# A Theory of Higher-Order Subtyping with Type Intervals

Sandro Stucki and Paolo G. Giarrusso. In Proceedings of the ACM on Programming Languages, 5(ICFP), pp. 69:1–69:30, ACM, 2021.

## Abstract

The calculus of Dependent Object Types (DOT) has enabled a more principled and robust implementation of Scala, but its support for type-level computation has proven insufficient. As a remedy, we propose $$F^\omega_{..}$$, a rigorous theoretical foundation for Scala’s higher-kinded types. $$F^\omega_{..}$$ extends $$F^\omega_{<:}$$ with interval kinds, which afford a unified treatment of important type- and kind-level abstraction mechanisms found in Scala, such as bounded quantification, bounded operator abstractions, translucent type definitions and first-class subtyping constraints. The result is a flexible and general theory of higher-order subtyping. We prove type and kind safety of $$F^\omega_{..}$$, as well as weak normalization of types and undecidability of subtyping. All our proofs are mechanized in Agda using a fully syntactic approach based on hereditary substitution.