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They express that inﬁnite context-sensitive rewrite sequences can involve not only the kind of recursion which is represented by the usual dependency pairs but also a new kind of recursion which is hidden inside 1 A symbol g ∈ F is deﬁned in R if there is a rule is of the form g( 1 , . . , k ) for some k ≥ 0. → r in R whose left-hand side 20 R. Guti´errez and S. Lucas (8) G (3) G (7) u (4) q qq 5 (5) y Ó ( |yy G l (2) (10) g U (6) bw r × ~ 9 # (11) n RF (12) qq y q5 × " |yy E (13) xt 9G (9) q Ø (1) (8) G (3) (14) w GU (7) w (1) (15) ØÐ (2) r Fig.

Then, the {(P \ P , R, S \ S , μ)} if (1) and (2) hold {(P, R, S, μ)} otherwise is sound and complete. 3 A binary relation R on terms is μ-monotonic if for all terms s, t, t1 , . . , tk , and k-ary symbols f, whenever s R t and i ∈ μ(f) we have f(t1 , . . , ti−1 , s, . . , tk ) R f(t1 , . . , ti−1 , t, . . , tk ). , if PX = ∅. Another advantage is that we can now remove rules from S. Furthermore, we can increase the power of this deﬁnition by considering the usable rules corresponding to P, instead of R as a whole (see [6,16]), and also by using argument ﬁlterings [9].

As remarked by Giesl and Kapur (see also Example 5 below) this is not true for arbitrary equational theories. The problem with Giesl and Kapur’s Deﬁnition 2 is that minimality is not preserved under E-equivalence. Example 2. Consider the following TRS R: f (x, x) → f (0, f (1, 2)) (1) where f ∈ ΣAC . Hence, ExtAC (R) only adds the following rule to R: f (f (x, x), y) → f (f (0, f (1, 2)), y) (2) Note that t = f (f (0, 1), f (0, f (1, 2))) is non-(ExtAC (R), AC)-terminating: f (f (0, 1), f (0, f (1, 2))) ∼A f (0, f (1, f (0, f (1, 2)))) ∼A f (0, f (f (1, 0), f (1, 2))) ∼C Λ f (0, f (f (0, 1), f (1, 2))) ∼A f (0, f (0, f (1, f (1, 2)))) ∼A f (f (0, 0), f (1, f (1, 2)))→ExtAC (R) f (f (0, f (1, 2)), f (1, f (1, 2))) →ExtAC (R),AC · · · Since f (0, 1) and f (0, f (1, 2)) are in (ExtAC (R), AC)-normal form, we have that t ∈ T∞,R,AC .