Occurrence Typing Modulo Theories

We present a new type system combining occurrence typing, previously used to type check programs in dynamically-typed languages such as Racket, JavaScript, and Ruby, with dependent refinement types. We demonstrate that the addition of refinement types allows the integration of arbitrary solver-backed reasoning about logical propositions from external theories. By building on occurrence typing, we can add our enriched type system as an extension of Typed Racket---adding dependency and refinement reuses the existing formalism while increasing its expressiveness. Dependent refinement types allow Typed Racket programmers to express rich type relationships, ranging from data structure invariants such as red-black tree balance to preconditions such as vector bounds. Refinements allow programmers to embed the propositions that occurrence typing in Typed Racket already reasons about into their types. Further, extending occurrence typing to refinements allows us to make the underlying formalism simpler and more powerful. In addition to presenting the design of our system, we present a formal model of the system, show how to integrate it with theories over both linear arithmetic and bitvectors, and evaluate the system in the context of the full Typed Racket implementation. Specifically, we take safe vector access as a case study, and examine all vector accesses in a 56,000 line corpus of Typed Racket programs. Our system is able to prove that 50% of these are safe with no new annotation, and with a few annotations and modifications, we can capture close to 80%.