic approach to formulating subtyping systems.