This paper explains the design and use of two equational proving tools, namely an inductive theorem prover -- to prove theorems about equational specifications with an initial algebra semantics -- and a Church-Rosser checker---to check whether such specifications satisfy the Church-Rosser property. These tools can be used to prove properties of order-sorted equational specifications in Cafe [11] and of membership equational logic specifications in Maude [7, 6]. The tools have been written entirely in Maude and are in fact executable specifications in rewriting logic of the formal inference systems that they implement.

Abstract. Abstraction reduces the problem of whether an infinite state system satisfies version. The most common abstractions are quotients of the original system. We present a simple method of defining quotient abstractions by means of equations collapsing the set of states. Our method yields the minimal quotient system together with a set of proof obligations that guarantee its executability and can be discharged with tools such as those in the Maude formal environment.