YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 199 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 89 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) QDP (11) QDPOrderProof [EQUIVALENT, 4541 ms] (12) QDP (13) DependencyGraphProof [EQUIVALENT, 0 ms] (14) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(b(x1)) -> c(d(x1)) d(d(x1)) -> b(e(x1)) b(x1) -> d(c(x1)) d(x1) -> x1 e(c(x1)) -> d(a(x1)) a(x1) -> e(d(x1)) Q is empty. ---------------------------------------- (1) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(b(x1))) -> a(c(d(x1))) b(a(b(x1))) -> b(c(d(x1))) c(a(b(x1))) -> c(c(d(x1))) d(a(b(x1))) -> d(c(d(x1))) e(a(b(x1))) -> e(c(d(x1))) a(d(d(x1))) -> a(b(e(x1))) b(d(d(x1))) -> b(b(e(x1))) c(d(d(x1))) -> c(b(e(x1))) d(d(d(x1))) -> d(b(e(x1))) e(d(d(x1))) -> e(b(e(x1))) a(b(x1)) -> a(d(c(x1))) b(b(x1)) -> b(d(c(x1))) c(b(x1)) -> c(d(c(x1))) d(b(x1)) -> d(d(c(x1))) e(b(x1)) -> e(d(c(x1))) a(d(x1)) -> a(x1) b(d(x1)) -> b(x1) c(d(x1)) -> c(x1) d(d(x1)) -> d(x1) e(d(x1)) -> e(x1) a(e(c(x1))) -> a(d(a(x1))) b(e(c(x1))) -> b(d(a(x1))) c(e(c(x1))) -> c(d(a(x1))) d(e(c(x1))) -> d(d(a(x1))) e(e(c(x1))) -> e(d(a(x1))) a(a(x1)) -> a(e(d(x1))) b(a(x1)) -> b(e(d(x1))) c(a(x1)) -> c(e(d(x1))) d(a(x1)) -> d(e(d(x1))) e(a(x1)) -> e(e(d(x1))) Q is empty. ---------------------------------------- (3) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled As Q is empty the root labeling was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{d_1}(d_{a_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{d_1}(d_{b_1}(x1))) a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{d_1}(d_{c_1}(x1))) a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{c_1}(c_{d_1}(d_{d_1}(x1))) a_{a_1}(a_{b_1}(b_{e_1}(x1))) -> a_{c_1}(c_{d_1}(d_{e_1}(x1))) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{d_1}(d_{a_1}(x1))) b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{d_1}(d_{b_1}(x1))) b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{d_1}(d_{c_1}(x1))) b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{c_1}(c_{d_1}(d_{d_1}(x1))) b_{a_1}(a_{b_1}(b_{e_1}(x1))) -> b_{c_1}(c_{d_1}(d_{e_1}(x1))) c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{d_1}(d_{a_1}(x1))) c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{d_1}(d_{b_1}(x1))) c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{d_1}(d_{c_1}(x1))) c_{a_1}(a_{b_1}(b_{d_1}(x1))) -> c_{c_1}(c_{d_1}(d_{d_1}(x1))) c_{a_1}(a_{b_1}(b_{e_1}(x1))) -> c_{c_1}(c_{d_1}(d_{e_1}(x1))) d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{c_1}(c_{d_1}(d_{a_1}(x1))) d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{c_1}(c_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{c_1}(c_{d_1}(d_{c_1}(x1))) d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{c_1}(c_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{b_1}(b_{e_1}(x1))) -> d_{c_1}(c_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(b_{a_1}(x1))) -> e_{c_1}(c_{d_1}(d_{a_1}(x1))) e_{a_1}(a_{b_1}(b_{b_1}(x1))) -> e_{c_1}(c_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{b_1}(b_{c_1}(x1))) -> e_{c_1}(c_{d_1}(d_{c_1}(x1))) e_{a_1}(a_{b_1}(b_{d_1}(x1))) -> e_{c_1}(c_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{b_1}(b_{e_1}(x1))) -> e_{c_1}(c_{d_1}(d_{e_1}(x1))) a_{d_1}(d_{d_1}(d_{a_1}(x1))) -> a_{b_1}(b_{e_1}(e_{a_1}(x1))) a_{d_1}(d_{d_1}(d_{b_1}(x1))) -> a_{b_1}(b_{e_1}(e_{b_1}(x1))) a_{d_1}(d_{d_1}(d_{c_1}(x1))) -> a_{b_1}(b_{e_1}(e_{c_1}(x1))) a_{d_1}(d_{d_1}(d_{d_1}(x1))) -> a_{b_1}(b_{e_1}(e_{d_1}(x1))) a_{d_1}(d_{d_1}(d_{e_1}(x1))) -> a_{b_1}(b_{e_1}(e_{e_1}(x1))) b_{d_1}(d_{d_1}(d_{a_1}(x1))) -> b_{b_1}(b_{e_1}(e_{a_1}(x1))) b_{d_1}(d_{d_1}(d_{b_1}(x1))) -> b_{b_1}(b_{e_1}(e_{b_1}(x1))) b_{d_1}(d_{d_1}(d_{c_1}(x1))) -> b_{b_1}(b_{e_1}(e_{c_1}(x1))) b_{d_1}(d_{d_1}(d_{d_1}(x1))) -> b_{b_1}(b_{e_1}(e_{d_1}(x1))) b_{d_1}(d_{d_1}(d_{e_1}(x1))) -> b_{b_1}(b_{e_1}(e_{e_1}(x1))) c_{d_1}(d_{d_1}(d_{a_1}(x1))) -> c_{b_1}(b_{e_1}(e_{a_1}(x1))) c_{d_1}(d_{d_1}(d_{b_1}(x1))) -> c_{b_1}(b_{e_1}(e_{b_1}(x1))) c_{d_1}(d_{d_1}(d_{c_1}(x1))) -> c_{b_1}(b_{e_1}(e_{c_1}(x1))) c_{d_1}(d_{d_1}(d_{d_1}(x1))) -> c_{b_1}(b_{e_1}(e_{d_1}(x1))) c_{d_1}(d_{d_1}(d_{e_1}(x1))) -> c_{b_1}(b_{e_1}(e_{e_1}(x1))) d_{d_1}(d_{d_1}(d_{a_1}(x1))) -> d_{b_1}(b_{e_1}(e_{a_1}(x1))) d_{d_1}(d_{d_1}(d_{b_1}(x1))) -> d_{b_1}(b_{e_1}(e_{b_1}(x1))) d_{d_1}(d_{d_1}(d_{c_1}(x1))) -> d_{b_1}(b_{e_1}(e_{c_1}(x1))) d_{d_1}(d_{d_1}(d_{d_1}(x1))) -> d_{b_1}(b_{e_1}(e_{d_1}(x1))) d_{d_1}(d_{d_1}(d_{e_1}(x1))) -> d_{b_1}(b_{e_1}(e_{e_1}(x1))) e_{d_1}(d_{d_1}(d_{a_1}(x1))) -> e_{b_1}(b_{e_1}(e_{a_1}(x1))) e_{d_1}(d_{d_1}(d_{b_1}(x1))) -> e_{b_1}(b_{e_1}(e_{b_1}(x1))) e_{d_1}(d_{d_1}(d_{c_1}(x1))) -> e_{b_1}(b_{e_1}(e_{c_1}(x1))) e_{d_1}(d_{d_1}(d_{d_1}(x1))) -> e_{b_1}(b_{e_1}(e_{d_1}(x1))) e_{d_1}(d_{d_1}(d_{e_1}(x1))) -> e_{b_1}(b_{e_1}(e_{e_1}(x1))) a_{b_1}(b_{a_1}(x1)) -> a_{d_1}(d_{c_1}(c_{a_1}(x1))) a_{b_1}(b_{b_1}(x1)) -> a_{d_1}(d_{c_1}(c_{b_1}(x1))) a_{b_1}(b_{c_1}(x1)) -> a_{d_1}(d_{c_1}(c_{c_1}(x1))) a_{b_1}(b_{d_1}(x1)) -> a_{d_1}(d_{c_1}(c_{d_1}(x1))) a_{b_1}(b_{e_1}(x1)) -> a_{d_1}(d_{c_1}(c_{e_1}(x1))) b_{b_1}(b_{a_1}(x1)) -> b_{d_1}(d_{c_1}(c_{a_1}(x1))) b_{b_1}(b_{b_1}(x1)) -> b_{d_1}(d_{c_1}(c_{b_1}(x1))) b_{b_1}(b_{c_1}(x1)) -> b_{d_1}(d_{c_1}(c_{c_1}(x1))) b_{b_1}(b_{d_1}(x1)) -> b_{d_1}(d_{c_1}(c_{d_1}(x1))) b_{b_1}(b_{e_1}(x1)) -> b_{d_1}(d_{c_1}(c_{e_1}(x1))) c_{b_1}(b_{a_1}(x1)) -> c_{d_1}(d_{c_1}(c_{a_1}(x1))) c_{b_1}(b_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) c_{b_1}(b_{c_1}(x1)) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) c_{b_1}(b_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) c_{b_1}(b_{e_1}(x1)) -> c_{d_1}(d_{c_1}(c_{e_1}(x1))) d_{b_1}(b_{a_1}(x1)) -> d_{d_1}(d_{c_1}(c_{a_1}(x1))) d_{b_1}(b_{b_1}(x1)) -> d_{d_1}(d_{c_1}(c_{b_1}(x1))) d_{b_1}(b_{c_1}(x1)) -> d_{d_1}(d_{c_1}(c_{c_1}(x1))) d_{b_1}(b_{d_1}(x1)) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{b_1}(b_{e_1}(x1)) -> d_{d_1}(d_{c_1}(c_{e_1}(x1))) e_{b_1}(b_{a_1}(x1)) -> e_{d_1}(d_{c_1}(c_{a_1}(x1))) e_{b_1}(b_{b_1}(x1)) -> e_{d_1}(d_{c_1}(c_{b_1}(x1))) e_{b_1}(b_{c_1}(x1)) -> e_{d_1}(d_{c_1}(c_{c_1}(x1))) e_{b_1}(b_{d_1}(x1)) -> e_{d_1}(d_{c_1}(c_{d_1}(x1))) e_{b_1}(b_{e_1}(x1)) -> e_{d_1}(d_{c_1}(c_{e_1}(x1))) a_{d_1}(d_{a_1}(x1)) -> a_{a_1}(x1) a_{d_1}(d_{b_1}(x1)) -> a_{b_1}(x1) a_{d_1}(d_{c_1}(x1)) -> a_{c_1}(x1) a_{d_1}(d_{d_1}(x1)) -> a_{d_1}(x1) a_{d_1}(d_{e_1}(x1)) -> a_{e_1}(x1) b_{d_1}(d_{a_1}(x1)) -> b_{a_1}(x1) b_{d_1}(d_{b_1}(x1)) -> b_{b_1}(x1) b_{d_1}(d_{c_1}(x1)) -> b_{c_1}(x1) b_{d_1}(d_{d_1}(x1)) -> b_{d_1}(x1) b_{d_1}(d_{e_1}(x1)) -> b_{e_1}(x1) c_{d_1}(d_{a_1}(x1)) -> c_{a_1}(x1) c_{d_1}(d_{b_1}(x1)) -> c_{b_1}(x1) c_{d_1}(d_{c_1}(x1)) -> c_{c_1}(x1) c_{d_1}(d_{d_1}(x1)) -> c_{d_1}(x1) c_{d_1}(d_{e_1}(x1)) -> c_{e_1}(x1) d_{d_1}(d_{a_1}(x1)) -> d_{a_1}(x1) d_{d_1}(d_{b_1}(x1)) -> d_{b_1}(x1) d_{d_1}(d_{c_1}(x1)) -> d_{c_1}(x1) d_{d_1}(d_{d_1}(x1)) -> d_{d_1}(x1) d_{d_1}(d_{e_1}(x1)) -> d_{e_1}(x1) e_{d_1}(d_{a_1}(x1)) -> e_{a_1}(x1) e_{d_1}(d_{b_1}(x1)) -> e_{b_1}(x1) e_{d_1}(d_{c_1}(x1)) -> e_{c_1}(x1) e_{d_1}(d_{d_1}(x1)) -> e_{d_1}(x1) e_{d_1}(d_{e_1}(x1)) -> e_{e_1}(x1) a_{e_1}(e_{c_1}(c_{a_1}(x1))) -> a_{d_1}(d_{a_1}(a_{a_1}(x1))) a_{e_1}(e_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(x1))) a_{e_1}(e_{c_1}(c_{c_1}(x1))) -> a_{d_1}(d_{a_1}(a_{c_1}(x1))) a_{e_1}(e_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{d_1}(x1))) a_{e_1}(e_{c_1}(c_{e_1}(x1))) -> a_{d_1}(d_{a_1}(a_{e_1}(x1))) b_{e_1}(e_{c_1}(c_{a_1}(x1))) -> b_{d_1}(d_{a_1}(a_{a_1}(x1))) b_{e_1}(e_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(x1))) b_{e_1}(e_{c_1}(c_{c_1}(x1))) -> b_{d_1}(d_{a_1}(a_{c_1}(x1))) b_{e_1}(e_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{d_1}(x1))) b_{e_1}(e_{c_1}(c_{e_1}(x1))) -> b_{d_1}(d_{a_1}(a_{e_1}(x1))) c_{e_1}(e_{c_1}(c_{a_1}(x1))) -> c_{d_1}(d_{a_1}(a_{a_1}(x1))) c_{e_1}(e_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(x1))) c_{e_1}(e_{c_1}(c_{c_1}(x1))) -> c_{d_1}(d_{a_1}(a_{c_1}(x1))) c_{e_1}(e_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{d_1}(x1))) c_{e_1}(e_{c_1}(c_{e_1}(x1))) -> c_{d_1}(d_{a_1}(a_{e_1}(x1))) d_{e_1}(e_{c_1}(c_{a_1}(x1))) -> d_{d_1}(d_{a_1}(a_{a_1}(x1))) d_{e_1}(e_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(x1))) d_{e_1}(e_{c_1}(c_{c_1}(x1))) -> d_{d_1}(d_{a_1}(a_{c_1}(x1))) d_{e_1}(e_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{d_1}(x1))) d_{e_1}(e_{c_1}(c_{e_1}(x1))) -> d_{d_1}(d_{a_1}(a_{e_1}(x1))) e_{e_1}(e_{c_1}(c_{a_1}(x1))) -> e_{d_1}(d_{a_1}(a_{a_1}(x1))) e_{e_1}(e_{c_1}(c_{b_1}(x1))) -> e_{d_1}(d_{a_1}(a_{b_1}(x1))) e_{e_1}(e_{c_1}(c_{c_1}(x1))) -> e_{d_1}(d_{a_1}(a_{c_1}(x1))) e_{e_1}(e_{c_1}(c_{d_1}(x1))) -> e_{d_1}(d_{a_1}(a_{d_1}(x1))) e_{e_1}(e_{c_1}(c_{e_1}(x1))) -> e_{d_1}(d_{a_1}(a_{e_1}(x1))) a_{a_1}(a_{a_1}(x1)) -> a_{e_1}(e_{d_1}(d_{a_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{e_1}(e_{d_1}(d_{b_1}(x1))) a_{a_1}(a_{c_1}(x1)) -> a_{e_1}(e_{d_1}(d_{c_1}(x1))) a_{a_1}(a_{d_1}(x1)) -> a_{e_1}(e_{d_1}(d_{d_1}(x1))) a_{a_1}(a_{e_1}(x1)) -> a_{e_1}(e_{d_1}(d_{e_1}(x1))) b_{a_1}(a_{a_1}(x1)) -> b_{e_1}(e_{d_1}(d_{a_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{e_1}(e_{d_1}(d_{b_1}(x1))) b_{a_1}(a_{c_1}(x1)) -> b_{e_1}(e_{d_1}(d_{c_1}(x1))) b_{a_1}(a_{d_1}(x1)) -> b_{e_1}(e_{d_1}(d_{d_1}(x1))) b_{a_1}(a_{e_1}(x1)) -> b_{e_1}(e_{d_1}(d_{e_1}(x1))) c_{a_1}(a_{a_1}(x1)) -> c_{e_1}(e_{d_1}(d_{a_1}(x1))) c_{a_1}(a_{b_1}(x1)) -> c_{e_1}(e_{d_1}(d_{b_1}(x1))) c_{a_1}(a_{c_1}(x1)) -> c_{e_1}(e_{d_1}(d_{c_1}(x1))) c_{a_1}(a_{d_1}(x1)) -> c_{e_1}(e_{d_1}(d_{d_1}(x1))) c_{a_1}(a_{e_1}(x1)) -> c_{e_1}(e_{d_1}(d_{e_1}(x1))) d_{a_1}(a_{a_1}(x1)) -> d_{e_1}(e_{d_1}(d_{a_1}(x1))) d_{a_1}(a_{b_1}(x1)) -> d_{e_1}(e_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{c_1}(x1)) -> d_{e_1}(e_{d_1}(d_{c_1}(x1))) d_{a_1}(a_{d_1}(x1)) -> d_{e_1}(e_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{e_1}(x1)) -> d_{e_1}(e_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{a_1}(x1)) -> e_{e_1}(e_{d_1}(d_{a_1}(x1))) e_{a_1}(a_{b_1}(x1)) -> e_{e_1}(e_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{c_1}(x1)) -> e_{e_1}(e_{d_1}(d_{c_1}(x1))) e_{a_1}(a_{d_1}(x1)) -> e_{e_1}(e_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{e_1}(x1)) -> e_{e_1}(e_{d_1}(d_{e_1}(x1))) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = 2 + x_1 POL(a_{b_1}(x_1)) = 10 + x_1 POL(a_{c_1}(x_1)) = 6 + x_1 POL(a_{d_1}(x_1)) = 5 + x_1 POL(a_{e_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = 4 + x_1 POL(b_{b_1}(x_1)) = 10 + x_1 POL(b_{c_1}(x_1)) = 8 + x_1 POL(b_{d_1}(x_1)) = 5 + x_1 POL(b_{e_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = 3 + x_1 POL(c_{b_1}(x_1)) = 10 + x_1 POL(c_{c_1}(x_1)) = 7 + x_1 POL(c_{d_1}(x_1)) = 5 + x_1 POL(c_{e_1}(x_1)) = x_1 POL(d_{a_1}(x_1)) = x_1 POL(d_{b_1}(x_1)) = 10 + x_1 POL(d_{c_1}(x_1)) = 5 + x_1 POL(d_{d_1}(x_1)) = 5 + x_1 POL(d_{e_1}(x_1)) = x_1 POL(e_{a_1}(x_1)) = x_1 POL(e_{b_1}(x_1)) = 5 + x_1 POL(e_{c_1}(x_1)) = 5 + x_1 POL(e_{d_1}(x_1)) = x_1 POL(e_{e_1}(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{d_1}(d_{a_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{d_1}(d_{b_1}(x1))) a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{d_1}(d_{c_1}(x1))) a_{a_1}(a_{b_1}(b_{d_1}(x1))) -> a_{c_1}(c_{d_1}(d_{d_1}(x1))) a_{a_1}(a_{b_1}(b_{e_1}(x1))) -> a_{c_1}(c_{d_1}(d_{e_1}(x1))) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{d_1}(d_{a_1}(x1))) b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{d_1}(d_{b_1}(x1))) b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{d_1}(d_{c_1}(x1))) b_{a_1}(a_{b_1}(b_{d_1}(x1))) -> b_{c_1}(c_{d_1}(d_{d_1}(x1))) b_{a_1}(a_{b_1}(b_{e_1}(x1))) -> b_{c_1}(c_{d_1}(d_{e_1}(x1))) c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{d_1}(d_{a_1}(x1))) c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{d_1}(d_{b_1}(x1))) c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{d_1}(d_{c_1}(x1))) c_{a_1}(a_{b_1}(b_{d_1}(x1))) -> c_{c_1}(c_{d_1}(d_{d_1}(x1))) c_{a_1}(a_{b_1}(b_{e_1}(x1))) -> c_{c_1}(c_{d_1}(d_{e_1}(x1))) d_{a_1}(a_{b_1}(b_{a_1}(x1))) -> d_{c_1}(c_{d_1}(d_{a_1}(x1))) d_{a_1}(a_{b_1}(b_{c_1}(x1))) -> d_{c_1}(c_{d_1}(d_{c_1}(x1))) e_{a_1}(a_{b_1}(b_{a_1}(x1))) -> e_{c_1}(c_{d_1}(d_{a_1}(x1))) e_{a_1}(a_{b_1}(b_{c_1}(x1))) -> e_{c_1}(c_{d_1}(d_{c_1}(x1))) a_{d_1}(d_{d_1}(d_{b_1}(x1))) -> a_{b_1}(b_{e_1}(e_{b_1}(x1))) a_{d_1}(d_{d_1}(d_{d_1}(x1))) -> a_{b_1}(b_{e_1}(e_{d_1}(x1))) b_{d_1}(d_{d_1}(d_{b_1}(x1))) -> b_{b_1}(b_{e_1}(e_{b_1}(x1))) b_{d_1}(d_{d_1}(d_{d_1}(x1))) -> b_{b_1}(b_{e_1}(e_{d_1}(x1))) c_{d_1}(d_{d_1}(d_{b_1}(x1))) -> c_{b_1}(b_{e_1}(e_{b_1}(x1))) c_{d_1}(d_{d_1}(d_{d_1}(x1))) -> c_{b_1}(b_{e_1}(e_{d_1}(x1))) d_{d_1}(d_{d_1}(d_{b_1}(x1))) -> d_{b_1}(b_{e_1}(e_{b_1}(x1))) d_{d_1}(d_{d_1}(d_{d_1}(x1))) -> d_{b_1}(b_{e_1}(e_{d_1}(x1))) e_{d_1}(d_{d_1}(d_{b_1}(x1))) -> e_{b_1}(b_{e_1}(e_{b_1}(x1))) e_{d_1}(d_{d_1}(d_{d_1}(x1))) -> e_{b_1}(b_{e_1}(e_{d_1}(x1))) a_{b_1}(b_{a_1}(x1)) -> a_{d_1}(d_{c_1}(c_{a_1}(x1))) a_{b_1}(b_{c_1}(x1)) -> a_{d_1}(d_{c_1}(c_{c_1}(x1))) b_{b_1}(b_{a_1}(x1)) -> b_{d_1}(d_{c_1}(c_{a_1}(x1))) b_{b_1}(b_{c_1}(x1)) -> b_{d_1}(d_{c_1}(c_{c_1}(x1))) c_{b_1}(b_{a_1}(x1)) -> c_{d_1}(d_{c_1}(c_{a_1}(x1))) c_{b_1}(b_{c_1}(x1)) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) d_{b_1}(b_{a_1}(x1)) -> d_{d_1}(d_{c_1}(c_{a_1}(x1))) d_{b_1}(b_{c_1}(x1)) -> d_{d_1}(d_{c_1}(c_{c_1}(x1))) e_{b_1}(b_{a_1}(x1)) -> e_{d_1}(d_{c_1}(c_{a_1}(x1))) e_{b_1}(b_{c_1}(x1)) -> e_{d_1}(d_{c_1}(c_{c_1}(x1))) a_{d_1}(d_{a_1}(x1)) -> a_{a_1}(x1) a_{d_1}(d_{b_1}(x1)) -> a_{b_1}(x1) a_{d_1}(d_{c_1}(x1)) -> a_{c_1}(x1) a_{d_1}(d_{d_1}(x1)) -> a_{d_1}(x1) a_{d_1}(d_{e_1}(x1)) -> a_{e_1}(x1) b_{d_1}(d_{a_1}(x1)) -> b_{a_1}(x1) b_{d_1}(d_{b_1}(x1)) -> b_{b_1}(x1) b_{d_1}(d_{c_1}(x1)) -> b_{c_1}(x1) b_{d_1}(d_{d_1}(x1)) -> b_{d_1}(x1) b_{d_1}(d_{e_1}(x1)) -> b_{e_1}(x1) c_{d_1}(d_{a_1}(x1)) -> c_{a_1}(x1) c_{d_1}(d_{b_1}(x1)) -> c_{b_1}(x1) c_{d_1}(d_{c_1}(x1)) -> c_{c_1}(x1) c_{d_1}(d_{d_1}(x1)) -> c_{d_1}(x1) c_{d_1}(d_{e_1}(x1)) -> c_{e_1}(x1) d_{d_1}(d_{a_1}(x1)) -> d_{a_1}(x1) d_{d_1}(d_{b_1}(x1)) -> d_{b_1}(x1) d_{d_1}(d_{c_1}(x1)) -> d_{c_1}(x1) d_{d_1}(d_{d_1}(x1)) -> d_{d_1}(x1) d_{d_1}(d_{e_1}(x1)) -> d_{e_1}(x1) e_{d_1}(d_{b_1}(x1)) -> e_{b_1}(x1) e_{d_1}(d_{d_1}(x1)) -> e_{d_1}(x1) a_{e_1}(e_{c_1}(c_{a_1}(x1))) -> a_{d_1}(d_{a_1}(a_{a_1}(x1))) a_{e_1}(e_{c_1}(c_{c_1}(x1))) -> a_{d_1}(d_{a_1}(a_{c_1}(x1))) b_{e_1}(e_{c_1}(c_{a_1}(x1))) -> b_{d_1}(d_{a_1}(a_{a_1}(x1))) b_{e_1}(e_{c_1}(c_{c_1}(x1))) -> b_{d_1}(d_{a_1}(a_{c_1}(x1))) c_{e_1}(e_{c_1}(c_{a_1}(x1))) -> c_{d_1}(d_{a_1}(a_{a_1}(x1))) c_{e_1}(e_{c_1}(c_{c_1}(x1))) -> c_{d_1}(d_{a_1}(a_{c_1}(x1))) d_{e_1}(e_{c_1}(c_{a_1}(x1))) -> d_{d_1}(d_{a_1}(a_{a_1}(x1))) d_{e_1}(e_{c_1}(c_{c_1}(x1))) -> d_{d_1}(d_{a_1}(a_{c_1}(x1))) e_{e_1}(e_{c_1}(c_{a_1}(x1))) -> e_{d_1}(d_{a_1}(a_{a_1}(x1))) e_{e_1}(e_{c_1}(c_{b_1}(x1))) -> e_{d_1}(d_{a_1}(a_{b_1}(x1))) e_{e_1}(e_{c_1}(c_{c_1}(x1))) -> e_{d_1}(d_{a_1}(a_{c_1}(x1))) e_{e_1}(e_{c_1}(c_{d_1}(x1))) -> e_{d_1}(d_{a_1}(a_{d_1}(x1))) e_{e_1}(e_{c_1}(c_{e_1}(x1))) -> e_{d_1}(d_{a_1}(a_{e_1}(x1))) a_{a_1}(a_{a_1}(x1)) -> a_{e_1}(e_{d_1}(d_{a_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{e_1}(e_{d_1}(d_{b_1}(x1))) a_{a_1}(a_{c_1}(x1)) -> a_{e_1}(e_{d_1}(d_{c_1}(x1))) a_{a_1}(a_{d_1}(x1)) -> a_{e_1}(e_{d_1}(d_{d_1}(x1))) a_{a_1}(a_{e_1}(x1)) -> a_{e_1}(e_{d_1}(d_{e_1}(x1))) b_{a_1}(a_{a_1}(x1)) -> b_{e_1}(e_{d_1}(d_{a_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{e_1}(e_{d_1}(d_{b_1}(x1))) b_{a_1}(a_{c_1}(x1)) -> b_{e_1}(e_{d_1}(d_{c_1}(x1))) b_{a_1}(a_{d_1}(x1)) -> b_{e_1}(e_{d_1}(d_{d_1}(x1))) b_{a_1}(a_{e_1}(x1)) -> b_{e_1}(e_{d_1}(d_{e_1}(x1))) c_{a_1}(a_{a_1}(x1)) -> c_{e_1}(e_{d_1}(d_{a_1}(x1))) c_{a_1}(a_{b_1}(x1)) -> c_{e_1}(e_{d_1}(d_{b_1}(x1))) c_{a_1}(a_{c_1}(x1)) -> c_{e_1}(e_{d_1}(d_{c_1}(x1))) c_{a_1}(a_{d_1}(x1)) -> c_{e_1}(e_{d_1}(d_{d_1}(x1))) c_{a_1}(a_{e_1}(x1)) -> c_{e_1}(e_{d_1}(d_{e_1}(x1))) d_{a_1}(a_{a_1}(x1)) -> d_{e_1}(e_{d_1}(d_{a_1}(x1))) d_{a_1}(a_{c_1}(x1)) -> d_{e_1}(e_{d_1}(d_{c_1}(x1))) e_{a_1}(a_{a_1}(x1)) -> e_{e_1}(e_{d_1}(d_{a_1}(x1))) e_{a_1}(a_{c_1}(x1)) -> e_{e_1}(e_{d_1}(d_{c_1}(x1))) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{c_1}(c_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{c_1}(c_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{b_1}(b_{e_1}(x1))) -> d_{c_1}(c_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(b_{b_1}(x1))) -> e_{c_1}(c_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{b_1}(b_{d_1}(x1))) -> e_{c_1}(c_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{b_1}(b_{e_1}(x1))) -> e_{c_1}(c_{d_1}(d_{e_1}(x1))) a_{d_1}(d_{d_1}(d_{a_1}(x1))) -> a_{b_1}(b_{e_1}(e_{a_1}(x1))) a_{d_1}(d_{d_1}(d_{c_1}(x1))) -> a_{b_1}(b_{e_1}(e_{c_1}(x1))) a_{d_1}(d_{d_1}(d_{e_1}(x1))) -> a_{b_1}(b_{e_1}(e_{e_1}(x1))) b_{d_1}(d_{d_1}(d_{a_1}(x1))) -> b_{b_1}(b_{e_1}(e_{a_1}(x1))) b_{d_1}(d_{d_1}(d_{c_1}(x1))) -> b_{b_1}(b_{e_1}(e_{c_1}(x1))) b_{d_1}(d_{d_1}(d_{e_1}(x1))) -> b_{b_1}(b_{e_1}(e_{e_1}(x1))) c_{d_1}(d_{d_1}(d_{a_1}(x1))) -> c_{b_1}(b_{e_1}(e_{a_1}(x1))) c_{d_1}(d_{d_1}(d_{c_1}(x1))) -> c_{b_1}(b_{e_1}(e_{c_1}(x1))) c_{d_1}(d_{d_1}(d_{e_1}(x1))) -> c_{b_1}(b_{e_1}(e_{e_1}(x1))) d_{d_1}(d_{d_1}(d_{a_1}(x1))) -> d_{b_1}(b_{e_1}(e_{a_1}(x1))) d_{d_1}(d_{d_1}(d_{c_1}(x1))) -> d_{b_1}(b_{e_1}(e_{c_1}(x1))) d_{d_1}(d_{d_1}(d_{e_1}(x1))) -> d_{b_1}(b_{e_1}(e_{e_1}(x1))) e_{d_1}(d_{d_1}(d_{a_1}(x1))) -> e_{b_1}(b_{e_1}(e_{a_1}(x1))) e_{d_1}(d_{d_1}(d_{c_1}(x1))) -> e_{b_1}(b_{e_1}(e_{c_1}(x1))) e_{d_1}(d_{d_1}(d_{e_1}(x1))) -> e_{b_1}(b_{e_1}(e_{e_1}(x1))) a_{b_1}(b_{b_1}(x1)) -> a_{d_1}(d_{c_1}(c_{b_1}(x1))) a_{b_1}(b_{d_1}(x1)) -> a_{d_1}(d_{c_1}(c_{d_1}(x1))) a_{b_1}(b_{e_1}(x1)) -> a_{d_1}(d_{c_1}(c_{e_1}(x1))) b_{b_1}(b_{b_1}(x1)) -> b_{d_1}(d_{c_1}(c_{b_1}(x1))) b_{b_1}(b_{d_1}(x1)) -> b_{d_1}(d_{c_1}(c_{d_1}(x1))) b_{b_1}(b_{e_1}(x1)) -> b_{d_1}(d_{c_1}(c_{e_1}(x1))) c_{b_1}(b_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) c_{b_1}(b_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) c_{b_1}(b_{e_1}(x1)) -> c_{d_1}(d_{c_1}(c_{e_1}(x1))) d_{b_1}(b_{b_1}(x1)) -> d_{d_1}(d_{c_1}(c_{b_1}(x1))) d_{b_1}(b_{d_1}(x1)) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{b_1}(b_{e_1}(x1)) -> d_{d_1}(d_{c_1}(c_{e_1}(x1))) e_{b_1}(b_{b_1}(x1)) -> e_{d_1}(d_{c_1}(c_{b_1}(x1))) e_{b_1}(b_{d_1}(x1)) -> e_{d_1}(d_{c_1}(c_{d_1}(x1))) e_{b_1}(b_{e_1}(x1)) -> e_{d_1}(d_{c_1}(c_{e_1}(x1))) e_{d_1}(d_{a_1}(x1)) -> e_{a_1}(x1) e_{d_1}(d_{c_1}(x1)) -> e_{c_1}(x1) e_{d_1}(d_{e_1}(x1)) -> e_{e_1}(x1) a_{e_1}(e_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(x1))) a_{e_1}(e_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{d_1}(x1))) a_{e_1}(e_{c_1}(c_{e_1}(x1))) -> a_{d_1}(d_{a_1}(a_{e_1}(x1))) b_{e_1}(e_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(x1))) b_{e_1}(e_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{d_1}(x1))) b_{e_1}(e_{c_1}(c_{e_1}(x1))) -> b_{d_1}(d_{a_1}(a_{e_1}(x1))) c_{e_1}(e_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(x1))) c_{e_1}(e_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{d_1}(x1))) c_{e_1}(e_{c_1}(c_{e_1}(x1))) -> c_{d_1}(d_{a_1}(a_{e_1}(x1))) d_{e_1}(e_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(x1))) d_{e_1}(e_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{d_1}(x1))) d_{e_1}(e_{c_1}(c_{e_1}(x1))) -> d_{d_1}(d_{a_1}(a_{e_1}(x1))) d_{a_1}(a_{b_1}(x1)) -> d_{e_1}(e_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{d_1}(x1)) -> d_{e_1}(e_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{e_1}(x1)) -> d_{e_1}(e_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(x1)) -> e_{e_1}(e_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{d_1}(x1)) -> e_{e_1}(e_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{e_1}(x1)) -> e_{e_1}(e_{d_1}(d_{e_1}(x1))) Q is empty. ---------------------------------------- (7) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: D_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{D_1}(d_{b_1}(x1)) D_{A_1}(a_{b_1}(b_{b_1}(x1))) -> D_{B_1}(x1) D_{A_1}(a_{b_1}(b_{d_1}(x1))) -> C_{D_1}(d_{d_1}(x1)) D_{A_1}(a_{b_1}(b_{d_1}(x1))) -> D_{D_1}(x1) D_{A_1}(a_{b_1}(b_{e_1}(x1))) -> C_{D_1}(d_{e_1}(x1)) D_{A_1}(a_{b_1}(b_{e_1}(x1))) -> D_{E_1}(x1) E_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{D_1}(d_{b_1}(x1)) E_{A_1}(a_{b_1}(b_{b_1}(x1))) -> D_{B_1}(x1) E_{A_1}(a_{b_1}(b_{d_1}(x1))) -> C_{D_1}(d_{d_1}(x1)) E_{A_1}(a_{b_1}(b_{d_1}(x1))) -> D_{D_1}(x1) E_{A_1}(a_{b_1}(b_{e_1}(x1))) -> C_{D_1}(d_{e_1}(x1)) E_{A_1}(a_{b_1}(b_{e_1}(x1))) -> D_{E_1}(x1) A_{D_1}(d_{d_1}(d_{a_1}(x1))) -> A_{B_1}(b_{e_1}(e_{a_1}(x1))) A_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) A_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) A_{D_1}(d_{d_1}(d_{c_1}(x1))) -> A_{B_1}(b_{e_1}(e_{c_1}(x1))) A_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) A_{D_1}(d_{d_1}(d_{e_1}(x1))) -> A_{B_1}(b_{e_1}(e_{e_1}(x1))) A_{D_1}(d_{d_1}(d_{e_1}(x1))) -> B_{E_1}(e_{e_1}(x1)) B_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{B_1}(b_{e_1}(e_{a_1}(x1))) B_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) B_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) B_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{B_1}(b_{e_1}(e_{c_1}(x1))) B_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) B_{D_1}(d_{d_1}(d_{e_1}(x1))) -> B_{B_1}(b_{e_1}(e_{e_1}(x1))) B_{D_1}(d_{d_1}(d_{e_1}(x1))) -> B_{E_1}(e_{e_1}(x1)) C_{D_1}(d_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{e_1}(e_{a_1}(x1))) C_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) C_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) C_{D_1}(d_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{e_1}(e_{c_1}(x1))) C_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) C_{D_1}(d_{d_1}(d_{e_1}(x1))) -> C_{B_1}(b_{e_1}(e_{e_1}(x1))) C_{D_1}(d_{d_1}(d_{e_1}(x1))) -> B_{E_1}(e_{e_1}(x1)) D_{D_1}(d_{d_1}(d_{a_1}(x1))) -> D_{B_1}(b_{e_1}(e_{a_1}(x1))) D_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) D_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) D_{D_1}(d_{d_1}(d_{c_1}(x1))) -> D_{B_1}(b_{e_1}(e_{c_1}(x1))) D_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) D_{D_1}(d_{d_1}(d_{e_1}(x1))) -> D_{B_1}(b_{e_1}(e_{e_1}(x1))) D_{D_1}(d_{d_1}(d_{e_1}(x1))) -> B_{E_1}(e_{e_1}(x1)) E_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{B_1}(b_{e_1}(e_{a_1}(x1))) E_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) E_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{D_1}(d_{d_1}(d_{c_1}(x1))) -> E_{B_1}(b_{e_1}(e_{c_1}(x1))) E_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) E_{D_1}(d_{d_1}(d_{e_1}(x1))) -> E_{B_1}(b_{e_1}(e_{e_1}(x1))) E_{D_1}(d_{d_1}(d_{e_1}(x1))) -> B_{E_1}(e_{e_1}(x1)) A_{B_1}(b_{b_1}(x1)) -> A_{D_1}(d_{c_1}(c_{b_1}(x1))) A_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) A_{B_1}(b_{d_1}(x1)) -> A_{D_1}(d_{c_1}(c_{d_1}(x1))) A_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) A_{B_1}(b_{e_1}(x1)) -> A_{D_1}(d_{c_1}(c_{e_1}(x1))) A_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) B_{B_1}(b_{b_1}(x1)) -> B_{D_1}(d_{c_1}(c_{b_1}(x1))) B_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) B_{B_1}(b_{d_1}(x1)) -> B_{D_1}(d_{c_1}(c_{d_1}(x1))) B_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) B_{B_1}(b_{e_1}(x1)) -> B_{D_1}(d_{c_1}(c_{e_1}(x1))) B_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) C_{B_1}(b_{b_1}(x1)) -> C_{D_1}(d_{c_1}(c_{b_1}(x1))) C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) C_{B_1}(b_{d_1}(x1)) -> C_{D_1}(d_{c_1}(c_{d_1}(x1))) C_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) C_{B_1}(b_{e_1}(x1)) -> C_{D_1}(d_{c_1}(c_{e_1}(x1))) C_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) D_{B_1}(b_{b_1}(x1)) -> D_{D_1}(d_{c_1}(c_{b_1}(x1))) D_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) D_{B_1}(b_{d_1}(x1)) -> D_{D_1}(d_{c_1}(c_{d_1}(x1))) D_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) D_{B_1}(b_{e_1}(x1)) -> D_{D_1}(d_{c_1}(c_{e_1}(x1))) D_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) E_{B_1}(b_{b_1}(x1)) -> E_{D_1}(d_{c_1}(c_{b_1}(x1))) E_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) E_{B_1}(b_{d_1}(x1)) -> E_{D_1}(d_{c_1}(c_{d_1}(x1))) E_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) E_{B_1}(b_{e_1}(x1)) -> E_{D_1}(d_{c_1}(c_{e_1}(x1))) E_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) E_{D_1}(d_{a_1}(x1)) -> E_{A_1}(x1) A_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{D_1}(d_{a_1}(a_{b_1}(x1))) A_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) A_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) A_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(d_{a_1}(a_{d_1}(x1))) A_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) A_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) A_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{D_1}(d_{a_1}(a_{e_1}(x1))) A_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) A_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) B_{E_1}(e_{c_1}(c_{b_1}(x1))) -> B_{D_1}(d_{a_1}(a_{b_1}(x1))) B_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) B_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) B_{E_1}(e_{c_1}(c_{d_1}(x1))) -> B_{D_1}(d_{a_1}(a_{d_1}(x1))) B_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) B_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) B_{E_1}(e_{c_1}(c_{e_1}(x1))) -> B_{D_1}(d_{a_1}(a_{e_1}(x1))) B_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) B_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) C_{E_1}(e_{c_1}(c_{b_1}(x1))) -> C_{D_1}(d_{a_1}(a_{b_1}(x1))) C_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) C_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) C_{E_1}(e_{c_1}(c_{d_1}(x1))) -> C_{D_1}(d_{a_1}(a_{d_1}(x1))) C_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) C_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) C_{E_1}(e_{c_1}(c_{e_1}(x1))) -> C_{D_1}(d_{a_1}(a_{e_1}(x1))) C_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) C_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) D_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{D_1}(d_{a_1}(a_{b_1}(x1))) D_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) D_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) D_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{D_1}(d_{a_1}(a_{d_1}(x1))) D_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) D_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) D_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{D_1}(d_{a_1}(a_{e_1}(x1))) D_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) D_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) D_{A_1}(a_{b_1}(x1)) -> D_{E_1}(e_{d_1}(d_{b_1}(x1))) D_{A_1}(a_{b_1}(x1)) -> E_{D_1}(d_{b_1}(x1)) D_{A_1}(a_{b_1}(x1)) -> D_{B_1}(x1) D_{A_1}(a_{d_1}(x1)) -> D_{E_1}(e_{d_1}(d_{d_1}(x1))) D_{A_1}(a_{d_1}(x1)) -> E_{D_1}(d_{d_1}(x1)) D_{A_1}(a_{d_1}(x1)) -> D_{D_1}(x1) D_{A_1}(a_{e_1}(x1)) -> D_{E_1}(e_{d_1}(d_{e_1}(x1))) D_{A_1}(a_{e_1}(x1)) -> E_{D_1}(d_{e_1}(x1)) D_{A_1}(a_{e_1}(x1)) -> D_{E_1}(x1) E_{A_1}(a_{b_1}(x1)) -> E_{D_1}(d_{b_1}(x1)) E_{A_1}(a_{b_1}(x1)) -> D_{B_1}(x1) E_{A_1}(a_{d_1}(x1)) -> E_{D_1}(d_{d_1}(x1)) E_{A_1}(a_{d_1}(x1)) -> D_{D_1}(x1) E_{A_1}(a_{e_1}(x1)) -> E_{D_1}(d_{e_1}(x1)) E_{A_1}(a_{e_1}(x1)) -> D_{E_1}(x1) The TRS R consists of the following rules: d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{c_1}(c_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{c_1}(c_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{b_1}(b_{e_1}(x1))) -> d_{c_1}(c_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(b_{b_1}(x1))) -> e_{c_1}(c_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{b_1}(b_{d_1}(x1))) -> e_{c_1}(c_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{b_1}(b_{e_1}(x1))) -> e_{c_1}(c_{d_1}(d_{e_1}(x1))) a_{d_1}(d_{d_1}(d_{a_1}(x1))) -> a_{b_1}(b_{e_1}(e_{a_1}(x1))) a_{d_1}(d_{d_1}(d_{c_1}(x1))) -> a_{b_1}(b_{e_1}(e_{c_1}(x1))) a_{d_1}(d_{d_1}(d_{e_1}(x1))) -> a_{b_1}(b_{e_1}(e_{e_1}(x1))) b_{d_1}(d_{d_1}(d_{a_1}(x1))) -> b_{b_1}(b_{e_1}(e_{a_1}(x1))) b_{d_1}(d_{d_1}(d_{c_1}(x1))) -> b_{b_1}(b_{e_1}(e_{c_1}(x1))) b_{d_1}(d_{d_1}(d_{e_1}(x1))) -> b_{b_1}(b_{e_1}(e_{e_1}(x1))) c_{d_1}(d_{d_1}(d_{a_1}(x1))) -> c_{b_1}(b_{e_1}(e_{a_1}(x1))) c_{d_1}(d_{d_1}(d_{c_1}(x1))) -> c_{b_1}(b_{e_1}(e_{c_1}(x1))) c_{d_1}(d_{d_1}(d_{e_1}(x1))) -> c_{b_1}(b_{e_1}(e_{e_1}(x1))) d_{d_1}(d_{d_1}(d_{a_1}(x1))) -> d_{b_1}(b_{e_1}(e_{a_1}(x1))) d_{d_1}(d_{d_1}(d_{c_1}(x1))) -> d_{b_1}(b_{e_1}(e_{c_1}(x1))) d_{d_1}(d_{d_1}(d_{e_1}(x1))) -> d_{b_1}(b_{e_1}(e_{e_1}(x1))) e_{d_1}(d_{d_1}(d_{a_1}(x1))) -> e_{b_1}(b_{e_1}(e_{a_1}(x1))) e_{d_1}(d_{d_1}(d_{c_1}(x1))) -> e_{b_1}(b_{e_1}(e_{c_1}(x1))) e_{d_1}(d_{d_1}(d_{e_1}(x1))) -> e_{b_1}(b_{e_1}(e_{e_1}(x1))) a_{b_1}(b_{b_1}(x1)) -> a_{d_1}(d_{c_1}(c_{b_1}(x1))) a_{b_1}(b_{d_1}(x1)) -> a_{d_1}(d_{c_1}(c_{d_1}(x1))) a_{b_1}(b_{e_1}(x1)) -> a_{d_1}(d_{c_1}(c_{e_1}(x1))) b_{b_1}(b_{b_1}(x1)) -> b_{d_1}(d_{c_1}(c_{b_1}(x1))) b_{b_1}(b_{d_1}(x1)) -> b_{d_1}(d_{c_1}(c_{d_1}(x1))) b_{b_1}(b_{e_1}(x1)) -> b_{d_1}(d_{c_1}(c_{e_1}(x1))) c_{b_1}(b_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) c_{b_1}(b_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) c_{b_1}(b_{e_1}(x1)) -> c_{d_1}(d_{c_1}(c_{e_1}(x1))) d_{b_1}(b_{b_1}(x1)) -> d_{d_1}(d_{c_1}(c_{b_1}(x1))) d_{b_1}(b_{d_1}(x1)) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{b_1}(b_{e_1}(x1)) -> d_{d_1}(d_{c_1}(c_{e_1}(x1))) e_{b_1}(b_{b_1}(x1)) -> e_{d_1}(d_{c_1}(c_{b_1}(x1))) e_{b_1}(b_{d_1}(x1)) -> e_{d_1}(d_{c_1}(c_{d_1}(x1))) e_{b_1}(b_{e_1}(x1)) -> e_{d_1}(d_{c_1}(c_{e_1}(x1))) e_{d_1}(d_{a_1}(x1)) -> e_{a_1}(x1) e_{d_1}(d_{c_1}(x1)) -> e_{c_1}(x1) e_{d_1}(d_{e_1}(x1)) -> e_{e_1}(x1) a_{e_1}(e_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(x1))) a_{e_1}(e_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{d_1}(x1))) a_{e_1}(e_{c_1}(c_{e_1}(x1))) -> a_{d_1}(d_{a_1}(a_{e_1}(x1))) b_{e_1}(e_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(x1))) b_{e_1}(e_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{d_1}(x1))) b_{e_1}(e_{c_1}(c_{e_1}(x1))) -> b_{d_1}(d_{a_1}(a_{e_1}(x1))) c_{e_1}(e_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(x1))) c_{e_1}(e_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{d_1}(x1))) c_{e_1}(e_{c_1}(c_{e_1}(x1))) -> c_{d_1}(d_{a_1}(a_{e_1}(x1))) d_{e_1}(e_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(x1))) d_{e_1}(e_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{d_1}(x1))) d_{e_1}(e_{c_1}(c_{e_1}(x1))) -> d_{d_1}(d_{a_1}(a_{e_1}(x1))) d_{a_1}(a_{b_1}(x1)) -> d_{e_1}(e_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{d_1}(x1)) -> d_{e_1}(e_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{e_1}(x1)) -> d_{e_1}(e_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(x1)) -> e_{e_1}(e_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{d_1}(x1)) -> e_{e_1}(e_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{e_1}(x1)) -> e_{e_1}(e_{d_1}(d_{e_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 21 less nodes. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: C_{D_1}(d_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{e_1}(e_{a_1}(x1))) C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) C_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) C_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) B_{E_1}(e_{c_1}(c_{b_1}(x1))) -> B_{D_1}(d_{a_1}(a_{b_1}(x1))) B_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{B_1}(b_{e_1}(e_{a_1}(x1))) B_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) C_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) C_{E_1}(e_{c_1}(c_{b_1}(x1))) -> C_{D_1}(d_{a_1}(a_{b_1}(x1))) C_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{D_1}(d_{b_1}(x1)) C_{D_1}(d_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{e_1}(e_{c_1}(x1))) C_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) B_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) D_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{D_1}(d_{b_1}(x1)) C_{D_1}(d_{d_1}(d_{e_1}(x1))) -> C_{B_1}(b_{e_1}(e_{e_1}(x1))) D_{A_1}(a_{b_1}(b_{b_1}(x1))) -> D_{B_1}(x1) D_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) D_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) D_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) C_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) D_{A_1}(a_{b_1}(b_{d_1}(x1))) -> C_{D_1}(d_{d_1}(x1)) D_{A_1}(a_{b_1}(b_{d_1}(x1))) -> D_{D_1}(x1) D_{D_1}(d_{d_1}(d_{a_1}(x1))) -> D_{B_1}(b_{e_1}(e_{a_1}(x1))) D_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) B_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) A_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) A_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) A_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) C_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) C_{E_1}(e_{c_1}(c_{d_1}(x1))) -> C_{D_1}(d_{a_1}(a_{d_1}(x1))) C_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{b_1}(b_{e_1}(x1))) -> C_{D_1}(d_{e_1}(x1)) D_{A_1}(a_{b_1}(b_{e_1}(x1))) -> D_{E_1}(x1) D_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{D_1}(d_{a_1}(a_{b_1}(x1))) D_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{A_1}(a_{b_1}(b_{b_1}(x1))) -> D_{B_1}(x1) E_{A_1}(a_{b_1}(b_{d_1}(x1))) -> C_{D_1}(d_{d_1}(x1)) E_{A_1}(a_{b_1}(b_{d_1}(x1))) -> D_{D_1}(x1) D_{D_1}(d_{d_1}(d_{c_1}(x1))) -> D_{B_1}(b_{e_1}(e_{c_1}(x1))) D_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) B_{E_1}(e_{c_1}(c_{d_1}(x1))) -> B_{D_1}(d_{a_1}(a_{d_1}(x1))) B_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) B_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{b_1}(x1)) -> D_{E_1}(e_{d_1}(d_{b_1}(x1))) D_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) D_{A_1}(a_{b_1}(x1)) -> E_{D_1}(d_{b_1}(x1)) E_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{B_1}(b_{e_1}(e_{a_1}(x1))) E_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) E_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) E_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) C_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) A_{D_1}(d_{d_1}(d_{a_1}(x1))) -> A_{B_1}(b_{e_1}(e_{a_1}(x1))) A_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) B_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) A_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{A_1}(a_{b_1}(b_{e_1}(x1))) -> C_{D_1}(d_{e_1}(x1)) E_{A_1}(a_{b_1}(b_{e_1}(x1))) -> D_{E_1}(x1) D_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) D_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{D_1}(d_{a_1}(a_{d_1}(x1))) D_{D_1}(d_{d_1}(d_{e_1}(x1))) -> D_{B_1}(b_{e_1}(e_{e_1}(x1))) D_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{b_1}(x1)) -> D_{B_1}(x1) D_{A_1}(a_{d_1}(x1)) -> D_{E_1}(e_{d_1}(d_{d_1}(x1))) D_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) A_{D_1}(d_{d_1}(d_{c_1}(x1))) -> A_{B_1}(b_{e_1}(e_{c_1}(x1))) A_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) B_{E_1}(e_{c_1}(c_{e_1}(x1))) -> B_{D_1}(d_{a_1}(a_{e_1}(x1))) B_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{A_1}(a_{b_1}(x1)) -> E_{D_1}(d_{b_1}(x1)) E_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) B_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) D_{A_1}(a_{d_1}(x1)) -> E_{D_1}(d_{d_1}(x1)) E_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{A_1}(a_{b_1}(x1)) -> D_{B_1}(x1) E_{A_1}(a_{d_1}(x1)) -> E_{D_1}(d_{d_1}(x1)) E_{D_1}(d_{d_1}(d_{c_1}(x1))) -> E_{B_1}(b_{e_1}(e_{c_1}(x1))) E_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) B_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) A_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{D_1}(d_{a_1}(a_{b_1}(x1))) A_{D_1}(d_{d_1}(d_{e_1}(x1))) -> A_{B_1}(b_{e_1}(e_{e_1}(x1))) A_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) D_{A_1}(a_{d_1}(x1)) -> D_{D_1}(x1) A_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) A_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(d_{a_1}(a_{d_1}(x1))) A_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) A_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) A_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{D_1}(d_{a_1}(a_{e_1}(x1))) A_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) D_{A_1}(a_{e_1}(x1)) -> D_{E_1}(e_{d_1}(d_{e_1}(x1))) D_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{D_1}(d_{a_1}(a_{e_1}(x1))) D_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) D_{A_1}(a_{e_1}(x1)) -> E_{D_1}(d_{e_1}(x1)) E_{D_1}(d_{d_1}(d_{e_1}(x1))) -> E_{B_1}(b_{e_1}(e_{e_1}(x1))) D_{A_1}(a_{e_1}(x1)) -> D_{E_1}(x1) D_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) A_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) E_{A_1}(a_{d_1}(x1)) -> D_{D_1}(x1) E_{A_1}(a_{e_1}(x1)) -> E_{D_1}(d_{e_1}(x1)) E_{A_1}(a_{e_1}(x1)) -> D_{E_1}(x1) B_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{B_1}(b_{e_1}(e_{c_1}(x1))) B_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) B_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) C_{E_1}(e_{c_1}(c_{e_1}(x1))) -> C_{D_1}(d_{a_1}(a_{e_1}(x1))) C_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) C_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) B_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) B_{D_1}(d_{d_1}(d_{e_1}(x1))) -> B_{B_1}(b_{e_1}(e_{e_1}(x1))) The TRS R consists of the following rules: d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{c_1}(c_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{c_1}(c_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{b_1}(b_{e_1}(x1))) -> d_{c_1}(c_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(b_{b_1}(x1))) -> e_{c_1}(c_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{b_1}(b_{d_1}(x1))) -> e_{c_1}(c_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{b_1}(b_{e_1}(x1))) -> e_{c_1}(c_{d_1}(d_{e_1}(x1))) a_{d_1}(d_{d_1}(d_{a_1}(x1))) -> a_{b_1}(b_{e_1}(e_{a_1}(x1))) a_{d_1}(d_{d_1}(d_{c_1}(x1))) -> a_{b_1}(b_{e_1}(e_{c_1}(x1))) a_{d_1}(d_{d_1}(d_{e_1}(x1))) -> a_{b_1}(b_{e_1}(e_{e_1}(x1))) b_{d_1}(d_{d_1}(d_{a_1}(x1))) -> b_{b_1}(b_{e_1}(e_{a_1}(x1))) b_{d_1}(d_{d_1}(d_{c_1}(x1))) -> b_{b_1}(b_{e_1}(e_{c_1}(x1))) b_{d_1}(d_{d_1}(d_{e_1}(x1))) -> b_{b_1}(b_{e_1}(e_{e_1}(x1))) c_{d_1}(d_{d_1}(d_{a_1}(x1))) -> c_{b_1}(b_{e_1}(e_{a_1}(x1))) c_{d_1}(d_{d_1}(d_{c_1}(x1))) -> c_{b_1}(b_{e_1}(e_{c_1}(x1))) c_{d_1}(d_{d_1}(d_{e_1}(x1))) -> c_{b_1}(b_{e_1}(e_{e_1}(x1))) d_{d_1}(d_{d_1}(d_{a_1}(x1))) -> d_{b_1}(b_{e_1}(e_{a_1}(x1))) d_{d_1}(d_{d_1}(d_{c_1}(x1))) -> d_{b_1}(b_{e_1}(e_{c_1}(x1))) d_{d_1}(d_{d_1}(d_{e_1}(x1))) -> d_{b_1}(b_{e_1}(e_{e_1}(x1))) e_{d_1}(d_{d_1}(d_{a_1}(x1))) -> e_{b_1}(b_{e_1}(e_{a_1}(x1))) e_{d_1}(d_{d_1}(d_{c_1}(x1))) -> e_{b_1}(b_{e_1}(e_{c_1}(x1))) e_{d_1}(d_{d_1}(d_{e_1}(x1))) -> e_{b_1}(b_{e_1}(e_{e_1}(x1))) a_{b_1}(b_{b_1}(x1)) -> a_{d_1}(d_{c_1}(c_{b_1}(x1))) a_{b_1}(b_{d_1}(x1)) -> a_{d_1}(d_{c_1}(c_{d_1}(x1))) a_{b_1}(b_{e_1}(x1)) -> a_{d_1}(d_{c_1}(c_{e_1}(x1))) b_{b_1}(b_{b_1}(x1)) -> b_{d_1}(d_{c_1}(c_{b_1}(x1))) b_{b_1}(b_{d_1}(x1)) -> b_{d_1}(d_{c_1}(c_{d_1}(x1))) b_{b_1}(b_{e_1}(x1)) -> b_{d_1}(d_{c_1}(c_{e_1}(x1))) c_{b_1}(b_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) c_{b_1}(b_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) c_{b_1}(b_{e_1}(x1)) -> c_{d_1}(d_{c_1}(c_{e_1}(x1))) d_{b_1}(b_{b_1}(x1)) -> d_{d_1}(d_{c_1}(c_{b_1}(x1))) d_{b_1}(b_{d_1}(x1)) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{b_1}(b_{e_1}(x1)) -> d_{d_1}(d_{c_1}(c_{e_1}(x1))) e_{b_1}(b_{b_1}(x1)) -> e_{d_1}(d_{c_1}(c_{b_1}(x1))) e_{b_1}(b_{d_1}(x1)) -> e_{d_1}(d_{c_1}(c_{d_1}(x1))) e_{b_1}(b_{e_1}(x1)) -> e_{d_1}(d_{c_1}(c_{e_1}(x1))) e_{d_1}(d_{a_1}(x1)) -> e_{a_1}(x1) e_{d_1}(d_{c_1}(x1)) -> e_{c_1}(x1) e_{d_1}(d_{e_1}(x1)) -> e_{e_1}(x1) a_{e_1}(e_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(x1))) a_{e_1}(e_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{d_1}(x1))) a_{e_1}(e_{c_1}(c_{e_1}(x1))) -> a_{d_1}(d_{a_1}(a_{e_1}(x1))) b_{e_1}(e_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(x1))) b_{e_1}(e_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{d_1}(x1))) b_{e_1}(e_{c_1}(c_{e_1}(x1))) -> b_{d_1}(d_{a_1}(a_{e_1}(x1))) c_{e_1}(e_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(x1))) c_{e_1}(e_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{d_1}(x1))) c_{e_1}(e_{c_1}(c_{e_1}(x1))) -> c_{d_1}(d_{a_1}(a_{e_1}(x1))) d_{e_1}(e_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(x1))) d_{e_1}(e_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{d_1}(x1))) d_{e_1}(e_{c_1}(c_{e_1}(x1))) -> d_{d_1}(d_{a_1}(a_{e_1}(x1))) d_{a_1}(a_{b_1}(x1)) -> d_{e_1}(e_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{d_1}(x1)) -> d_{e_1}(e_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{e_1}(x1)) -> d_{e_1}(e_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(x1)) -> e_{e_1}(e_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{d_1}(x1)) -> e_{e_1}(e_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{e_1}(x1)) -> e_{e_1}(e_{d_1}(d_{e_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) C_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) C_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) B_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{B_1}(b_{e_1}(e_{a_1}(x1))) B_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) C_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) C_{E_1}(e_{c_1}(c_{b_1}(x1))) -> C_{D_1}(d_{a_1}(a_{b_1}(x1))) C_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{D_1}(d_{b_1}(x1)) C_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) D_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{D_1}(d_{b_1}(x1)) D_{A_1}(a_{b_1}(b_{b_1}(x1))) -> D_{B_1}(x1) D_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) D_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) D_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) C_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) D_{A_1}(a_{b_1}(b_{d_1}(x1))) -> C_{D_1}(d_{d_1}(x1)) D_{A_1}(a_{b_1}(b_{d_1}(x1))) -> D_{D_1}(x1) D_{D_1}(d_{d_1}(d_{a_1}(x1))) -> D_{B_1}(b_{e_1}(e_{a_1}(x1))) D_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) B_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) A_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) A_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) C_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) C_{E_1}(e_{c_1}(c_{d_1}(x1))) -> C_{D_1}(d_{a_1}(a_{d_1}(x1))) C_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{b_1}(b_{e_1}(x1))) -> C_{D_1}(d_{e_1}(x1)) D_{A_1}(a_{b_1}(b_{e_1}(x1))) -> D_{E_1}(x1) D_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{A_1}(a_{b_1}(b_{b_1}(x1))) -> D_{B_1}(x1) E_{A_1}(a_{b_1}(b_{d_1}(x1))) -> C_{D_1}(d_{d_1}(x1)) E_{A_1}(a_{b_1}(b_{d_1}(x1))) -> D_{D_1}(x1) D_{D_1}(d_{d_1}(d_{c_1}(x1))) -> D_{B_1}(b_{e_1}(e_{c_1}(x1))) D_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) B_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) D_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) E_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) E_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) E_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) C_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) A_{D_1}(d_{d_1}(d_{a_1}(x1))) -> A_{B_1}(b_{e_1}(e_{a_1}(x1))) A_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) B_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) E_{A_1}(a_{b_1}(b_{e_1}(x1))) -> C_{D_1}(d_{e_1}(x1)) E_{A_1}(a_{b_1}(b_{e_1}(x1))) -> D_{E_1}(x1) D_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) D_{D_1}(d_{d_1}(d_{e_1}(x1))) -> D_{B_1}(b_{e_1}(e_{e_1}(x1))) D_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{b_1}(x1)) -> D_{B_1}(x1) D_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) A_{D_1}(d_{d_1}(d_{c_1}(x1))) -> A_{B_1}(b_{e_1}(e_{c_1}(x1))) A_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) B_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{A_1}(a_{b_1}(x1)) -> E_{D_1}(d_{b_1}(x1)) E_{D_1}(d_{d_1}(d_{a_1}(x1))) -> B_{E_1}(e_{a_1}(x1)) E_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) E_{A_1}(a_{b_1}(x1)) -> D_{B_1}(x1) E_{A_1}(a_{d_1}(x1)) -> E_{D_1}(d_{d_1}(x1)) E_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) A_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{D_1}(d_{a_1}(a_{b_1}(x1))) A_{D_1}(d_{d_1}(d_{e_1}(x1))) -> A_{B_1}(b_{e_1}(e_{e_1}(x1))) A_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) A_{E_1}(e_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) A_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(d_{a_1}(a_{d_1}(x1))) A_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) A_{E_1}(e_{c_1}(c_{d_1}(x1))) -> A_{D_1}(x1) A_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{D_1}(d_{a_1}(a_{e_1}(x1))) A_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) D_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) D_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) A_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) E_{A_1}(a_{d_1}(x1)) -> D_{D_1}(x1) E_{A_1}(a_{e_1}(x1)) -> E_{D_1}(d_{e_1}(x1)) E_{A_1}(a_{e_1}(x1)) -> D_{E_1}(x1) B_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{B_1}(b_{e_1}(e_{c_1}(x1))) B_{B_1}(b_{d_1}(x1)) -> C_{D_1}(x1) C_{E_1}(e_{c_1}(c_{e_1}(x1))) -> C_{D_1}(d_{a_1}(a_{e_1}(x1))) C_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) C_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) B_{D_1}(d_{d_1}(d_{c_1}(x1))) -> B_{E_1}(e_{c_1}(x1)) B_{D_1}(d_{d_1}(d_{e_1}(x1))) -> B_{B_1}(b_{e_1}(e_{e_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( B_{E_1}_1(x_1) ) = max{0, 2x_1 - 2} POL( D_{E_1}_1(x_1) ) = 2x_1 POL( A_{B_1}_1(x_1) ) = 2x_1 POL( b_{e_1}_1(x_1) ) = x_1 POL( A_{D_1}_1(x_1) ) = max{0, 2x_1 - 1} POL( d_{e_1}_1(x_1) ) = x_1 POL( B_{B_1}_1(x_1) ) = 2x_1 POL( B_{D_1}_1(x_1) ) = 2x_1 POL( C_{B_1}_1(x_1) ) = 2x_1 + 2 POL( C_{D_1}_1(x_1) ) = 2x_1 POL( D_{B_1}_1(x_1) ) = 2x_1 + 2 POL( D_{D_1}_1(x_1) ) = 2x_1 + 2 POL( E_{B_1}_1(x_1) ) = 2x_1 + 2 POL( b_{d_1}_1(x_1) ) = x_1 + 1 POL( d_{b_1}_1(x_1) ) = x_1 + 2 POL( D_{A_1}_1(x_1) ) = 2x_1 POL( a_{b_1}_1(x_1) ) = x_1 + 2 POL( a_{d_1}_1(x_1) ) = x_1 + 1 POL( E_{D_1}_1(x_1) ) = 2x_1 POL( d_{d_1}_1(x_1) ) = x_1 + 1 POL( d_{a_1}_1(x_1) ) = x_1 POL( c_{b_1}_1(x_1) ) = x_1 + 2 POL( d_{c_1}_1(x_1) ) = x_1 + 1 POL( e_{a_1}_1(x_1) ) = x_1 POL( b_{b_1}_1(x_1) ) = x_1 + 2 POL( e_{c_1}_1(x_1) ) = x_1 + 1 POL( c_{d_1}_1(x_1) ) = x_1 + 1 POL( e_{e_1}_1(x_1) ) = max{0, -2} POL( e_{d_1}_1(x_1) ) = x_1 POL( a_{e_1}_1(x_1) ) = x_1 POL( e_{b_1}_1(x_1) ) = x_1 + 1 POL( c_{e_1}_1(x_1) ) = x_1 POL( C_{E_1}_1(x_1) ) = 2x_1 POL( E_{A_1}_1(x_1) ) = 2x_1 + 1 POL( A_{E_1}_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: e_{a_1}(a_{b_1}(b_{b_1}(x1))) -> e_{c_1}(c_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{b_1}(b_{d_1}(x1))) -> e_{c_1}(c_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{b_1}(b_{e_1}(x1))) -> e_{c_1}(c_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(x1)) -> e_{e_1}(e_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{d_1}(x1)) -> e_{e_1}(e_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{e_1}(x1)) -> e_{e_1}(e_{d_1}(d_{e_1}(x1))) b_{e_1}(e_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(x1))) b_{e_1}(e_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{d_1}(x1))) b_{e_1}(e_{c_1}(c_{e_1}(x1))) -> b_{d_1}(d_{a_1}(a_{e_1}(x1))) a_{b_1}(b_{b_1}(x1)) -> a_{d_1}(d_{c_1}(c_{b_1}(x1))) a_{b_1}(b_{d_1}(x1)) -> a_{d_1}(d_{c_1}(c_{d_1}(x1))) a_{b_1}(b_{e_1}(x1)) -> a_{d_1}(d_{c_1}(c_{e_1}(x1))) d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{c_1}(c_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{c_1}(c_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{b_1}(b_{e_1}(x1))) -> d_{c_1}(c_{d_1}(d_{e_1}(x1))) d_{a_1}(a_{b_1}(x1)) -> d_{e_1}(e_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{d_1}(x1)) -> d_{e_1}(e_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{e_1}(x1)) -> d_{e_1}(e_{d_1}(d_{e_1}(x1))) d_{b_1}(b_{b_1}(x1)) -> d_{d_1}(d_{c_1}(c_{b_1}(x1))) d_{b_1}(b_{d_1}(x1)) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{b_1}(b_{e_1}(x1)) -> d_{d_1}(d_{c_1}(c_{e_1}(x1))) d_{d_1}(d_{d_1}(d_{a_1}(x1))) -> d_{b_1}(b_{e_1}(e_{a_1}(x1))) d_{d_1}(d_{d_1}(d_{c_1}(x1))) -> d_{b_1}(b_{e_1}(e_{c_1}(x1))) d_{d_1}(d_{d_1}(d_{e_1}(x1))) -> d_{b_1}(b_{e_1}(e_{e_1}(x1))) a_{d_1}(d_{d_1}(d_{a_1}(x1))) -> a_{b_1}(b_{e_1}(e_{a_1}(x1))) a_{d_1}(d_{d_1}(d_{c_1}(x1))) -> a_{b_1}(b_{e_1}(e_{c_1}(x1))) a_{d_1}(d_{d_1}(d_{e_1}(x1))) -> a_{b_1}(b_{e_1}(e_{e_1}(x1))) d_{e_1}(e_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(x1))) d_{e_1}(e_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{d_1}(x1))) d_{e_1}(e_{c_1}(c_{e_1}(x1))) -> d_{d_1}(d_{a_1}(a_{e_1}(x1))) e_{d_1}(d_{d_1}(d_{a_1}(x1))) -> e_{b_1}(b_{e_1}(e_{a_1}(x1))) e_{d_1}(d_{d_1}(d_{c_1}(x1))) -> e_{b_1}(b_{e_1}(e_{c_1}(x1))) e_{d_1}(d_{d_1}(d_{e_1}(x1))) -> e_{b_1}(b_{e_1}(e_{e_1}(x1))) e_{d_1}(d_{a_1}(x1)) -> e_{a_1}(x1) e_{d_1}(d_{c_1}(x1)) -> e_{c_1}(x1) e_{d_1}(d_{e_1}(x1)) -> e_{e_1}(x1) a_{e_1}(e_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(x1))) a_{e_1}(e_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{d_1}(x1))) a_{e_1}(e_{c_1}(c_{e_1}(x1))) -> a_{d_1}(d_{a_1}(a_{e_1}(x1))) c_{b_1}(b_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) c_{b_1}(b_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) c_{b_1}(b_{e_1}(x1)) -> c_{d_1}(d_{c_1}(c_{e_1}(x1))) c_{d_1}(d_{d_1}(d_{a_1}(x1))) -> c_{b_1}(b_{e_1}(e_{a_1}(x1))) c_{d_1}(d_{d_1}(d_{c_1}(x1))) -> c_{b_1}(b_{e_1}(e_{c_1}(x1))) c_{d_1}(d_{d_1}(d_{e_1}(x1))) -> c_{b_1}(b_{e_1}(e_{e_1}(x1))) b_{d_1}(d_{d_1}(d_{a_1}(x1))) -> b_{b_1}(b_{e_1}(e_{a_1}(x1))) b_{d_1}(d_{d_1}(d_{c_1}(x1))) -> b_{b_1}(b_{e_1}(e_{c_1}(x1))) b_{d_1}(d_{d_1}(d_{e_1}(x1))) -> b_{b_1}(b_{e_1}(e_{e_1}(x1))) c_{e_1}(e_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(x1))) c_{e_1}(e_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{d_1}(x1))) c_{e_1}(e_{c_1}(c_{e_1}(x1))) -> c_{d_1}(d_{a_1}(a_{e_1}(x1))) e_{b_1}(b_{b_1}(x1)) -> e_{d_1}(d_{c_1}(c_{b_1}(x1))) e_{b_1}(b_{d_1}(x1)) -> e_{d_1}(d_{c_1}(c_{d_1}(x1))) e_{b_1}(b_{e_1}(x1)) -> e_{d_1}(d_{c_1}(c_{e_1}(x1))) b_{b_1}(b_{b_1}(x1)) -> b_{d_1}(d_{c_1}(c_{b_1}(x1))) b_{b_1}(b_{d_1}(x1)) -> b_{d_1}(d_{c_1}(c_{d_1}(x1))) b_{b_1}(b_{e_1}(x1)) -> b_{d_1}(d_{c_1}(c_{e_1}(x1))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: C_{D_1}(d_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{e_1}(e_{a_1}(x1))) B_{E_1}(e_{c_1}(c_{b_1}(x1))) -> B_{D_1}(d_{a_1}(a_{b_1}(x1))) C_{D_1}(d_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{e_1}(e_{c_1}(x1))) B_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) C_{D_1}(d_{d_1}(d_{e_1}(x1))) -> C_{B_1}(b_{e_1}(e_{e_1}(x1))) A_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) D_{E_1}(e_{c_1}(c_{b_1}(x1))) -> D_{D_1}(d_{a_1}(a_{b_1}(x1))) B_{E_1}(e_{c_1}(c_{d_1}(x1))) -> B_{D_1}(d_{a_1}(a_{d_1}(x1))) B_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{b_1}(x1)) -> D_{E_1}(e_{d_1}(d_{b_1}(x1))) D_{A_1}(a_{b_1}(x1)) -> E_{D_1}(d_{b_1}(x1)) E_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{B_1}(b_{e_1}(e_{a_1}(x1))) A_{D_1}(d_{d_1}(d_{a_1}(x1))) -> E_{A_1}(x1) D_{E_1}(e_{c_1}(c_{d_1}(x1))) -> D_{D_1}(d_{a_1}(a_{d_1}(x1))) D_{A_1}(a_{d_1}(x1)) -> D_{E_1}(e_{d_1}(d_{d_1}(x1))) B_{E_1}(e_{c_1}(c_{e_1}(x1))) -> B_{D_1}(d_{a_1}(a_{e_1}(x1))) B_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{A_1}(a_{e_1}(x1)) D_{A_1}(a_{d_1}(x1)) -> E_{D_1}(d_{d_1}(x1)) E_{D_1}(d_{d_1}(d_{c_1}(x1))) -> E_{B_1}(b_{e_1}(e_{c_1}(x1))) B_{E_1}(e_{c_1}(c_{e_1}(x1))) -> A_{E_1}(x1) D_{A_1}(a_{d_1}(x1)) -> D_{D_1}(x1) D_{A_1}(a_{e_1}(x1)) -> D_{E_1}(e_{d_1}(d_{e_1}(x1))) D_{E_1}(e_{c_1}(c_{e_1}(x1))) -> D_{D_1}(d_{a_1}(a_{e_1}(x1))) D_{A_1}(a_{e_1}(x1)) -> E_{D_1}(d_{e_1}(x1)) E_{D_1}(d_{d_1}(d_{e_1}(x1))) -> E_{B_1}(b_{e_1}(e_{e_1}(x1))) D_{A_1}(a_{e_1}(x1)) -> D_{E_1}(x1) B_{B_1}(b_{e_1}(x1)) -> C_{E_1}(x1) The TRS R consists of the following rules: d_{a_1}(a_{b_1}(b_{b_1}(x1))) -> d_{c_1}(c_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{b_1}(b_{d_1}(x1))) -> d_{c_1}(c_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{b_1}(b_{e_1}(x1))) -> d_{c_1}(c_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(b_{b_1}(x1))) -> e_{c_1}(c_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{b_1}(b_{d_1}(x1))) -> e_{c_1}(c_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{b_1}(b_{e_1}(x1))) -> e_{c_1}(c_{d_1}(d_{e_1}(x1))) a_{d_1}(d_{d_1}(d_{a_1}(x1))) -> a_{b_1}(b_{e_1}(e_{a_1}(x1))) a_{d_1}(d_{d_1}(d_{c_1}(x1))) -> a_{b_1}(b_{e_1}(e_{c_1}(x1))) a_{d_1}(d_{d_1}(d_{e_1}(x1))) -> a_{b_1}(b_{e_1}(e_{e_1}(x1))) b_{d_1}(d_{d_1}(d_{a_1}(x1))) -> b_{b_1}(b_{e_1}(e_{a_1}(x1))) b_{d_1}(d_{d_1}(d_{c_1}(x1))) -> b_{b_1}(b_{e_1}(e_{c_1}(x1))) b_{d_1}(d_{d_1}(d_{e_1}(x1))) -> b_{b_1}(b_{e_1}(e_{e_1}(x1))) c_{d_1}(d_{d_1}(d_{a_1}(x1))) -> c_{b_1}(b_{e_1}(e_{a_1}(x1))) c_{d_1}(d_{d_1}(d_{c_1}(x1))) -> c_{b_1}(b_{e_1}(e_{c_1}(x1))) c_{d_1}(d_{d_1}(d_{e_1}(x1))) -> c_{b_1}(b_{e_1}(e_{e_1}(x1))) d_{d_1}(d_{d_1}(d_{a_1}(x1))) -> d_{b_1}(b_{e_1}(e_{a_1}(x1))) d_{d_1}(d_{d_1}(d_{c_1}(x1))) -> d_{b_1}(b_{e_1}(e_{c_1}(x1))) d_{d_1}(d_{d_1}(d_{e_1}(x1))) -> d_{b_1}(b_{e_1}(e_{e_1}(x1))) e_{d_1}(d_{d_1}(d_{a_1}(x1))) -> e_{b_1}(b_{e_1}(e_{a_1}(x1))) e_{d_1}(d_{d_1}(d_{c_1}(x1))) -> e_{b_1}(b_{e_1}(e_{c_1}(x1))) e_{d_1}(d_{d_1}(d_{e_1}(x1))) -> e_{b_1}(b_{e_1}(e_{e_1}(x1))) a_{b_1}(b_{b_1}(x1)) -> a_{d_1}(d_{c_1}(c_{b_1}(x1))) a_{b_1}(b_{d_1}(x1)) -> a_{d_1}(d_{c_1}(c_{d_1}(x1))) a_{b_1}(b_{e_1}(x1)) -> a_{d_1}(d_{c_1}(c_{e_1}(x1))) b_{b_1}(b_{b_1}(x1)) -> b_{d_1}(d_{c_1}(c_{b_1}(x1))) b_{b_1}(b_{d_1}(x1)) -> b_{d_1}(d_{c_1}(c_{d_1}(x1))) b_{b_1}(b_{e_1}(x1)) -> b_{d_1}(d_{c_1}(c_{e_1}(x1))) c_{b_1}(b_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) c_{b_1}(b_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) c_{b_1}(b_{e_1}(x1)) -> c_{d_1}(d_{c_1}(c_{e_1}(x1))) d_{b_1}(b_{b_1}(x1)) -> d_{d_1}(d_{c_1}(c_{b_1}(x1))) d_{b_1}(b_{d_1}(x1)) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{b_1}(b_{e_1}(x1)) -> d_{d_1}(d_{c_1}(c_{e_1}(x1))) e_{b_1}(b_{b_1}(x1)) -> e_{d_1}(d_{c_1}(c_{b_1}(x1))) e_{b_1}(b_{d_1}(x1)) -> e_{d_1}(d_{c_1}(c_{d_1}(x1))) e_{b_1}(b_{e_1}(x1)) -> e_{d_1}(d_{c_1}(c_{e_1}(x1))) e_{d_1}(d_{a_1}(x1)) -> e_{a_1}(x1) e_{d_1}(d_{c_1}(x1)) -> e_{c_1}(x1) e_{d_1}(d_{e_1}(x1)) -> e_{e_1}(x1) a_{e_1}(e_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(x1))) a_{e_1}(e_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{d_1}(x1))) a_{e_1}(e_{c_1}(c_{e_1}(x1))) -> a_{d_1}(d_{a_1}(a_{e_1}(x1))) b_{e_1}(e_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(x1))) b_{e_1}(e_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{d_1}(x1))) b_{e_1}(e_{c_1}(c_{e_1}(x1))) -> b_{d_1}(d_{a_1}(a_{e_1}(x1))) c_{e_1}(e_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(x1))) c_{e_1}(e_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{d_1}(x1))) c_{e_1}(e_{c_1}(c_{e_1}(x1))) -> c_{d_1}(d_{a_1}(a_{e_1}(x1))) d_{e_1}(e_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(x1))) d_{e_1}(e_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{d_1}(x1))) d_{e_1}(e_{c_1}(c_{e_1}(x1))) -> d_{d_1}(d_{a_1}(a_{e_1}(x1))) d_{a_1}(a_{b_1}(x1)) -> d_{e_1}(e_{d_1}(d_{b_1}(x1))) d_{a_1}(a_{d_1}(x1)) -> d_{e_1}(e_{d_1}(d_{d_1}(x1))) d_{a_1}(a_{e_1}(x1)) -> d_{e_1}(e_{d_1}(d_{e_1}(x1))) e_{a_1}(a_{b_1}(x1)) -> e_{e_1}(e_{d_1}(d_{b_1}(x1))) e_{a_1}(a_{d_1}(x1)) -> e_{e_1}(e_{d_1}(d_{d_1}(x1))) e_{a_1}(a_{e_1}(x1)) -> e_{e_1}(e_{d_1}(d_{e_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 27 less nodes. ---------------------------------------- (14) TRUE