A calculus of constructions with explicit subtyping

Abstract : The calculus of constructions can be extended with an infinite hierarchy of universes and cumulative subtyping. In this hierarchy, each universe is contained in a higher universe. Subtyping is usually left implicit in the typing rules. We present an alternative version of the calculus of constructions where subtyping is explicit. This new system avoids problems related to coercions and dependent types by using the Tarski style of universes and by introducing additional equations to reflect equality.
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Ali Assaf. A calculus of constructions with explicit subtyping. 20th International Conference on Types for Proofs and Programs (TYPES 2014), May 2014, Institut Henri Poincaré, Paris, France. ⟨hal-01097401v2⟩

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