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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) FlatCCProof [EQUIVALENT, 0 ms] (4) QTRS (5) RootLabelingProof [EQUIVALENT, 0 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 35 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 0 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 475 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 1 ms] (16) AND (17) QDP (18) UsableRulesProof [EQUIVALENT, 0 ms] (19) QDP (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] (21) YES (22) QDP (23) QDPOrderProof [EQUIVALENT, 61 ms] (24) QDP (25) QDPOrderProof [EQUIVALENT, 474 ms] (26) QDP (27) QDPOrderProof [EQUIVALENT, 6647 ms] (28) QDP (29) QDPOrderProof [EQUIVALENT, 50 ms] (30) QDP (31) QDPOrderProof [EQUIVALENT, 41 ms] (32) QDP (33) PisEmptyProof [EQUIVALENT, 0 ms] (34) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(c(x1)) -> a(x1) d(a(x1)) -> a(c(b(c(d(x1))))) a(c(b(c(x1)))) -> c(b(c(c(x1)))) c(x1) -> b(a(a(x1))) d(c(x1)) -> a(c(d(a(x1)))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c(a(x1)) -> a(x1) a(d(x1)) -> d(c(b(c(a(x1))))) c(b(c(a(x1)))) -> c(c(b(c(x1)))) c(x1) -> a(a(b(x1))) c(d(x1)) -> a(d(c(a(x1)))) Q is empty. ---------------------------------------- (3) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c(b(c(a(x1)))) -> c(c(b(c(x1)))) c(c(a(x1))) -> c(a(x1)) a(c(a(x1))) -> a(a(x1)) d(c(a(x1))) -> d(a(x1)) b(c(a(x1))) -> b(a(x1)) c(a(d(x1))) -> c(d(c(b(c(a(x1)))))) a(a(d(x1))) -> a(d(c(b(c(a(x1)))))) d(a(d(x1))) -> d(d(c(b(c(a(x1)))))) b(a(d(x1))) -> b(d(c(b(c(a(x1)))))) c(c(x1)) -> c(a(a(b(x1)))) a(c(x1)) -> a(a(a(b(x1)))) d(c(x1)) -> d(a(a(b(x1)))) b(c(x1)) -> b(a(a(b(x1)))) c(c(d(x1))) -> c(a(d(c(a(x1))))) a(c(d(x1))) -> a(a(d(c(a(x1))))) d(c(d(x1))) -> d(a(d(c(a(x1))))) b(c(d(x1))) -> b(a(d(c(a(x1))))) Q is empty. ---------------------------------------- (5) 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. ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{b_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{b_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{b_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) b_{a_1}(a_{d_1}(d_{c_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) b_{a_1}(a_{d_1}(d_{b_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) b_{a_1}(a_{d_1}(d_{a_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) b_{a_1}(a_{d_1}(d_{d_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{d_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{d_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{d_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{b_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{b_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{b_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{b_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{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)) = 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)) = 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)) = x_1 POL(c_{c_1}(x_1)) = x_1 POL(c_{d_1}(x_1)) = 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)) = 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_{b_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) a_{a_1}(a_{d_1}(d_{b_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) d_{a_1}(a_{d_1}(d_{b_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) b_{a_1}(a_{d_1}(d_{c_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) b_{a_1}(a_{d_1}(d_{b_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) b_{a_1}(a_{d_1}(d_{a_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) b_{a_1}(a_{d_1}(d_{d_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{d_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) a_{c_1}(c_{d_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) d_{c_1}(c_{d_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) b_{c_1}(c_{d_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) c_{c_1}(c_{d_1}(d_{b_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) a_{c_1}(c_{d_1}(d_{b_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) d_{c_1}(c_{d_1}(d_{b_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) b_{c_1}(c_{d_1}(d_{b_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 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: C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(c_{c_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(x1) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{b_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{a_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(x1) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> B_{C_1}(c_{d_1}(x1)) A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) A_{C_1}(c_{a_1}(a_{d_1}(x1))) -> A_{A_1}(a_{d_1}(x1)) D_{C_1}(c_{a_1}(a_{c_1}(x1))) -> D_{A_1}(a_{c_1}(x1)) D_{C_1}(c_{a_1}(a_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) D_{C_1}(c_{a_1}(a_{a_1}(x1))) -> D_{A_1}(a_{a_1}(x1)) D_{C_1}(c_{a_1}(a_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) C_{C_1}(c_{c_1}(x1)) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) C_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{b_1}(b_{c_1}(x1))) C_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) C_{C_1}(c_{b_1}(x1)) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) C_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) C_{C_1}(c_{a_1}(x1)) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) C_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) A_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) A_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{b_1}(b_{c_1}(x1))) A_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) A_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) A_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) A_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) D_{C_1}(c_{c_1}(x1)) -> D_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) D_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{b_1}(b_{c_1}(x1))) D_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) D_{C_1}(c_{b_1}(x1)) -> D_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) D_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) D_{C_1}(c_{a_1}(x1)) -> D_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) D_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) B_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{b_1}(b_{c_1}(x1))) B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) B_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) B_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 23 less nodes. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: C_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{C_1}(c_{a_1}(a_{c_1}(x1))) -> D_{A_1}(a_{c_1}(x1)) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) D_{C_1}(c_{a_1}(a_{a_1}(x1))) -> D_{A_1}(a_{a_1}(x1)) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) D_{C_1}(c_{a_1}(a_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(c_{c_1}(x1)) B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) D_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(x1) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{a_1}(a_{d_1}(x1))) -> A_{A_1}(a_{d_1}(x1)) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{b_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{a_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(x1) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> B_{C_1}(c_{d_1}(x1)) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{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. B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{A_1}(x_1)) = x_1 POL(A_{C_1}(x_1)) = x_1 POL(B_{C_1}(x_1)) = x_1 POL(C_{A_1}(x_1)) = x_1 POL(C_{B_1}(x_1)) = x_1 POL(C_{C_1}(x_1)) = x_1 POL(D_{A_1}(x_1)) = x_1 POL(D_{C_1}(x_1)) = x_1 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)) = 1 + x_1 POL(b_{a_1}(x_1)) = 0 POL(b_{b_1}(x_1)) = 0 POL(b_{c_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = 0 POL(c_{c_1}(x_1)) = x_1 POL(c_{d_1}(x_1)) = 1 + x_1 POL(d_{a_1}(x_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_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: C_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) D_{C_1}(c_{a_1}(a_{c_1}(x1))) -> D_{A_1}(a_{c_1}(x1)) D_{C_1}(c_{a_1}(a_{a_1}(x1))) -> D_{A_1}(a_{a_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) D_{C_1}(c_{a_1}(a_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(c_{c_1}(x1)) A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) D_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(x1) A_{C_1}(c_{a_1}(a_{d_1}(x1))) -> A_{A_1}(a_{d_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) A_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{b_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{a_1}(x1)) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(x1) C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> B_{C_1}(c_{d_1}(x1)) C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 27 less nodes. ---------------------------------------- (16) Complex Obligation (AND) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (21) YES ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(C_{B_1}(x_1)) = x_1 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(a_{d_1}(x_1)) = 0 POL(b_{a_1}(x_1)) = 0 POL(b_{b_1}(x_1)) = 0 POL(b_{c_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = 0 POL(c_{c_1}(x_1)) = x_1 POL(c_{d_1}(x_1)) = 0 POL(d_{a_1}(x_1)) = 0 POL(d_{c_1}(x_1)) = 0 POL(d_{d_1}(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( C_{B_1}_1(x_1) ) = 2x_1 POL( b_{c_1}_1(x_1) ) = x_1 POL( a_{d_1}_1(x_1) ) = 1 POL( b_{a_1}_1(x_1) ) = max{0, -2} POL( d_{c_1}_1(x_1) ) = max{0, -2} POL( c_{b_1}_1(x_1) ) = 0 POL( c_{a_1}_1(x_1) ) = 2x_1 POL( a_{c_1}_1(x_1) ) = 2x_1 POL( c_{c_1}_1(x_1) ) = 2x_1 POL( a_{b_1}_1(x_1) ) = x_1 POL( a_{a_1}_1(x_1) ) = 2x_1 POL( c_{d_1}_1(x_1) ) = 1 POL( b_{b_1}_1(x_1) ) = 0 POL( d_{a_1}_1(x_1) ) = max{0, x_1 - 2} POL( d_{d_1}_1(x_1) ) = max{0, x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(C_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[0A], [1A], [1A]] + [[0A, 0A, 0A], [-I, 0A, 0A], [1A, 0A, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [-I], [1A]] + [[0A, -I, -I], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(c_{b_1}(x_1)) = [[0A], [1A], [1A]] + [[0A, -I, -I], [0A, 0A, 0A], [1A, 0A, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[1A], [-I], [0A]] + [[1A, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, 1A], [0A, 1A, 1A], [1A, 0A, 0A]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[0A], [1A], [1A]] + [[0A, 0A, -I], [1A, 0A, 0A], [1A, 1A, 0A]] * x_1 >>> <<< POL(a_{d_1}(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(c_{d_1}(x_1)) = [[0A], [1A], [1A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [0A, -I, 0A], [0A, -I, -I]] * x_1 >>> <<< POL(d_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(d_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 1A], [0A, 1A, 0A], [0A, 1A, -I]] * x_1 >>> <<< POL(d_{d_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [-I, 0A, 0A], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(C_{B_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = 0 POL(a_{b_1}(x_1)) = x_1 POL(a_{c_1}(x_1)) = 0 POL(a_{d_1}(x_1)) = 0 POL(b_{a_1}(x_1)) = 0 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_{b_1}(x_1)) = 0 POL(c_{c_1}(x_1)) = x_1 POL(c_{d_1}(x_1)) = 0 POL(d_{a_1}(x_1)) = 0 POL(d_{c_1}(x_1)) = 0 POL(d_{d_1}(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(C_{B_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = 0 POL(a_{b_1}(x_1)) = 1 POL(a_{c_1}(x_1)) = 0 POL(a_{d_1}(x_1)) = 0 POL(b_{a_1}(x_1)) = 0 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_{b_1}(x_1)) = 0 POL(c_{c_1}(x_1)) = x_1 POL(c_{d_1}(x_1)) = 0 POL(d_{a_1}(x_1)) = 0 POL(d_{c_1}(x_1)) = 0 POL(d_{d_1}(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) ---------------------------------------- (32) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (34) YES