Rewriting modulo a rewrite system

. We introduce rewriting with two sets of rules, the first interpreted equationally and the second not. A semantic view considers equational rules as defining an equational theory and reduction rules as defining a rewrite relation modulo this theory. An operational view considers both sets of rules as similar. We introduce sufficient properties for these two views to be equivalent (up to different notions of equivalence). The paper ends with a collection of example showing the effectiveness of this approach. Rewriting can be viewed simultaneously as the most basic symbol-manipulating method, and as a very expressive specification framework, given the expressive power of rewriting modulo equations. It is a primary candidate to the role of a general logical framework [Mes92, MOM93]. Historically, rewriting has been given an equational semantics, saying that a rewrite rule u \\Gamma! v is interpreted as u is equal to v. This is the case for instance when defining functions or solving the w...