Inductive Reasoning with Equality Predicates, Contextual Rewriting and Variant-Based Simplification

Abstract

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 20 inference rules, but only 9 of them require user interaction; the remaining 11 can be automated as simplification rules. In this way, a substantial fraction of the proof effort can be automated. The inference rules are based on advanced equational reasoning techniques, including: 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.

Equational Unification and Matching, and Symbolic Reachability Analysis in Maude 3.2 (System Description)

“…(4) Theorem Proving. The new Maude Inductive Theorem Prover under construction [34], as well as Maude's Invariant Analyzer [43] and Reachability Logic Theorem Prover [45] all use equational unification and narrowing modulo equations; so all will benefit from the new features. ( 5) Theory Transformations based on equational unification, e.g., partial evaluation [4], ground confluence methods [17] or program termination methods [25,26] could likewise become more efficient.…”