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200–209. es/muterm/ 14. : Proving Termination of ContextSensitive Rewriting with MU-TERM. Electronic Notes in Theoretical Computer Science 188, 105–115 (2007) 15. : Automating the Dependency Pair Method. Information and Computation 199, 172–199 (2005) 16. : Usable Rules for Context-Sensitive Rewrite Systems. In: Voronkov, A. ) RTA 2008. LNCS, vol. 5117, pp. 126–141. Springer, Heidelberg (2008) 17. : Polynomials over the Reals in Proofs of Termination: from Theory to Practice. RAIRO Theoretical Informatics and Applications 39(3), 547–586 (2005) 18.
Then, given a non-μ-terminating term t we can always ﬁnd a subterm t0 ∈ T∞,μ of t which starts a minimal infinite μ-rewrite sequence of the form t0 Λ >Λ Λ >Λ >Λ −→∗R,μ σ1 ( 1 ) →R,μ σ1 (r1 ) μ t1 −→∗R,μ σ2 ( 2 ) →R,μ σ2 (r2 ) μ t2 −→∗R,μ · · · where ti , σi ( i ) ∈ M∞,μ for all i > 0 . Theorem 1 below tells us that we have two possibilities: – The minimal non-μ-terminating terms ti ∈ M∞,μ in the sequence are partially introduced by a μ-replacing nonvariable subterm of the right-hand sides ri of the rules i → ri .
Eq N in null = false . eq N in M S = (N == M) or N in S . endfm Fig. 1. Example in Maude syntax  of axioms that fall outside their scope. For instance, they could not be applied to prove termination of the TRS in Figure 1, (speciﬁed in Maude with selfexplanatory syntax; we would not care about sort information here) where we have a (free) combination of associative and commutative axioms which we call an A∨C-rewrite theory in this paper. Furthermore, the Dependency Pair Framework (DP-framework ), which is the basis of state-of-the-art tools for proving termination of (diﬀerent variants of) term rewriting has not yet been adapted to the AC case.