Refinement types decorate types with assertions that enable automatic verification. Like assertions, refinements are limited to binders that are in scope, and hence, cannot express higher-order specifications. Ghost variables circumvent this limitation but are prohibitively tedious to use as the programmer must divine and explicate their values at all call-sites. We introduce Implicit Refinement Types which turn ghost variables into implicit pair and function types, in a way that lets the refinement typechecker automatically synthesize their values at compile time. Implicit Refinement Types further take advantage of refinement type information, allowing them to be used as a lightweight verification tool, rather than merely as a technique to automate programming tasks. We evaluate the utility of Implicit Refinement Types by showing how they enable the modular specification and automatic verification of various higher-order examples including stateful protocols, access control, and resource usage.

We introduce Refinement Reflection , a new framework for building SMT-based deductive verifiers. The key idea is to reflect the code implementing a user-defined function into the function’s (output) refinement type. As a consequence, at uses of the function, the function definition is instantiated in the SMT logic in a precise fashion that permits decidable verification. Reflection allows the user to write equational proofs of programs just by writing other programs using pattern-matching and recursion to perform case-splitting and induction. Thus, via the propositions-as-types principle, we show that reflection permits the specification of arbitrary functional correctness properties. Finally, we introduce a proof-search algorithm called Proof by Logical Evaluation that uses techniques from model checking and abstract interpretation, to completely automate equational reasoning. We have implemented reflection in Liquid Haskell and used it to verify that the widely used instances of the Monoid, Applicative, Functor, and Monad typeclasses actually satisfy key algebraic laws required to make the clients safe, and have used reflection to build the first library that actually verifies assumptions about associativity and ordering that are crucial for safe deterministic parallelism.