Computing correctly with inductive relations

Inductive relations are the predominant way of writing specifications in mechanized proof developments. Compared to purely functional specifications, they enjoy increased expressive power and facilitate more compositional reasoning. However, inductive relations also come with a significant drawback: they can’t be used for computation. In this paper, we present a unifying framework for extracting three different kinds of computational content from inductively defined relations: semi-decision procedures, enumerators, and random generators. We show how three different instantiations of the same algorithm can be used to generate all three classes of computational definitions inside the logic of the Coq proof assistant. For each derived computation, we also derive mechanized proofs that it is sound and complete with respect to the original inductive relation, using Ltac2, Coq’s new metaprogramming facility. We implement our framework on top of the QuickChick testing tool for Coq, and demonstrate that it covers most cases of interest by extracting computations for the inductive relations found in the Software Foundations series. Finally, we evaluate the practicality and the efficiency of our approach with small case studies in randomized property-based testing and proof by computational reflection.