Reasoning about logical systems in the Coq proof assistant

Abstract

The theory of institutions provides an abstract mathematical framework for specifying logical systems and their semantic relationships. Institutions are based on category theory and have deep roots in a well-developed branch of algebraic specification. However, there are no machine-assisted proofs of correctness for institution-theoretic constructions—chiefly satisfaction conditions for institutions and their (co)morphisms—making them difficult to incorporate into mainstream formal methods. This paper therefore provides the details of our approach to formalizing a fragment of the theory of institutions in the Coq proof assistant. We instantiate this framework with the institutions FOPEQ for first-order predicate logic and EVT for the Event-B specification language, and define some institution-independent constructions, all of which serve as an illustration and evaluation of the overall approach.