Layer Systems for Proving Confluence

We introduce layer systems for proving generalizations of the modularity of confluence for first-order rewrite systems. Layer systems specify how terms can be divided into layers. We establish structural conditions on those systems that imply confluence. Our abstract framework covers known results like modularity, many-sorted persistence, layer-preservation and currying. We present a counterexample to an extension of persistence to order-sorted rewriting and derive new sufficient conditions for the extension to hold. All our proofs are constructive.