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By Jean-Paul Tremblay

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Example 9. Consider again the TRS of Example 4. The dependency graph (which can be computed with the estimations mentioned in the preceding section) contains one SCC: {1, 3, 4, 5}. The constraints generated by Corollary 8 cannot be solved automatically. In order to cope with this problem, we propose a new recursive approach to compute and solve SCCs. More precisely, if S is the current SCC then we first compute (see the next section) an argument filtering π and a reduction pair ( , >) such that π(R ∪ S) ⊆ ∪ > and π(S) ∩ > = ∅.

Im ], where in Fπ the arity of f is m. Every argument filtering π induces a mapping from T (F, V) to T (Fπ , V), also denoted by π: π(t) = t if t is a variable, π(t) = π(ti ) if t = f (t1 , . . , tn ) and π(f ) = i, and π(t) = f (π(ti1 ), . . , π(tim )) if t = f (t1 , . . , tn ) and π(f ) = [i1 , . . , im ]. Thus, an argument filtering is used to replace function symbols by one of their arguments or to eliminate certain arguments of function symbols. In Section 5 we consider argument filterings that are partially defined.

Jd−1 . For every rule α : fd (s∗ , y ∗ ) → C[fd (t∗1 , y ∗ ), . . , fd (t∗n , y ∗ )] ∈ / Exc fd−1 ,fd , we have r[s∗ ] →∗R/T D[r[t∗i1 ], . . , r[t∗ik ]] for some i1 , . . , ik ∈ {1, . . , n} and context D. ,fd (α)}. Our notion of compatibility extends the one in [12] considerably (see [9] for a detailed comparison). In particular, we extended compatibility by exceptions Exc and in a term f1 (p∗1 , f2 (x∗ , q2∗ ), q1∗ ) with a compatibility sequence f1 , f2 and induction variables x∗ , we permitted defined symbols in the terms p∗1 , q1∗ , q2∗ .

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