Abstract. mu-term is a tool which can be used to verify a number of termination properties of (variants of) Term Rewriting Systems (TRSs): termination of rewriting, termination of innermost rewriting, termination of order-sorted rewriting, termination of context-sensitive rewriting, termination of innermost context-sensitive rewriting and termination of rewriting modulo specific axioms. Such termination properties are essential to prove termination of programs in sophisticated rewriting-based programming languages. Specific methods have been developed and implemented in mu-term in order to efficiently deal with most of them. In this paper, we report on these new features of the tool. 1

We introduce the notion of well-founded recursive order-sorted equational logic (OS) theories modulo axioms. Such theories define functions by well-founded recursion and are inherently terminating. Moreover, for well-founded recursive theories important properties such as confluence and sufficient completeness are modular for so-called fair extensions. This enables us to incrementally check these properties for hierarchies of such theories that occur naturally in modular rule-based functional programs. Well-founded recursive OS theories modulo axioms contain only commutativity and associativity-commutativity axioms. In order to support arbitrary combinations of associativity, commutativity and identity axioms, we show how to eliminate identity and (under certain conditions) associativity (without commutativity) axioms by theory transformations in the last part of the paper.

Abstract. We introduce the notion of well-founded recursive order-sorted equational logic (OS) theories modulo axioms. Such theories de ne functions by well-founded recursion and are inherently terminating. Moreover, for well-founded recursive theories important properties such as con uence and su cient completeness are modular for so-called fair extensions. This enables us to incrementally check these properties for hierarchies of such theories that occur naturally in modular rule-based functional programs. Well-founded recursive OS theories modulo axioms contain only commutativity and associativity-commutativity axioms. In order to support arbitrary combinations of associativity, commutativity and identity axioms, we show how to eliminate identity and (under certain conditions) associativity (without commutativity) axioms by theory transformations in the last part of the paper. 1