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.

One of the key goals of rewriting logic from its beginning has been to provide a semantic and logical framework in which many models of computation and languages can be naturally represented. There is by now very extensive evidence supporting the claim that rewriting logic is indeed a very flexible and simple logical and semantic framework. From a language design point of view the obvious question to ask is: how can a rewriting logic language best support logical and semantic framework applications, so that it becomes a metalanguage in which a very wide variety of logics and languages can be both semantically defined, and implemented? Our answer is: by being reflective. This paper discusses our latest language design and implementation work on Maude as a reflective metalanguage in which entire environments---including syntax definition, parsing, pretty printing, execution, and input/output---can be defined for a language or logic L of choice. 1

Maude's language design and implementation make systematic use of the fact that rewriting logic is reflective. This makes the metatheory of rewriting logic accessible to the user in a clear and principled way, and makes possible many advanced metaprogramming applications, including user-definable strategy languages, language extensions by new module composition operations, development of theorem proving tools, and reifications of other languages and logics within rewriting logic. A naive implementation of reflection can be computationally very expensive. We explain the semantic principles and implementation techniques through which efficient ways of performing reflective computations are achieved in Maude through its predefined META-LEVEL module.

This document explains the design and use of a Church-Rosser checker tool, which checks whether an equational specification satisfies the Church-Rosser property. This tool can be used to prove the Church-Rosser property of order-sorted equational specifications in Maude [12, 8, 6]. The tool has been written entirely in Maude and is in fact an executable specification in rewriting logic [29] of the formal inference system that it implements. The fact that rewriting logic is reflective [13, 5], and that Maude efficiently supports reective rewriting logic computations [7, 6] is systematically exploited in the design of the tool.

This document describes the executable formal specification of Full Maude -- a version of Maude supporting a rich module algebra with module hierarchies, parameterization, views, theories, module expressions, and object-oriented modules -- which is in fact its implementation in the Maude 1.00 system

A categorical four-rule deduction system for equational logics is presented. We show that under reasonable niteness requirements this system is complete with respect to equational satisfaction abstracted as injectivity. The generality of the presented framework allows one to derive conditional equations as well at no extra cost. In fact, our deduction system is also complete for conditional equations, a new result at the author's knowledge.

This paper introduces the novel concept of inclusive institution as a foundational framework for studying logic-independent module compositionality, defines specification modules as specifications allowing both public and private signatures, and shows that an internal property of modules, called conservatism, is crucial for compositional semantics.