/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 72 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 77 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) AND (11) QDP (12) UsableRulesProof [EQUIVALENT, 0 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) QDP (17) QDPOrderProof [EQUIVALENT, 170 ms] (18) QDP (19) DependencyGraphProof [EQUIVALENT, 0 ms] (20) AND (21) QDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) QDP (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] (25) YES (26) QDP (27) UsableRulesProof [EQUIVALENT, 1 ms] (28) QDP (29) QDPSizeChangeProof [EQUIVALENT, 0 ms] (30) YES (31) QDP (32) UsableRulesProof [EQUIVALENT, 0 ms] (33) QDP (34) QDPSizeChangeProof [EQUIVALENT, 0 ms] (35) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(b(c(x1))) -> c(b(a(a(c(b(x1)))))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 Q is empty. ---------------------------------------- (1) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(b(c(x1)))) -> a(c(b(a(a(c(b(x1))))))) b(a(b(c(x1)))) -> b(c(b(a(a(c(b(x1))))))) c(a(b(c(x1)))) -> c(c(b(a(a(c(b(x1))))))) a(a(x1)) -> a(x1) b(a(x1)) -> b(x1) c(a(x1)) -> c(x1) a(b(x1)) -> a(x1) b(b(x1)) -> b(x1) c(b(x1)) -> c(x1) a(c(x1)) -> a(x1) b(c(x1)) -> b(x1) c(c(x1)) -> c(x1) Q is empty. ---------------------------------------- (3) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled As Q is empty the root labeling was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(a_{c_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = x_1 POL(b_{c_1}(x_1)) = 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)) = 1 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_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_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(x1)) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(x1) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(x1)) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{B_1}(x1) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(x1)) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(x1)) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(x1) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1))) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(x1)) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{B_1}(x1) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(x1)) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(x1)) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(x1) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{b_1}(x1))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(x1)) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{B_1}(x1) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(x1)) B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) C_{B_1}(b_{a_1}(x1)) -> C_{A_1}(x1) C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) A_{C_1}(c_{b_1}(x1)) -> A_{B_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_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 2 SCCs with 19 less nodes. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) 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. ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) 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: *A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) C_{B_1}(b_{a_1}(x1)) -> C_{A_1}(x1) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(x1)) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(x1) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(x1)) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(x1) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(x1)) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(x1)) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(x1)) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(x1) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(x1)) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C_{B_1}(b_{a_1}(x1)) -> C_{A_1}(x1) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(x1)) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(x1) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(x1)) A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(x1) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(x1)) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(x1)) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(x1)) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> B_{A_1}(x1) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(x1)) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{A_1}(x_1)) = 1 + x_1 POL(B_{A_1}(x_1)) = 1 + x_1 POL(C_{A_1}(x_1)) = 1 + x_1 POL(C_{B_1}(x_1)) = 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(b_{a_1}(x_1)) = 1 + x_1 POL(b_{b_1}(x_1)) = x_1 POL(b_{c_1}(x_1)) = 1 + x_1 POL(c_{a_1}(x_1)) = 1 + x_1 POL(c_{b_1}(x_1)) = x_1 POL(c_{c_1}(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))))) C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))))) C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 9 less nodes. ---------------------------------------- (20) Complex Obligation (AND) ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) 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. ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) 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: *C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (25) YES ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) 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. ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) 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_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (30) YES ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))))) c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))))) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) 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. ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) 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: *C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (35) YES