/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 10 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 115 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 3 ms] (8) QTRS (9) QTRSRRRProof [EQUIVALENT, 3 ms] (10) QTRS (11) DependencyPairsProof [EQUIVALENT, 5 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 28 ms] (14) QDP (15) PisEmptyProof [EQUIVALENT, 0 ms] (16) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b(c(a(x1))) -> a(b(x1)) b(b(b(x1))) -> c(a(c(x1))) c(d(x1)) -> d(c(x1)) c(d(b(x1))) -> d(c(c(x1))) d(c(x1)) -> b(b(b(x1))) c(b(x1)) -> d(a(x1)) d(b(c(x1))) -> a(a(x1)) d(a(x1)) -> b(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: b(b(c(a(x1)))) -> b(a(b(x1))) c(b(c(a(x1)))) -> c(a(b(x1))) a(b(c(a(x1)))) -> a(a(b(x1))) d(b(c(a(x1)))) -> d(a(b(x1))) b(b(b(b(x1)))) -> b(c(a(c(x1)))) c(b(b(b(x1)))) -> c(c(a(c(x1)))) a(b(b(b(x1)))) -> a(c(a(c(x1)))) d(b(b(b(x1)))) -> d(c(a(c(x1)))) b(c(d(x1))) -> b(d(c(x1))) c(c(d(x1))) -> c(d(c(x1))) a(c(d(x1))) -> a(d(c(x1))) d(c(d(x1))) -> d(d(c(x1))) b(c(d(b(x1)))) -> b(d(c(c(x1)))) c(c(d(b(x1)))) -> c(d(c(c(x1)))) a(c(d(b(x1)))) -> a(d(c(c(x1)))) d(c(d(b(x1)))) -> d(d(c(c(x1)))) b(d(c(x1))) -> b(b(b(b(x1)))) c(d(c(x1))) -> c(b(b(b(x1)))) a(d(c(x1))) -> a(b(b(b(x1)))) d(d(c(x1))) -> d(b(b(b(x1)))) b(c(b(x1))) -> b(d(a(x1))) c(c(b(x1))) -> c(d(a(x1))) a(c(b(x1))) -> a(d(a(x1))) d(c(b(x1))) -> d(d(a(x1))) b(d(b(c(x1)))) -> b(a(a(x1))) c(d(b(c(x1)))) -> c(a(a(x1))) a(d(b(c(x1)))) -> a(a(a(x1))) d(d(b(c(x1)))) -> d(a(a(x1))) b(d(a(x1))) -> b(b(x1)) c(d(a(x1))) -> c(b(x1)) a(d(a(x1))) -> a(b(x1)) d(d(a(x1))) -> d(b(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: b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1))) b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(x1))) b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1))) b_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{d_1}(x1))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{d_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{d_1}(x1))) d_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> d_{a_1}(a_{b_1}(b_{b_1}(x1))) d_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> d_{a_1}(a_{b_1}(b_{c_1}(x1))) d_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> d_{a_1}(a_{b_1}(b_{a_1}(x1))) d_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> d_{a_1}(a_{b_1}(b_{d_1}(x1))) b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{d_1}(x1)))) c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{d_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{d_1}(x1)))) d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> d_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) d_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> d_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) d_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> d_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) d_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> d_{c_1}(c_{a_1}(a_{c_1}(c_{d_1}(x1)))) b_{c_1}(c_{d_1}(d_{b_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(x1))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{d_1}(d_{c_1}(c_{c_1}(x1))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{d_1}(d_{c_1}(c_{a_1}(x1))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{d_1}(d_{c_1}(c_{d_1}(x1))) c_{c_1}(c_{d_1}(d_{b_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{a_1}(x1))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) a_{c_1}(c_{d_1}(d_{b_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(x1))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{c_1}(x1))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{a_1}(x1))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{c_1}(c_{d_1}(d_{b_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(x1))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{c_1}(x1))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{a_1}(x1))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) b_{c_1}(c_{d_1}(d_{b_1}(b_{b_1}(x1)))) -> b_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{d_1}(d_{b_1}(b_{c_1}(x1)))) -> b_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{d_1}(d_{b_1}(b_{a_1}(x1)))) -> b_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{d_1}(d_{b_1}(b_{d_1}(x1)))) -> b_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{d_1}(d_{b_1}(b_{b_1}(x1)))) -> c_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{d_1}(d_{b_1}(b_{c_1}(x1)))) -> c_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{d_1}(d_{b_1}(b_{a_1}(x1)))) -> c_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{b_1}(b_{d_1}(x1)))) -> c_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x1)))) a_{c_1}(c_{d_1}(d_{b_1}(b_{b_1}(x1)))) -> a_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{d_1}(d_{b_1}(b_{c_1}(x1)))) -> a_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x1)))) a_{c_1}(c_{d_1}(d_{b_1}(b_{a_1}(x1)))) -> a_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{d_1}(d_{b_1}(b_{d_1}(x1)))) -> a_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x1)))) d_{c_1}(c_{d_1}(d_{b_1}(b_{b_1}(x1)))) -> d_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x1)))) d_{c_1}(c_{d_1}(d_{b_1}(b_{c_1}(x1)))) -> d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x1)))) d_{c_1}(c_{d_1}(d_{b_1}(b_{a_1}(x1)))) -> d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x1)))) d_{c_1}(c_{d_1}(d_{b_1}(b_{d_1}(x1)))) -> d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x1)))) b_{d_1}(d_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{d_1}(d_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) b_{d_1}(d_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{d_1}(d_{c_1}(c_{d_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) c_{d_1}(d_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) c_{d_1}(d_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) c_{d_1}(d_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) c_{d_1}(d_{c_1}(c_{d_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) a_{d_1}(d_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{d_1}(d_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) a_{d_1}(d_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{d_1}(d_{c_1}(c_{d_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) d_{d_1}(d_{c_1}(c_{b_1}(x1))) -> d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) d_{d_1}(d_{c_1}(c_{c_1}(x1))) -> d_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) d_{d_1}(d_{c_1}(c_{a_1}(x1))) -> d_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) d_{d_1}(d_{c_1}(c_{d_1}(x1))) -> d_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(x1))) b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{d_1}(d_{a_1}(a_{c_1}(x1))) b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{d_1}(d_{a_1}(a_{a_1}(x1))) b_{c_1}(c_{b_1}(b_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{d_1}(x1))) c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(x1))) c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{d_1}(d_{a_1}(a_{c_1}(x1))) c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{d_1}(d_{a_1}(a_{a_1}(x1))) c_{c_1}(c_{b_1}(b_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{d_1}(x1))) a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(x1))) a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{d_1}(d_{a_1}(a_{c_1}(x1))) a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{d_1}(d_{a_1}(a_{a_1}(x1))) a_{c_1}(c_{b_1}(b_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{d_1}(x1))) d_{c_1}(c_{b_1}(b_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(x1))) d_{c_1}(c_{b_1}(b_{c_1}(x1))) -> d_{d_1}(d_{a_1}(a_{c_1}(x1))) d_{c_1}(c_{b_1}(b_{a_1}(x1))) -> d_{d_1}(d_{a_1}(a_{a_1}(x1))) d_{c_1}(c_{b_1}(b_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{d_1}(x1))) b_{d_1}(d_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1))) b_{d_1}(d_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(x1))) b_{d_1}(d_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1))) b_{d_1}(d_{b_1}(b_{c_1}(c_{d_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{d_1}(x1))) c_{d_1}(d_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1))) c_{d_1}(d_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1))) c_{d_1}(d_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1))) c_{d_1}(d_{b_1}(b_{c_1}(c_{d_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{d_1}(x1))) a_{d_1}(d_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1))) a_{d_1}(d_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(x1))) a_{d_1}(d_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1))) a_{d_1}(d_{b_1}(b_{c_1}(c_{d_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{d_1}(x1))) d_{d_1}(d_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> d_{a_1}(a_{a_1}(a_{b_1}(x1))) d_{d_1}(d_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> d_{a_1}(a_{a_1}(a_{c_1}(x1))) d_{d_1}(d_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> d_{a_1}(a_{a_1}(a_{a_1}(x1))) d_{d_1}(d_{b_1}(b_{c_1}(c_{d_1}(x1)))) -> d_{a_1}(a_{a_1}(a_{d_1}(x1))) b_{d_1}(d_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) b_{d_1}(d_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1)) b_{d_1}(d_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1)) b_{d_1}(d_{a_1}(a_{d_1}(x1))) -> b_{b_1}(b_{d_1}(x1)) c_{d_1}(d_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) c_{d_1}(d_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1)) c_{d_1}(d_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1)) c_{d_1}(d_{a_1}(a_{d_1}(x1))) -> c_{b_1}(b_{d_1}(x1)) a_{d_1}(d_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) a_{d_1}(d_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1)) a_{d_1}(d_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1)) a_{d_1}(d_{a_1}(a_{d_1}(x1))) -> a_{b_1}(b_{d_1}(x1)) d_{d_1}(d_{a_1}(a_{b_1}(x1))) -> d_{b_1}(b_{b_1}(x1)) d_{d_1}(d_{a_1}(a_{c_1}(x1))) -> d_{b_1}(b_{c_1}(x1)) d_{d_1}(d_{a_1}(a_{a_1}(x1))) -> d_{b_1}(b_{a_1}(x1)) d_{d_1}(d_{a_1}(a_{d_1}(x1))) -> d_{b_1}(b_{d_1}(x1)) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = 6 + 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)) = 6 + x_1 POL(b_{c_1}(x_1)) = x_1 POL(b_{d_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = 21 + x_1 POL(c_{c_1}(x_1)) = 15 + x_1 POL(c_{d_1}(x_1)) = 17 + x_1 POL(d_{a_1}(x_1)) = 17 + x_1 POL(d_{b_1}(x_1)) = 49 + x_1 POL(d_{c_1}(x_1)) = 31 + x_1 POL(d_{d_1}(x_1)) = 33 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: b_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{d_1}(x1))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{d_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{d_1}(x1))) d_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> d_{a_1}(a_{b_1}(b_{b_1}(x1))) d_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> d_{a_1}(a_{b_1}(b_{c_1}(x1))) d_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> d_{a_1}(a_{b_1}(b_{a_1}(x1))) d_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> d_{a_1}(a_{b_1}(b_{d_1}(x1))) b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{d_1}(x1)))) c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{d_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{d_1}(x1)))) d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> d_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) d_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> d_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) d_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> d_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) d_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) -> d_{c_1}(c_{a_1}(a_{c_1}(c_{d_1}(x1)))) b_{c_1}(c_{d_1}(d_{b_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(x1))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{d_1}(d_{c_1}(c_{c_1}(x1))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{d_1}(d_{c_1}(c_{a_1}(x1))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{d_1}(d_{c_1}(c_{d_1}(x1))) c_{c_1}(c_{d_1}(d_{b_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{a_1}(x1))) a_{c_1}(c_{d_1}(d_{b_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(x1))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{c_1}(x1))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{a_1}(x1))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{c_1}(c_{d_1}(d_{b_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(x1))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{a_1}(x1))) b_{c_1}(c_{d_1}(d_{b_1}(b_{b_1}(x1)))) -> b_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{d_1}(d_{b_1}(b_{c_1}(x1)))) -> b_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{d_1}(d_{b_1}(b_{a_1}(x1)))) -> b_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{d_1}(d_{b_1}(b_{d_1}(x1)))) -> b_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{d_1}(d_{b_1}(b_{b_1}(x1)))) -> c_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{d_1}(d_{b_1}(b_{c_1}(x1)))) -> c_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{d_1}(d_{b_1}(b_{a_1}(x1)))) -> c_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{b_1}(b_{d_1}(x1)))) -> c_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x1)))) a_{c_1}(c_{d_1}(d_{b_1}(b_{b_1}(x1)))) -> a_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{d_1}(d_{b_1}(b_{c_1}(x1)))) -> a_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x1)))) a_{c_1}(c_{d_1}(d_{b_1}(b_{a_1}(x1)))) -> a_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{d_1}(d_{b_1}(b_{d_1}(x1)))) -> a_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x1)))) d_{c_1}(c_{d_1}(d_{b_1}(b_{b_1}(x1)))) -> d_{d_1}(d_{c_1}(c_{c_1}(c_{b_1}(x1)))) d_{c_1}(c_{d_1}(d_{b_1}(b_{c_1}(x1)))) -> d_{d_1}(d_{c_1}(c_{c_1}(c_{c_1}(x1)))) d_{c_1}(c_{d_1}(d_{b_1}(b_{a_1}(x1)))) -> d_{d_1}(d_{c_1}(c_{c_1}(c_{a_1}(x1)))) d_{c_1}(c_{d_1}(d_{b_1}(b_{d_1}(x1)))) -> d_{d_1}(d_{c_1}(c_{c_1}(c_{d_1}(x1)))) b_{d_1}(d_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{d_1}(d_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) b_{d_1}(d_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{d_1}(d_{c_1}(c_{d_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) c_{d_1}(d_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) c_{d_1}(d_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) c_{d_1}(d_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) c_{d_1}(d_{c_1}(c_{d_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) a_{d_1}(d_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{d_1}(d_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) a_{d_1}(d_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{d_1}(d_{c_1}(c_{d_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) d_{d_1}(d_{c_1}(c_{b_1}(x1))) -> d_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) d_{d_1}(d_{c_1}(c_{c_1}(x1))) -> d_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) d_{d_1}(d_{c_1}(c_{a_1}(x1))) -> d_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) d_{d_1}(d_{c_1}(c_{d_1}(x1))) -> d_{b_1}(b_{b_1}(b_{b_1}(b_{d_1}(x1)))) b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(x1))) b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{d_1}(d_{a_1}(a_{c_1}(x1))) b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{d_1}(d_{a_1}(a_{a_1}(x1))) b_{c_1}(c_{b_1}(b_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{d_1}(x1))) c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(x1))) c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{d_1}(d_{a_1}(a_{c_1}(x1))) c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{d_1}(d_{a_1}(a_{a_1}(x1))) c_{c_1}(c_{b_1}(b_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{d_1}(x1))) a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(x1))) a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{d_1}(d_{a_1}(a_{c_1}(x1))) a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{d_1}(d_{a_1}(a_{a_1}(x1))) a_{c_1}(c_{b_1}(b_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{d_1}(x1))) d_{c_1}(c_{b_1}(b_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(x1))) d_{c_1}(c_{b_1}(b_{c_1}(x1))) -> d_{d_1}(d_{a_1}(a_{c_1}(x1))) d_{c_1}(c_{b_1}(b_{a_1}(x1))) -> d_{d_1}(d_{a_1}(a_{a_1}(x1))) d_{c_1}(c_{b_1}(b_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{d_1}(x1))) b_{d_1}(d_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1))) b_{d_1}(d_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(x1))) b_{d_1}(d_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1))) b_{d_1}(d_{b_1}(b_{c_1}(c_{d_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{d_1}(x1))) c_{d_1}(d_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1))) c_{d_1}(d_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1))) c_{d_1}(d_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1))) c_{d_1}(d_{b_1}(b_{c_1}(c_{d_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{d_1}(x1))) a_{d_1}(d_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(x1))) a_{d_1}(d_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(x1))) a_{d_1}(d_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(x1))) a_{d_1}(d_{b_1}(b_{c_1}(c_{d_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{d_1}(x1))) d_{d_1}(d_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> d_{a_1}(a_{a_1}(a_{b_1}(x1))) d_{d_1}(d_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> d_{a_1}(a_{a_1}(a_{c_1}(x1))) d_{d_1}(d_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> d_{a_1}(a_{a_1}(a_{a_1}(x1))) d_{d_1}(d_{b_1}(b_{c_1}(c_{d_1}(x1)))) -> d_{a_1}(a_{a_1}(a_{d_1}(x1))) b_{d_1}(d_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) b_{d_1}(d_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1)) b_{d_1}(d_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1)) b_{d_1}(d_{a_1}(a_{d_1}(x1))) -> b_{b_1}(b_{d_1}(x1)) c_{d_1}(d_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) c_{d_1}(d_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1)) c_{d_1}(d_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1)) c_{d_1}(d_{a_1}(a_{d_1}(x1))) -> c_{b_1}(b_{d_1}(x1)) a_{d_1}(d_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) a_{d_1}(d_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1)) a_{d_1}(d_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1)) a_{d_1}(d_{a_1}(a_{d_1}(x1))) -> a_{b_1}(b_{d_1}(x1)) d_{d_1}(d_{a_1}(a_{b_1}(x1))) -> d_{b_1}(b_{b_1}(x1)) d_{d_1}(d_{a_1}(a_{c_1}(x1))) -> d_{b_1}(b_{c_1}(x1)) d_{d_1}(d_{a_1}(a_{a_1}(x1))) -> d_{b_1}(b_{a_1}(x1)) d_{d_1}(d_{a_1}(a_{d_1}(x1))) -> d_{b_1}(b_{d_1}(x1)) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1))) b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(x1))) b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{c_1}(x1))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) 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)) = 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(c_{a_1}(x_1)) = x_1 POL(c_{c_1}(x_1)) = x_1 POL(c_{d_1}(x_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: b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1))) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1))) b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{c_1}(x1))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) Q is empty. ---------------------------------------- (9) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{b_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)) = 1 + x_1 POL(c_{a_1}(x_1)) = x_1 POL(c_{c_1}(x_1)) = x_1 POL(c_{d_1}(x_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: b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1))) b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1))) ---------------------------------------- (10) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{c_1}(x1))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) Q is empty. ---------------------------------------- (11) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{c_1}(x1)) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{C_1}(x1) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{d_1}(x1)) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{c_1}(x1)) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{C_1}(x1) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{d_1}(x1)) The TRS R consists of the following rules: c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{c_1}(x1))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{c_1}(x1)) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{C_1}(x1) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{d_1}(x1)) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{c_1}(x1)) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{C_1}(x1) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{d_1}(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(C_{C_1}(x_1)) = x_1 POL(D_{C_1}(x_1)) = x_1 POL(c_{c_1}(x_1)) = 1 + x_1 POL(c_{d_1}(x_1)) = 1 + x_1 POL(d_{c_1}(x_1)) = 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: c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{c_1}(x1))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) ---------------------------------------- (14) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{c_1}(x1))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{d_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (16) YES