From Amber to Coercion Constraints

Subtyping is a common tool in the design of type systems that finds its roots in the η-expansion of arrow types and the notion of type containment obtained by closing System F by η-expansion. Although strongly related, subtyping and type containment still significantly differ from one another when put into practice. We introduce coercion constraints to relate and generalize subtyping and type containment as well as all variants of Fbounded quantification and instance-bounded quantification used for firstorder type inference in the presence of second-order types. We obtain a type system with a clearer separation between computational and erasable parts of terms. 1 The different flavors of subtyping Subtyping in Amber Nowadays, subtyping is a well-understood concept and tool in the design of type systems, but it has not always been so. The origin of subtyping goes back to the 60’s when type conversions between base types or type classes were introduced in programming languages, but the first formalization of subtyping is by Reynolds (1980). Subtyping was introduced in the strongly typed functional language Amber by Cardelli (1984). By contrast with simple type conversions between base types, subtyping in Amber can be propagated through arrow types covariantly on the codomains of functions and contravariantly on their domains. The language Amber had only subtyping but no parametric polymorphism: its typing rules are those of the simply typed λ-calculus extended with a subtyping rule Sub: Sub Γ ` a : τ τ . σ Γ ` a : σ Bot