YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) FlatCCProof [EQUIVALENT, 0 ms] (4) QTRS (5) RootLabelingProof [EQUIVALENT, 0 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 206 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) QDP (11) QDPOrderProof [EQUIVALENT, 327 ms] (12) QDP (13) DependencyGraphProof [EQUIVALENT, 0 ms] (14) QDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) QDP (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] (18) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(b(b(b(b(a(a(x1)))))))) -> a(a(c(c(a(a(b(b(x1)))))))) a(a(c(c(x1)))) -> c(c(c(c(a(a(x1)))))) c(c(c(c(c(c(x1)))))) -> b(b(c(c(b(b(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: a(a(b(b(b(b(a(a(x1)))))))) -> b(b(a(a(c(c(a(a(x1)))))))) c(c(a(a(x1)))) -> a(a(c(c(c(c(x1)))))) c(c(c(c(c(c(x1)))))) -> b(b(c(c(b(b(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: a(a(a(b(b(b(b(a(a(x1))))))))) -> a(b(b(a(a(c(c(a(a(x1))))))))) b(a(a(b(b(b(b(a(a(x1))))))))) -> b(b(b(a(a(c(c(a(a(x1))))))))) c(a(a(b(b(b(b(a(a(x1))))))))) -> c(b(b(a(a(c(c(a(a(x1))))))))) a(c(c(a(a(x1))))) -> a(a(a(c(c(c(c(x1))))))) b(c(c(a(a(x1))))) -> b(a(a(c(c(c(c(x1))))))) c(c(c(a(a(x1))))) -> c(a(a(c(c(c(c(x1))))))) a(c(c(c(c(c(c(x1))))))) -> a(b(b(c(c(b(b(x1))))))) b(c(c(c(c(c(c(x1))))))) -> b(b(b(c(c(b(b(x1))))))) c(c(c(c(c(c(c(x1))))))) -> c(b(b(c(c(b(b(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: a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) Q is empty. ---------------------------------------- (7) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{a_1}(x1)) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{b_1}(x1)) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{a_1}(x1)) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{b_1}(x1)) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{a_1}(x1)) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{b_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{b_1}(x1)) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{A_1}(x1) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{A_1}(x1) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{A_1}(x1) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 48 less nodes. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{A_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{A_1}(x1) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{A_1}(x1) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))))) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(x1))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{c_1}(x1))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_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_{A_1}(x_1)) = x_1 POL(B_{C_1}(x_1)) = x_1 POL(C_{A_1}(x_1)) = x_1 POL(C_{C_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(a_{c_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = 1 + x_1 POL(b_{b_1}(x_1)) = x_1 POL(b_{c_1}(x_1)) = 0 POL(c_{a_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = 0 POL(c_{c_1}(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{A_1}(x1) C_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{A_1}(x1) A_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> B_{A_1}(x1) B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes. ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) 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. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: B_{C_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{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. ---------------------------------------- (17) 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}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))))) -> B_{C_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (18) YES