Abstract Using Isabelle/HOL, we verify a union-find data structure with an explain operation due to Nieuwenhuis and Oliveras. We devise a simpler, more naive version of the explain operation whose soundness and completeness is easy to verify. Then, we prove the original formulation of the explain operation to be equal to our version. Finally, we refine this data structure to Imperative HOL, enabling us to export efficient imperative code. The formalisation provides a stepping stone towards the verification of proof-producing congruence closure algorithms which are a core ingredient of Satisfiability Modulo Theories (SMT) solvers.

Abstract Using Isabelle/HOL, we verify the state-of-the-art decision procedure for multi-level syllogistic with singleton (MLSS for short), which is a quantifier-free fragment of set theory. We formalise its syntax and semantics as well as a sound and complete tableau calculus for it. We also provide an executable specification of a decision procedure that exhaustively applies the rules of the calculus and prove its termination. Furthermore, we extend the calculus with a lightweight type system that paves the way for an integration of the procedure into Isabelle/HOL.