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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) FlatCCProof [EQUIVALENT, 0 ms] (4) QTRS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) QTRSRRRProof [EQUIVALENT, 48 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 0 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) UsableRulesProof [EQUIVALENT, 0 ms] (15) QDP (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] (17) YES (18) QDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) QDP (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] (22) YES (23) QDP (24) UsableRulesProof [EQUIVALENT, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES (28) QDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) QDP (31) QDPSizeChangeProof [EQUIVALENT, 0 ms] (32) YES (33) QDP (34) UsableRulesProof [EQUIVALENT, 0 ms] (35) QDP (36) QDPSizeChangeProof [EQUIVALENT, 0 ms] (37) YES (38) QDP (39) QDPOrderProof [EQUIVALENT, 107 ms] (40) QDP (41) UsableRulesProof [EQUIVALENT, 0 ms] (42) QDP (43) QDPSizeChangeProof [EQUIVALENT, 0 ms] (44) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(b(c(x1))) -> c(c(b(b(a(a(x1)))))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c(b(a(x1))) -> a(a(b(b(c(c(x1)))))) a(x1) -> x1 b(x1) -> x1 c(x1) -> x1 Q is empty. ---------------------------------------- (3) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c(c(b(a(x1)))) -> c(a(a(b(b(c(c(x1))))))) b(c(b(a(x1)))) -> b(a(a(b(b(c(c(x1))))))) a(c(b(a(x1)))) -> a(a(a(b(b(c(c(x1))))))) c(a(x1)) -> c(x1) b(a(x1)) -> b(x1) a(a(x1)) -> a(x1) c(b(x1)) -> c(x1) b(b(x1)) -> b(x1) a(b(x1)) -> a(x1) c(c(x1)) -> c(x1) b(c(x1)) -> b(x1) a(c(x1)) -> a(x1) Q is empty. ---------------------------------------- (5) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled As Q is empty the root labeling was sound AND complete. ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))))) a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = 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)) = x_1 POL(c_{a_1}(x_1)) = 1 + x_1 POL(c_{b_1}(x_1)) = 2 + x_1 POL(c_{c_1}(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))))) a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))))) a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) Q is empty. ---------------------------------------- (9) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))) C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))) C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1))) C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1)) C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1))) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{a_1}(x1)) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(x1) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))))) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1)) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))))) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{a_1}(x1)) B_{C_1}(c_{b_1}(b_{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_{a_1}(x1)) -> C_{A_1}(x1) B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) B_{C_1}(c_{a_1}(x1)) -> B_{A_1}(x1) A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) The TRS R consists of the following rules: c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 22 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) The TRS R consists of the following rules: c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) 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. ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) 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_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (17) YES ---------------------------------------- (18) 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: c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) 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. ---------------------------------------- (20) 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. ---------------------------------------- (21) 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 ---------------------------------------- (22) YES ---------------------------------------- (23) 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: c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) 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. ---------------------------------------- (25) 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. ---------------------------------------- (26) 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 ---------------------------------------- (27) YES ---------------------------------------- (28) 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: c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) 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. ---------------------------------------- (30) 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. ---------------------------------------- (31) 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 ---------------------------------------- (32) YES ---------------------------------------- (33) 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: c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) 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. ---------------------------------------- (35) 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. ---------------------------------------- (36) 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 ---------------------------------------- (37) YES ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1)) C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1)) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{a_1}(x1)) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{a_1}(x1)) B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) The TRS R consists of the following rules: c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{b_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1)) C_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1)) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{a_1}(x1))) B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{a_1}(x1)) C_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{a_1}(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( B_{C_1}_1(x_1) ) = 2x_1 + 1 POL( C_{C_1}_1(x_1) ) = 2x_1 + 1 POL( c_{b_1}_1(x_1) ) = 2x_1 POL( b_{b_1}_1(x_1) ) = x_1 POL( c_{c_1}_1(x_1) ) = x_1 POL( b_{a_1}_1(x_1) ) = x_1 + 2 POL( a_{b_1}_1(x_1) ) = 2x_1 POL( c_{a_1}_1(x_1) ) = x_1 POL( a_{a_1}_1(x_1) ) = x_1 POL( b_{c_1}_1(x_1) ) = x_1 + 2 POL( a_{c_1}_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) The TRS R consists of the following rules: c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))))) c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) 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. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (44) YES