/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [SOUND, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 124 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 6 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 0 ms] (10) QDP (11) QDPOrderProof [EQUIVALENT, 251 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(d(x)) -> d(c(b(a(x)))) b(c(x)) -> c(d(a(b(x)))) a(c(x)) -> x b(d(x)) -> x The set Q consists of the following terms: a(d(x0)) b(c(x0)) a(c(x0)) b(d(x0)) ---------------------------------------- (1) FlatCCProof (SOUND) We used flat context closure [ROOTLAB] ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(d(x))) -> a(d(c(b(a(x))))) d(a(d(x))) -> d(d(c(b(a(x))))) c(a(d(x))) -> c(d(c(b(a(x))))) b(a(d(x))) -> b(d(c(b(a(x))))) a(b(c(x))) -> a(c(d(a(b(x))))) d(b(c(x))) -> d(c(d(a(b(x))))) c(b(c(x))) -> c(c(d(a(b(x))))) b(b(c(x))) -> b(c(d(a(b(x))))) a(a(c(x))) -> a(x) d(a(c(x))) -> d(x) c(a(c(x))) -> c(x) b(a(c(x))) -> b(x) a(b(d(x))) -> a(x) d(b(d(x))) -> d(x) c(b(d(x))) -> c(x) b(b(d(x))) -> b(x) 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_{d_1}(d_{a_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) a_{a_1}(a_{d_1}(d_{d_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) a_{a_1}(a_{d_1}(d_{c_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) a_{a_1}(a_{d_1}(d_{b_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) d_{a_1}(a_{d_1}(d_{a_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) d_{a_1}(a_{d_1}(d_{d_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) d_{a_1}(a_{d_1}(d_{c_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) d_{a_1}(a_{d_1}(d_{b_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) c_{a_1}(a_{d_1}(d_{a_1}(x))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) c_{a_1}(a_{d_1}(d_{d_1}(x))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) c_{a_1}(a_{d_1}(d_{c_1}(x))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) c_{a_1}(a_{d_1}(d_{b_1}(x))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) b_{a_1}(a_{d_1}(d_{a_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) b_{a_1}(a_{d_1}(d_{d_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) b_{a_1}(a_{d_1}(d_{c_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) b_{a_1}(a_{d_1}(d_{b_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) a_{b_1}(b_{c_1}(c_{a_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x))))) a_{b_1}(b_{c_1}(c_{d_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) a_{b_1}(b_{c_1}(c_{c_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) a_{b_1}(b_{c_1}(c_{b_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) d_{b_1}(b_{c_1}(c_{a_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x))))) d_{b_1}(b_{c_1}(c_{d_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) d_{b_1}(b_{c_1}(c_{c_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) d_{b_1}(b_{c_1}(c_{b_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) c_{b_1}(b_{c_1}(c_{a_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x))))) c_{b_1}(b_{c_1}(c_{d_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) c_{b_1}(b_{c_1}(c_{c_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) c_{b_1}(b_{c_1}(c_{b_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) b_{b_1}(b_{c_1}(c_{a_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x))))) b_{b_1}(b_{c_1}(c_{d_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) b_{b_1}(b_{c_1}(c_{c_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) b_{b_1}(b_{c_1}(c_{b_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) a_{a_1}(a_{c_1}(c_{a_1}(x))) -> a_{a_1}(x) a_{a_1}(a_{c_1}(c_{d_1}(x))) -> a_{d_1}(x) a_{a_1}(a_{c_1}(c_{c_1}(x))) -> a_{c_1}(x) a_{a_1}(a_{c_1}(c_{b_1}(x))) -> a_{b_1}(x) d_{a_1}(a_{c_1}(c_{a_1}(x))) -> d_{a_1}(x) d_{a_1}(a_{c_1}(c_{d_1}(x))) -> d_{d_1}(x) d_{a_1}(a_{c_1}(c_{c_1}(x))) -> d_{c_1}(x) d_{a_1}(a_{c_1}(c_{b_1}(x))) -> d_{b_1}(x) c_{a_1}(a_{c_1}(c_{a_1}(x))) -> c_{a_1}(x) c_{a_1}(a_{c_1}(c_{d_1}(x))) -> c_{d_1}(x) c_{a_1}(a_{c_1}(c_{c_1}(x))) -> c_{c_1}(x) c_{a_1}(a_{c_1}(c_{b_1}(x))) -> c_{b_1}(x) b_{a_1}(a_{c_1}(c_{a_1}(x))) -> b_{a_1}(x) b_{a_1}(a_{c_1}(c_{d_1}(x))) -> b_{d_1}(x) b_{a_1}(a_{c_1}(c_{c_1}(x))) -> b_{c_1}(x) b_{a_1}(a_{c_1}(c_{b_1}(x))) -> b_{b_1}(x) a_{b_1}(b_{d_1}(d_{a_1}(x))) -> a_{a_1}(x) a_{b_1}(b_{d_1}(d_{d_1}(x))) -> a_{d_1}(x) a_{b_1}(b_{d_1}(d_{c_1}(x))) -> a_{c_1}(x) a_{b_1}(b_{d_1}(d_{b_1}(x))) -> a_{b_1}(x) d_{b_1}(b_{d_1}(d_{a_1}(x))) -> d_{a_1}(x) d_{b_1}(b_{d_1}(d_{d_1}(x))) -> d_{d_1}(x) d_{b_1}(b_{d_1}(d_{c_1}(x))) -> d_{c_1}(x) d_{b_1}(b_{d_1}(d_{b_1}(x))) -> d_{b_1}(x) c_{b_1}(b_{d_1}(d_{a_1}(x))) -> c_{a_1}(x) c_{b_1}(b_{d_1}(d_{d_1}(x))) -> c_{d_1}(x) c_{b_1}(b_{d_1}(d_{c_1}(x))) -> c_{c_1}(x) c_{b_1}(b_{d_1}(d_{b_1}(x))) -> c_{b_1}(x) b_{b_1}(b_{d_1}(d_{a_1}(x))) -> b_{a_1}(x) b_{b_1}(b_{d_1}(d_{d_1}(x))) -> b_{d_1}(x) b_{b_1}(b_{d_1}(d_{c_1}(x))) -> b_{c_1}(x) b_{b_1}(b_{d_1}(d_{b_1}(x))) -> b_{b_1}(x) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = x_1 POL(a_{c_1}(x_1)) = x_1 POL(a_{d_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 1 + x_1 POL(b_{c_1}(x_1)) = 1 + x_1 POL(b_{d_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = 1 + x_1 POL(c_{b_1}(x_1)) = 1 + x_1 POL(c_{c_1}(x_1)) = 1 + x_1 POL(c_{d_1}(x_1)) = x_1 POL(d_{a_1}(x_1)) = 1 + x_1 POL(d_{b_1}(x_1)) = x_1 POL(d_{c_1}(x_1)) = x_1 POL(d_{d_1}(x_1)) = 1 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: c_{a_1}(a_{d_1}(d_{a_1}(x))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) c_{a_1}(a_{d_1}(d_{d_1}(x))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) c_{a_1}(a_{d_1}(d_{c_1}(x))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) c_{a_1}(a_{d_1}(d_{b_1}(x))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) a_{b_1}(b_{c_1}(c_{a_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x))))) d_{b_1}(b_{c_1}(c_{a_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x))))) c_{b_1}(b_{c_1}(c_{a_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x))))) b_{b_1}(b_{c_1}(c_{a_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x))))) a_{a_1}(a_{c_1}(c_{a_1}(x))) -> a_{a_1}(x) a_{a_1}(a_{c_1}(c_{c_1}(x))) -> a_{c_1}(x) a_{a_1}(a_{c_1}(c_{b_1}(x))) -> a_{b_1}(x) d_{a_1}(a_{c_1}(c_{a_1}(x))) -> d_{a_1}(x) d_{a_1}(a_{c_1}(c_{c_1}(x))) -> d_{c_1}(x) d_{a_1}(a_{c_1}(c_{b_1}(x))) -> d_{b_1}(x) c_{a_1}(a_{c_1}(c_{a_1}(x))) -> c_{a_1}(x) c_{a_1}(a_{c_1}(c_{d_1}(x))) -> c_{d_1}(x) c_{a_1}(a_{c_1}(c_{c_1}(x))) -> c_{c_1}(x) c_{a_1}(a_{c_1}(c_{b_1}(x))) -> c_{b_1}(x) b_{a_1}(a_{c_1}(c_{a_1}(x))) -> b_{a_1}(x) c_{b_1}(b_{d_1}(d_{a_1}(x))) -> c_{a_1}(x) c_{b_1}(b_{d_1}(d_{d_1}(x))) -> c_{d_1}(x) b_{b_1}(b_{d_1}(d_{a_1}(x))) -> b_{a_1}(x) b_{b_1}(b_{d_1}(d_{d_1}(x))) -> b_{d_1}(x) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{d_1}(d_{a_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) a_{a_1}(a_{d_1}(d_{d_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) a_{a_1}(a_{d_1}(d_{c_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) a_{a_1}(a_{d_1}(d_{b_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) d_{a_1}(a_{d_1}(d_{a_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) d_{a_1}(a_{d_1}(d_{d_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) d_{a_1}(a_{d_1}(d_{c_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) d_{a_1}(a_{d_1}(d_{b_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) b_{a_1}(a_{d_1}(d_{a_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) b_{a_1}(a_{d_1}(d_{d_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) b_{a_1}(a_{d_1}(d_{c_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) b_{a_1}(a_{d_1}(d_{b_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) a_{b_1}(b_{c_1}(c_{d_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) a_{b_1}(b_{c_1}(c_{c_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) a_{b_1}(b_{c_1}(c_{b_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) d_{b_1}(b_{c_1}(c_{d_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) d_{b_1}(b_{c_1}(c_{c_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) d_{b_1}(b_{c_1}(c_{b_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) c_{b_1}(b_{c_1}(c_{d_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) c_{b_1}(b_{c_1}(c_{c_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) c_{b_1}(b_{c_1}(c_{b_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) b_{b_1}(b_{c_1}(c_{d_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) b_{b_1}(b_{c_1}(c_{c_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) b_{b_1}(b_{c_1}(c_{b_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) a_{a_1}(a_{c_1}(c_{d_1}(x))) -> a_{d_1}(x) d_{a_1}(a_{c_1}(c_{d_1}(x))) -> d_{d_1}(x) b_{a_1}(a_{c_1}(c_{d_1}(x))) -> b_{d_1}(x) b_{a_1}(a_{c_1}(c_{c_1}(x))) -> b_{c_1}(x) b_{a_1}(a_{c_1}(c_{b_1}(x))) -> b_{b_1}(x) a_{b_1}(b_{d_1}(d_{a_1}(x))) -> a_{a_1}(x) a_{b_1}(b_{d_1}(d_{d_1}(x))) -> a_{d_1}(x) a_{b_1}(b_{d_1}(d_{c_1}(x))) -> a_{c_1}(x) a_{b_1}(b_{d_1}(d_{b_1}(x))) -> a_{b_1}(x) d_{b_1}(b_{d_1}(d_{a_1}(x))) -> d_{a_1}(x) d_{b_1}(b_{d_1}(d_{d_1}(x))) -> d_{d_1}(x) d_{b_1}(b_{d_1}(d_{c_1}(x))) -> d_{c_1}(x) d_{b_1}(b_{d_1}(d_{b_1}(x))) -> d_{b_1}(x) c_{b_1}(b_{d_1}(d_{c_1}(x))) -> c_{c_1}(x) c_{b_1}(b_{d_1}(d_{b_1}(x))) -> c_{b_1}(x) b_{b_1}(b_{d_1}(d_{c_1}(x))) -> b_{c_1}(x) b_{b_1}(b_{d_1}(d_{b_1}(x))) -> b_{b_1}(x) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = x_1 POL(a_{c_1}(x_1)) = x_1 POL(a_{d_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = x_1 POL(b_{c_1}(x_1)) = x_1 POL(b_{d_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = x_1 POL(c_{c_1}(x_1)) = x_1 POL(c_{d_1}(x_1)) = x_1 POL(d_{a_1}(x_1)) = x_1 POL(d_{b_1}(x_1)) = 1 + x_1 POL(d_{c_1}(x_1)) = x_1 POL(d_{d_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_{d_1}(d_{b_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) d_{a_1}(a_{d_1}(d_{b_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) b_{a_1}(a_{d_1}(d_{b_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x))))) d_{b_1}(b_{c_1}(c_{d_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) d_{b_1}(b_{c_1}(c_{c_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) d_{b_1}(b_{c_1}(c_{b_1}(x))) -> d_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) a_{b_1}(b_{d_1}(d_{b_1}(x))) -> a_{b_1}(x) d_{b_1}(b_{d_1}(d_{a_1}(x))) -> d_{a_1}(x) d_{b_1}(b_{d_1}(d_{d_1}(x))) -> d_{d_1}(x) d_{b_1}(b_{d_1}(d_{c_1}(x))) -> d_{c_1}(x) d_{b_1}(b_{d_1}(d_{b_1}(x))) -> d_{b_1}(x) c_{b_1}(b_{d_1}(d_{b_1}(x))) -> c_{b_1}(x) b_{b_1}(b_{d_1}(d_{b_1}(x))) -> b_{b_1}(x) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{d_1}(d_{a_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) a_{a_1}(a_{d_1}(d_{d_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) a_{a_1}(a_{d_1}(d_{c_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) d_{a_1}(a_{d_1}(d_{a_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) d_{a_1}(a_{d_1}(d_{d_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) d_{a_1}(a_{d_1}(d_{c_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) b_{a_1}(a_{d_1}(d_{a_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) b_{a_1}(a_{d_1}(d_{d_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) b_{a_1}(a_{d_1}(d_{c_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) a_{b_1}(b_{c_1}(c_{d_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) a_{b_1}(b_{c_1}(c_{c_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) a_{b_1}(b_{c_1}(c_{b_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) c_{b_1}(b_{c_1}(c_{d_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) c_{b_1}(b_{c_1}(c_{c_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) c_{b_1}(b_{c_1}(c_{b_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) b_{b_1}(b_{c_1}(c_{d_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) b_{b_1}(b_{c_1}(c_{c_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) b_{b_1}(b_{c_1}(c_{b_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) a_{a_1}(a_{c_1}(c_{d_1}(x))) -> a_{d_1}(x) d_{a_1}(a_{c_1}(c_{d_1}(x))) -> d_{d_1}(x) b_{a_1}(a_{c_1}(c_{d_1}(x))) -> b_{d_1}(x) b_{a_1}(a_{c_1}(c_{c_1}(x))) -> b_{c_1}(x) b_{a_1}(a_{c_1}(c_{b_1}(x))) -> b_{b_1}(x) a_{b_1}(b_{d_1}(d_{a_1}(x))) -> a_{a_1}(x) a_{b_1}(b_{d_1}(d_{d_1}(x))) -> a_{d_1}(x) a_{b_1}(b_{d_1}(d_{c_1}(x))) -> a_{c_1}(x) c_{b_1}(b_{d_1}(d_{c_1}(x))) -> c_{c_1}(x) b_{b_1}(b_{d_1}(d_{c_1}(x))) -> b_{c_1}(x) Q is empty. ---------------------------------------- (9) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{d_1}(d_{a_1}(x))) -> C_{B_1}(b_{a_1}(a_{a_1}(x))) A_{A_1}(a_{d_1}(d_{a_1}(x))) -> B_{A_1}(a_{a_1}(x)) A_{A_1}(a_{d_1}(d_{a_1}(x))) -> A_{A_1}(x) A_{A_1}(a_{d_1}(d_{d_1}(x))) -> C_{B_1}(b_{a_1}(a_{d_1}(x))) A_{A_1}(a_{d_1}(d_{d_1}(x))) -> B_{A_1}(a_{d_1}(x)) A_{A_1}(a_{d_1}(d_{c_1}(x))) -> C_{B_1}(b_{a_1}(a_{c_1}(x))) A_{A_1}(a_{d_1}(d_{c_1}(x))) -> B_{A_1}(a_{c_1}(x)) D_{A_1}(a_{d_1}(d_{a_1}(x))) -> C_{B_1}(b_{a_1}(a_{a_1}(x))) D_{A_1}(a_{d_1}(d_{a_1}(x))) -> B_{A_1}(a_{a_1}(x)) D_{A_1}(a_{d_1}(d_{a_1}(x))) -> A_{A_1}(x) D_{A_1}(a_{d_1}(d_{d_1}(x))) -> C_{B_1}(b_{a_1}(a_{d_1}(x))) D_{A_1}(a_{d_1}(d_{d_1}(x))) -> B_{A_1}(a_{d_1}(x)) D_{A_1}(a_{d_1}(d_{c_1}(x))) -> C_{B_1}(b_{a_1}(a_{c_1}(x))) D_{A_1}(a_{d_1}(d_{c_1}(x))) -> B_{A_1}(a_{c_1}(x)) B_{A_1}(a_{d_1}(d_{a_1}(x))) -> C_{B_1}(b_{a_1}(a_{a_1}(x))) B_{A_1}(a_{d_1}(d_{a_1}(x))) -> B_{A_1}(a_{a_1}(x)) B_{A_1}(a_{d_1}(d_{a_1}(x))) -> A_{A_1}(x) B_{A_1}(a_{d_1}(d_{d_1}(x))) -> C_{B_1}(b_{a_1}(a_{d_1}(x))) B_{A_1}(a_{d_1}(d_{d_1}(x))) -> B_{A_1}(a_{d_1}(x)) B_{A_1}(a_{d_1}(d_{c_1}(x))) -> C_{B_1}(b_{a_1}(a_{c_1}(x))) B_{A_1}(a_{d_1}(d_{c_1}(x))) -> B_{A_1}(a_{c_1}(x)) A_{B_1}(b_{c_1}(c_{d_1}(x))) -> D_{A_1}(a_{b_1}(b_{d_1}(x))) A_{B_1}(b_{c_1}(c_{d_1}(x))) -> A_{B_1}(b_{d_1}(x)) A_{B_1}(b_{c_1}(c_{c_1}(x))) -> D_{A_1}(a_{b_1}(b_{c_1}(x))) A_{B_1}(b_{c_1}(c_{c_1}(x))) -> A_{B_1}(b_{c_1}(x)) A_{B_1}(b_{c_1}(c_{b_1}(x))) -> D_{A_1}(a_{b_1}(b_{b_1}(x))) A_{B_1}(b_{c_1}(c_{b_1}(x))) -> A_{B_1}(b_{b_1}(x)) A_{B_1}(b_{c_1}(c_{b_1}(x))) -> B_{B_1}(x) C_{B_1}(b_{c_1}(c_{d_1}(x))) -> D_{A_1}(a_{b_1}(b_{d_1}(x))) C_{B_1}(b_{c_1}(c_{d_1}(x))) -> A_{B_1}(b_{d_1}(x)) C_{B_1}(b_{c_1}(c_{c_1}(x))) -> D_{A_1}(a_{b_1}(b_{c_1}(x))) C_{B_1}(b_{c_1}(c_{c_1}(x))) -> A_{B_1}(b_{c_1}(x)) C_{B_1}(b_{c_1}(c_{b_1}(x))) -> D_{A_1}(a_{b_1}(b_{b_1}(x))) C_{B_1}(b_{c_1}(c_{b_1}(x))) -> A_{B_1}(b_{b_1}(x)) C_{B_1}(b_{c_1}(c_{b_1}(x))) -> B_{B_1}(x) B_{B_1}(b_{c_1}(c_{d_1}(x))) -> D_{A_1}(a_{b_1}(b_{d_1}(x))) B_{B_1}(b_{c_1}(c_{d_1}(x))) -> A_{B_1}(b_{d_1}(x)) B_{B_1}(b_{c_1}(c_{c_1}(x))) -> D_{A_1}(a_{b_1}(b_{c_1}(x))) B_{B_1}(b_{c_1}(c_{c_1}(x))) -> A_{B_1}(b_{c_1}(x)) B_{B_1}(b_{c_1}(c_{b_1}(x))) -> D_{A_1}(a_{b_1}(b_{b_1}(x))) B_{B_1}(b_{c_1}(c_{b_1}(x))) -> A_{B_1}(b_{b_1}(x)) B_{B_1}(b_{c_1}(c_{b_1}(x))) -> B_{B_1}(x) B_{A_1}(a_{c_1}(c_{b_1}(x))) -> B_{B_1}(x) A_{B_1}(b_{d_1}(d_{a_1}(x))) -> A_{A_1}(x) The TRS R consists of the following rules: a_{a_1}(a_{d_1}(d_{a_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) a_{a_1}(a_{d_1}(d_{d_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) a_{a_1}(a_{d_1}(d_{c_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) d_{a_1}(a_{d_1}(d_{a_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) d_{a_1}(a_{d_1}(d_{d_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) d_{a_1}(a_{d_1}(d_{c_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) b_{a_1}(a_{d_1}(d_{a_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) b_{a_1}(a_{d_1}(d_{d_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) b_{a_1}(a_{d_1}(d_{c_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) a_{b_1}(b_{c_1}(c_{d_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) a_{b_1}(b_{c_1}(c_{c_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) a_{b_1}(b_{c_1}(c_{b_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) c_{b_1}(b_{c_1}(c_{d_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) c_{b_1}(b_{c_1}(c_{c_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) c_{b_1}(b_{c_1}(c_{b_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) b_{b_1}(b_{c_1}(c_{d_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) b_{b_1}(b_{c_1}(c_{c_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) b_{b_1}(b_{c_1}(c_{b_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) a_{a_1}(a_{c_1}(c_{d_1}(x))) -> a_{d_1}(x) d_{a_1}(a_{c_1}(c_{d_1}(x))) -> d_{d_1}(x) b_{a_1}(a_{c_1}(c_{d_1}(x))) -> b_{d_1}(x) b_{a_1}(a_{c_1}(c_{c_1}(x))) -> b_{c_1}(x) b_{a_1}(a_{c_1}(c_{b_1}(x))) -> b_{b_1}(x) a_{b_1}(b_{d_1}(d_{a_1}(x))) -> a_{a_1}(x) a_{b_1}(b_{d_1}(d_{d_1}(x))) -> a_{d_1}(x) a_{b_1}(b_{d_1}(d_{c_1}(x))) -> a_{c_1}(x) c_{b_1}(b_{d_1}(d_{c_1}(x))) -> c_{c_1}(x) b_{b_1}(b_{d_1}(d_{c_1}(x))) -> b_{c_1}(x) 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. A_{A_1}(a_{d_1}(d_{a_1}(x))) -> C_{B_1}(b_{a_1}(a_{a_1}(x))) A_{A_1}(a_{d_1}(d_{a_1}(x))) -> B_{A_1}(a_{a_1}(x)) A_{A_1}(a_{d_1}(d_{a_1}(x))) -> A_{A_1}(x) A_{A_1}(a_{d_1}(d_{d_1}(x))) -> C_{B_1}(b_{a_1}(a_{d_1}(x))) A_{A_1}(a_{d_1}(d_{d_1}(x))) -> B_{A_1}(a_{d_1}(x)) A_{A_1}(a_{d_1}(d_{c_1}(x))) -> C_{B_1}(b_{a_1}(a_{c_1}(x))) A_{A_1}(a_{d_1}(d_{c_1}(x))) -> B_{A_1}(a_{c_1}(x)) D_{A_1}(a_{d_1}(d_{a_1}(x))) -> B_{A_1}(a_{a_1}(x)) D_{A_1}(a_{d_1}(d_{a_1}(x))) -> A_{A_1}(x) D_{A_1}(a_{d_1}(d_{d_1}(x))) -> B_{A_1}(a_{d_1}(x)) D_{A_1}(a_{d_1}(d_{c_1}(x))) -> B_{A_1}(a_{c_1}(x)) B_{A_1}(a_{d_1}(d_{a_1}(x))) -> B_{A_1}(a_{a_1}(x)) B_{A_1}(a_{d_1}(d_{a_1}(x))) -> A_{A_1}(x) B_{A_1}(a_{d_1}(d_{d_1}(x))) -> B_{A_1}(a_{d_1}(x)) B_{A_1}(a_{d_1}(d_{c_1}(x))) -> B_{A_1}(a_{c_1}(x)) A_{B_1}(b_{c_1}(c_{d_1}(x))) -> D_{A_1}(a_{b_1}(b_{d_1}(x))) A_{B_1}(b_{c_1}(c_{d_1}(x))) -> A_{B_1}(b_{d_1}(x)) A_{B_1}(b_{c_1}(c_{c_1}(x))) -> D_{A_1}(a_{b_1}(b_{c_1}(x))) A_{B_1}(b_{c_1}(c_{c_1}(x))) -> A_{B_1}(b_{c_1}(x)) A_{B_1}(b_{c_1}(c_{b_1}(x))) -> D_{A_1}(a_{b_1}(b_{b_1}(x))) A_{B_1}(b_{c_1}(c_{b_1}(x))) -> A_{B_1}(b_{b_1}(x)) A_{B_1}(b_{c_1}(c_{b_1}(x))) -> B_{B_1}(x) C_{B_1}(b_{c_1}(c_{d_1}(x))) -> D_{A_1}(a_{b_1}(b_{d_1}(x))) C_{B_1}(b_{c_1}(c_{d_1}(x))) -> A_{B_1}(b_{d_1}(x)) C_{B_1}(b_{c_1}(c_{c_1}(x))) -> D_{A_1}(a_{b_1}(b_{c_1}(x))) C_{B_1}(b_{c_1}(c_{c_1}(x))) -> A_{B_1}(b_{c_1}(x)) C_{B_1}(b_{c_1}(c_{b_1}(x))) -> D_{A_1}(a_{b_1}(b_{b_1}(x))) C_{B_1}(b_{c_1}(c_{b_1}(x))) -> A_{B_1}(b_{b_1}(x)) C_{B_1}(b_{c_1}(c_{b_1}(x))) -> B_{B_1}(x) B_{B_1}(b_{c_1}(c_{d_1}(x))) -> D_{A_1}(a_{b_1}(b_{d_1}(x))) B_{B_1}(b_{c_1}(c_{c_1}(x))) -> D_{A_1}(a_{b_1}(b_{c_1}(x))) B_{B_1}(b_{c_1}(c_{b_1}(x))) -> D_{A_1}(a_{b_1}(b_{b_1}(x))) B_{B_1}(b_{c_1}(c_{b_1}(x))) -> B_{B_1}(x) B_{A_1}(a_{c_1}(c_{b_1}(x))) -> B_{B_1}(x) A_{B_1}(b_{d_1}(d_{a_1}(x))) -> A_{A_1}(x) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{A_1}(x_1)) = 1 + x_1 POL(A_{B_1}(x_1)) = 1 + x_1 POL(B_{A_1}(x_1)) = x_1 POL(B_{B_1}(x_1)) = x_1 POL(C_{B_1}(x_1)) = 1 + x_1 POL(D_{A_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = x_1 POL(a_{c_1}(x_1)) = x_1 POL(a_{d_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 1 + x_1 POL(b_{c_1}(x_1)) = 1 + x_1 POL(b_{d_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = 1 + x_1 POL(c_{c_1}(x_1)) = 1 + x_1 POL(c_{d_1}(x_1)) = x_1 POL(d_{a_1}(x_1)) = 1 + x_1 POL(d_{c_1}(x_1)) = x_1 POL(d_{d_1}(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a_{a_1}(a_{d_1}(d_{a_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) a_{a_1}(a_{d_1}(d_{d_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) a_{a_1}(a_{d_1}(d_{c_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) a_{a_1}(a_{c_1}(c_{d_1}(x))) -> a_{d_1}(x) b_{a_1}(a_{d_1}(d_{a_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) b_{a_1}(a_{d_1}(d_{d_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) b_{a_1}(a_{d_1}(d_{c_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) b_{a_1}(a_{c_1}(c_{d_1}(x))) -> b_{d_1}(x) b_{a_1}(a_{c_1}(c_{c_1}(x))) -> b_{c_1}(x) b_{a_1}(a_{c_1}(c_{b_1}(x))) -> b_{b_1}(x) a_{b_1}(b_{d_1}(d_{a_1}(x))) -> a_{a_1}(x) a_{b_1}(b_{d_1}(d_{d_1}(x))) -> a_{d_1}(x) a_{b_1}(b_{d_1}(d_{c_1}(x))) -> a_{c_1}(x) a_{b_1}(b_{c_1}(c_{d_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) a_{b_1}(b_{c_1}(c_{c_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) a_{b_1}(b_{c_1}(c_{b_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) b_{b_1}(b_{c_1}(c_{d_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) b_{b_1}(b_{c_1}(c_{c_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) b_{b_1}(b_{c_1}(c_{b_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) b_{b_1}(b_{d_1}(d_{c_1}(x))) -> b_{c_1}(x) c_{b_1}(b_{c_1}(c_{d_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) c_{b_1}(b_{c_1}(c_{c_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) c_{b_1}(b_{c_1}(c_{b_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) d_{a_1}(a_{d_1}(d_{a_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) d_{a_1}(a_{d_1}(d_{d_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) d_{a_1}(a_{d_1}(d_{c_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) c_{b_1}(b_{d_1}(d_{c_1}(x))) -> c_{c_1}(x) d_{a_1}(a_{c_1}(c_{d_1}(x))) -> d_{d_1}(x) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: D_{A_1}(a_{d_1}(d_{a_1}(x))) -> C_{B_1}(b_{a_1}(a_{a_1}(x))) D_{A_1}(a_{d_1}(d_{d_1}(x))) -> C_{B_1}(b_{a_1}(a_{d_1}(x))) D_{A_1}(a_{d_1}(d_{c_1}(x))) -> C_{B_1}(b_{a_1}(a_{c_1}(x))) B_{A_1}(a_{d_1}(d_{a_1}(x))) -> C_{B_1}(b_{a_1}(a_{a_1}(x))) B_{A_1}(a_{d_1}(d_{d_1}(x))) -> C_{B_1}(b_{a_1}(a_{d_1}(x))) B_{A_1}(a_{d_1}(d_{c_1}(x))) -> C_{B_1}(b_{a_1}(a_{c_1}(x))) B_{B_1}(b_{c_1}(c_{d_1}(x))) -> A_{B_1}(b_{d_1}(x)) B_{B_1}(b_{c_1}(c_{c_1}(x))) -> A_{B_1}(b_{c_1}(x)) B_{B_1}(b_{c_1}(c_{b_1}(x))) -> A_{B_1}(b_{b_1}(x)) The TRS R consists of the following rules: a_{a_1}(a_{d_1}(d_{a_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) a_{a_1}(a_{d_1}(d_{d_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) a_{a_1}(a_{d_1}(d_{c_1}(x))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) d_{a_1}(a_{d_1}(d_{a_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) d_{a_1}(a_{d_1}(d_{d_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) d_{a_1}(a_{d_1}(d_{c_1}(x))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) b_{a_1}(a_{d_1}(d_{a_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x))))) b_{a_1}(a_{d_1}(d_{d_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{d_1}(x))))) b_{a_1}(a_{d_1}(d_{c_1}(x))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x))))) a_{b_1}(b_{c_1}(c_{d_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) a_{b_1}(b_{c_1}(c_{c_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) a_{b_1}(b_{c_1}(c_{b_1}(x))) -> a_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) c_{b_1}(b_{c_1}(c_{d_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) c_{b_1}(b_{c_1}(c_{c_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) c_{b_1}(b_{c_1}(c_{b_1}(x))) -> c_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) b_{b_1}(b_{c_1}(c_{d_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x))))) b_{b_1}(b_{c_1}(c_{c_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x))))) b_{b_1}(b_{c_1}(c_{b_1}(x))) -> b_{c_1}(c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x))))) a_{a_1}(a_{c_1}(c_{d_1}(x))) -> a_{d_1}(x) d_{a_1}(a_{c_1}(c_{d_1}(x))) -> d_{d_1}(x) b_{a_1}(a_{c_1}(c_{d_1}(x))) -> b_{d_1}(x) b_{a_1}(a_{c_1}(c_{c_1}(x))) -> b_{c_1}(x) b_{a_1}(a_{c_1}(c_{b_1}(x))) -> b_{b_1}(x) a_{b_1}(b_{d_1}(d_{a_1}(x))) -> a_{a_1}(x) a_{b_1}(b_{d_1}(d_{d_1}(x))) -> a_{d_1}(x) a_{b_1}(b_{d_1}(d_{c_1}(x))) -> a_{c_1}(x) c_{b_1}(b_{d_1}(d_{c_1}(x))) -> c_{c_1}(x) b_{b_1}(b_{d_1}(d_{c_1}(x))) -> b_{c_1}(x) 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 9 less nodes. ---------------------------------------- (14) TRUE