Type inference in systems of recursive types with subtyping

We present general methods for performing type inference and deciding subtyping in languages with recursive types. Our type inference algorithm generalizes a common idea of previous work: type inference is reduced to a constraint satisfaction problem, whose satisfiability can be decided by a process of closure and consistency checking. We prove a general correctness theorem for this style of type inference. We define subtyping co-inductively, and we prove by co-induction that a closed and consistent constraint set has a solution. Our theorem makes it easier to find new type inference algorithms. For example, we provide definitions of closure and consistency for recursive types with a greatest type, but not a least type; we show that the definitions satisfy the conditions of our theorem; and the theorem immediately provides a type inference algorithm, thereby solving an open problem.