The Rewster: Type Preserving Rewrite Rules for the Coq Proof Assistant

In dependently typed proof assistants, users can declare axioms to extend the ambient logic locally with new principles and propositional equalities governing them.Additionally, rewrite rules have recently been proposed to allow users to extend the logic with new definitional equalities, enabling them to handle new principles with a computational behaviour.While axioms can only break consistency, the addition of arbitrary rewrite rules can break other important metatheoretical properties such as type preservation.In this paper, we present an implementation of rewrite rules on top of the Coq proof assistant, together with a modular criterion to ensure that the added rewrite rules preserve typing. This criterion, based on bidirectional type checking, is formally expressed in PCUIC --- the type theory of Coq recently developed in the MetaCoq project.