Ranjit Jhala

Practical checkers based on refinement types use the combination of implicit semantic subtyping and parametric polymorphism to simplify the specification and automate the verification of sophisticated properties of programs. However, a formal metatheoretic accounting of the soundness of refinement type systems using this combination has proved elusive. We present λ RF , a core refinement calculus that combines semantic subtyping and parametric polymorphism. We develop a metatheory for this calculus and prove soundness of the type system. Finally, we give two mechanizations of our metatheory. First, we introduce data propositions , a novel feature that enables encoding derivation trees for inductively defined judgments as refined data types, and use them to show that LiquidHaskell’s refinement types can be used for mechanization. Second, we mechanize our results in Coq, which comes with stronger soundness guarantees than LiquidHaskell, thereby laying the foundations for mechanizing the metatheory of LiquidHaskell.

Several practical tools for automatically verifying functional programs (e.g., Liquid Haskell and Leon for Scala programs) rely on a heuristic based on unrolling recursive function definitions followed by quantifier-free reasoning using SMT solvers. We uncover foundational theoretical properties of this heuristic, revealing that it can be generalized and formalized as a technique that is in fact complete for reasoning with combined First-Order theories of algebraic datatypes and background theories, where background theories support decidable quantifier-free reasoning. The theory developed in this paper explains the efficacy of these heuristics when they succeed, explain why they fail when they fail, and the precise role that user help plays in making proofs succeed.