Type Refinements for Compiler Correctness

Type refinements, introduced by Freeman and Pfenning and explored by Davies and Dunfield, unify the ontological and epistemic views of typing. Types tell us what programming language constructs exist, whereas refinements express properties of the values of a type. Here we show that refinements are very useful in compiler correctness proofs, wherein it often arises that two expressions that are inequivalent in general are equivalent in the particular context in which they occur. Refinements serve to restrict the contexts sufficiently so that the desired equivalence holds. For example, an expression might be replaced by a more efficient one, even though it is not generally equivalent to the original, but is interchangeable in any context satisfying a specified refinement of the type of those expressions. We study here in detail a particular problem of compiler correctness, namely the correctness of compiling polymorphism (generics) to dynamic types by treating values of variable type as values of a universal dynamic type. Although this technique is widely used (for example, to compile Java generics), no proof of its correctness has been given to date. Surprisingly, standard arguments based on logical relations do not suffice, precisely because it is necessary to record deeper invariants about the compiled code than is expressible in their types alone. We show that refinements provide an elegant solution to this problem by capturing the required invariants so that a critical invertibility property that is false is general can be proved to hold in the contexts that arise in the translated code. This proof not only establishes the correctness of this compilation method, but also exemplifies the importance of refinements for compiler correctness proofs more generally.