Contractive Signatures with Recursive Types, Type Parameters, and Abstract Types

Although theories of equivalence or subtyping for recursive types have been extensively investigated, sophisticated interaction between recursive types and abstract types has gained little attention. The key idea behind type theories for recursive types is to use syntactic contractiveness, meaning every mu-bound variable occurs only under a type constructor such as -> or *. This syntactic contractiveness guarantees the existence of the unique solution of recursive equations and thus has been considered necessary for designing a sound theory for recursive types. However, in an advanced type system, such as OCaml, with recursive types, type parameters, and abstract types, we cannot easily define the syntactic contractiveness of types. In this paper, we investigate a sound type system for recursive types, type parameters, and abstract types. In particular, we develop a new semantic notion of contractiveness for types and signatures using mixed induction and coinduction, and show that our type system is sound with respect to the standard call-by-value operational semantics, which eliminates signature sealings. Moreover we show that while non-contractive types in signatures lead to unsoundness of the type system, they may be allowed in modules. We have also formalized the whole system and its type soundness proof in Coq.