The Rewster: Type Preserving Rewrite Rules for the Rocq Prover
Résumé
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 preservation of typing. In this paper, we present an implementation of rewrite rules on top of the Rocq Prover, together with two modular criteria to ensure that the added rewrite rules preserve confluence and typing. These criterion, based on the triangle property for the former and bidirectional type checking for the latter, are formally expressed in PCUIC-the type theory of Rocq recently developed in the MetaRocq project.
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