How to Safely Use Extensionality in Liquid Haskell

Refinement type checkers are a powerful way to reason about functional programs. For example, one can prove properties of a slow, specification implementation and port the proofs to an optimized pure implementation that behaves the same. But to reason about higher-order programs, we must reason about equalities between functions: we need a consistent encoding of functional extensionality. A natural but naive phrasing of the functional extensionality axiom (funext) is inconsistent in refinement type systems with semantic subtyping and polymorphism: if we assume funext, then we can prove false. We demonstrate the inconsistency and develop a new approach to equality in Liquid Haskell: we define a propositional equality in a library we call PEq. Using PEq avoids the inconsistency while proving useful equalities at higher types; we demonstrate its use in several case studies. We validate PEq by building a model and developing its metatheory. Additionally, we prove metaproperties of PEq inside Liquid Haskell itself using an unnamed folklore technique, which we dub ‘classy induction’.