Equational Formulae with Membership Constraints

We propose a set of transformation rules for rst order formulae whose atoms are either equations between terms or \membership constraints" t 2. can be interpreted as a regular tree language (is called a sort in the algebraic speciication community) or as a tree language in any class of languages which satisses some adequate closure and decidability properties. This set of rules is proved to be correct, terminating and complete. This provides a quantiier elimination procedure: for every regular tree language L, the rst order theory of some structure deening L is decidable. This extends the results of Mal'cev (1971), Maher (1988), Comon and Lescanne (1989). We also show how to apply our results to automatic inductive proofs in equational theories.