Automated Deduction in the B Set Theory using Deduction Modulo

We introduce a new encoding of the set theory of the B method based on deduction modulo. The theory of deduction modulo is an extension of predicate calculus that includes rewriting on both terms and propositions, which is well suited for proof search in axiomatic theories, as it turns many axioms into rewrite rules. We also present Zenon Modulo and iProver Modulo, two automated theorem provers that rely on deduction modulo and that are intended to deal with this B set theory modulo. These two tools have backends based on the Dedukti universal proof checker, which also relies on deduction modulo, and which allow us to certify the correctness of the proofs produced by these two tools. Finally, we provide some experimental results obtained on a benchmark consisting of derived properties of the B set theory, which show a significant gain of the tools based on deduction modulo compared to other first order automated theorem provers. In addition, to show the effectiveness of our approach, we describe an example of proof, whose proof is only found by the tools based on deduction modulo.