A Theory of Higher-Order Subtyping with Type Intervals (Extended Version)

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 𝜔 •• , a rigorous theoretical foundation for Scala's higher-kinded types. 𝐹 𝜔 •• extends 𝐹 𝜔 <: 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 𝜔•• , 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.