Inductive reasoning with equality predicates, contextual rewriting and variant-based simplification

An inductive inference system for proving validity of formulas in the initial algebra $T_E$ of an order-sorted equational theory $E$ is presented. It has 21 inference rules. Only 9 of them require user interaction; the remaining 12 can be automated as simplification rules. In this way, a substantial fraction of the proof effort can be automated. Other rules can be automated by tactics. The inference rules are based on advanced equational reasoning techniques, including: equational proof search, equationally defined equality predicates, narrowing, constructor variant unification, variant satisfiability, order-sorted congruence closure, contextual rewriting, ordered rewriting, and recursive path orderings. All these techniques work modulo axioms B, for B any combination of associativity and/or commutativity and/or identity axioms. Most of these inference rules have already been implemented in Maude's NuITP inductive theorem prover.