29.89/8.61 YES 29.89/8.62 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 29.89/8.62 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.89/8.62 29.89/8.62 29.89/8.62 Termination w.r.t. Q of the given QTRS could be proven: 29.89/8.62 29.89/8.62 (0) QTRS 29.89/8.62 (1) QTRS Reverse [EQUIVALENT, 0 ms] 29.89/8.62 (2) QTRS 29.89/8.62 (3) FlatCCProof [EQUIVALENT, 0 ms] 29.89/8.62 (4) QTRS 29.89/8.62 (5) RootLabelingProof [EQUIVALENT, 0 ms] 29.89/8.62 (6) QTRS 29.89/8.62 (7) DependencyPairsProof [EQUIVALENT, 178 ms] 29.89/8.62 (8) QDP 29.89/8.62 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 29.89/8.62 (10) QDP 29.89/8.62 (11) QDPOrderProof [EQUIVALENT, 460 ms] 29.89/8.62 (12) QDP 29.89/8.62 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 29.89/8.62 (14) QDP 29.89/8.62 (15) UsableRulesProof [EQUIVALENT, 0 ms] 29.89/8.62 (16) QDP 29.89/8.62 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 29.89/8.62 (18) YES 29.89/8.62 29.89/8.62 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (0) 29.89/8.62 Obligation: 29.89/8.62 Q restricted rewrite system: 29.89/8.62 The TRS R consists of the following rules: 29.89/8.62 29.89/8.62 a(a(b(b(b(b(a(a(x1)))))))) -> a(a(c(c(a(a(b(b(x1)))))))) 29.89/8.62 a(a(c(c(x1)))) -> c(c(c(c(a(a(x1)))))) 29.89/8.62 c(c(c(c(c(c(x1)))))) -> b(b(c(c(b(b(x1)))))) 29.89/8.62 29.89/8.62 Q is empty. 29.89/8.62 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (1) QTRS Reverse (EQUIVALENT) 29.89/8.62 We applied the QTRS Reverse Processor [REVERSE]. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (2) 29.89/8.62 Obligation: 29.89/8.62 Q restricted rewrite system: 29.89/8.62 The TRS R consists of the following rules: 29.89/8.62 29.89/8.62 a(a(b(b(b(b(a(a(x1)))))))) -> b(b(a(a(c(c(a(a(x1)))))))) 29.89/8.62 c(c(a(a(x1)))) -> a(a(c(c(c(c(x1)))))) 29.89/8.62 c(c(c(c(c(c(x1)))))) -> b(b(c(c(b(b(x1)))))) 29.89/8.62 29.89/8.62 Q is empty. 29.89/8.62 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (3) FlatCCProof (EQUIVALENT) 29.89/8.62 We used flat context closure [ROOTLAB] 29.89/8.62 As Q is empty the flat context closure was sound AND complete. 29.89/8.62 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (4) 29.89/8.62 Obligation: 29.89/8.62 Q restricted rewrite system: 29.89/8.62 The TRS R consists of the following rules: 29.89/8.62 29.89/8.62 a(a(a(b(b(b(b(a(a(x1))))))))) -> a(b(b(a(a(c(c(a(a(x1))))))))) 29.89/8.62 b(a(a(b(b(b(b(a(a(x1))))))))) -> b(b(b(a(a(c(c(a(a(x1))))))))) 29.89/8.62 c(a(a(b(b(b(b(a(a(x1))))))))) -> c(b(b(a(a(c(c(a(a(x1))))))))) 29.89/8.62 a(c(c(a(a(x1))))) -> a(a(a(c(c(c(c(x1))))))) 29.89/8.62 b(c(c(a(a(x1))))) -> b(a(a(c(c(c(c(x1))))))) 29.89/8.62 c(c(c(a(a(x1))))) -> c(a(a(c(c(c(c(x1))))))) 29.89/8.62 a(c(c(c(c(c(c(x1))))))) -> a(b(b(c(c(b(b(x1))))))) 29.89/8.62 b(c(c(c(c(c(c(x1))))))) -> b(b(b(c(c(b(b(x1))))))) 29.89/8.62 c(c(c(c(c(c(c(x1))))))) -> c(b(b(c(c(b(b(x1))))))) 29.89/8.62 29.89/8.62 Q is empty. 29.89/8.62 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (5) RootLabelingProof (EQUIVALENT) 29.89/8.62 We used plain root labeling [ROOTLAB] with the following heuristic: 29.89/8.62 LabelAll: All function symbols get labeled 29.89/8.62 29.89/8.62 As Q is empty the root labeling was sound AND complete. 29.89/8.62 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (6) 29.89/8.62 Obligation: 29.89/8.62 Q restricted rewrite system: 29.89/8.62 The TRS R consists of the following rules: 29.89/8.62 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 29.89/8.62 Q is empty. 29.89/8.62 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (7) DependencyPairsProof (EQUIVALENT) 29.89/8.62 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (8) 29.89/8.62 Obligation: 29.89/8.62 Q DP problem: 29.89/8.62 The TRS P consists of the following rules: 29.89/8.62 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{a_1}(x1)) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{b_1}(x1)) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{a_1}(x1)) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{b_1}(x1)) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{C_1}(c_{a_1}(x1)) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{C_1}(c_{b_1}(x1)) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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) 29.89/8.62 29.89/8.62 The TRS R consists of the following rules: 29.89/8.62 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 29.89/8.62 Q is empty. 29.89/8.62 We have to consider all minimal (P,Q,R)-chains. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (9) DependencyGraphProof (EQUIVALENT) 29.89/8.62 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 48 less nodes. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (10) 29.89/8.62 Obligation: 29.89/8.62 Q DP problem: 29.89/8.62 The TRS P consists of the following rules: 29.89/8.62 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))))) 29.89/8.62 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))))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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))) 29.89/8.62 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)))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))))))) 29.89/8.62 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))))) 29.89/8.62 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))) 29.89/8.62 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)))))) 29.89/8.62 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))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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))))) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) 29.89/8.62 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)))))) 29.89/8.62 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))))))) 29.89/8.62 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))) 29.89/8.62 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))))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))))) 29.89/8.62 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))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))))) 29.89/8.62 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))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) 29.89/8.62 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))))))) 29.89/8.62 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))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) 29.89/8.62 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) 29.89/8.62 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))) 29.89/8.62 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) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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) 29.89/8.62 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) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) 29.89/8.62 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) 29.89/8.62 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) 29.89/8.62 29.89/8.62 The TRS R consists of the following rules: 29.89/8.62 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 29.89/8.62 Q is empty. 29.89/8.62 We have to consider all minimal (P,Q,R)-chains. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (11) QDPOrderProof (EQUIVALENT) 29.89/8.62 We use the reduction pair processor [LPAR04,JAR06]. 29.89/8.62 29.89/8.62 29.89/8.62 The following pairs can be oriented strictly and are deleted. 29.89/8.62 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))))) 29.89/8.62 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))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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))) 29.89/8.62 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)))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))))) 29.89/8.62 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))) 29.89/8.62 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)))))) 29.89/8.62 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))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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))))) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) 29.89/8.62 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)))))) 29.89/8.62 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))))))) 29.89/8.62 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))) 29.89/8.62 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))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))))) 29.89/8.62 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))) 29.89/8.62 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))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))))) 29.89/8.62 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))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) 29.89/8.62 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))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 A_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) 29.89/8.62 C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) 29.89/8.62 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))) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(x1) 29.89/8.62 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)))))) 29.89/8.62 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))))) 29.89/8.62 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)))) 29.89/8.62 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))) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(x1)) 29.89/8.62 B_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{C_1}(x1) 29.89/8.62 The remaining pairs can at least be oriented weakly. 29.89/8.62 Used ordering: Polynomial interpretation [POLO]: 29.89/8.62 29.89/8.62 POL(A_{A_1}(x_1)) = x_1 29.89/8.62 POL(A_{C_1}(x_1)) = x_1 29.89/8.62 POL(B_{A_1}(x_1)) = x_1 29.89/8.62 POL(B_{C_1}(x_1)) = x_1 29.89/8.62 POL(C_{A_1}(x_1)) = x_1 29.89/8.62 POL(C_{C_1}(x_1)) = x_1 29.89/8.62 POL(a_{a_1}(x_1)) = 1 + x_1 29.89/8.62 POL(a_{b_1}(x_1)) = 1 + x_1 29.89/8.62 POL(a_{c_1}(x_1)) = 1 + x_1 29.89/8.62 POL(b_{a_1}(x_1)) = 1 + x_1 29.89/8.62 POL(b_{b_1}(x_1)) = x_1 29.89/8.62 POL(b_{c_1}(x_1)) = 0 29.89/8.62 POL(c_{a_1}(x_1)) = x_1 29.89/8.62 POL(c_{b_1}(x_1)) = 0 29.89/8.62 POL(c_{c_1}(x_1)) = x_1 29.89/8.62 29.89/8.62 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.89/8.62 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 29.89/8.62 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (12) 29.89/8.62 Obligation: 29.89/8.62 Q DP problem: 29.89/8.62 The TRS P consists of the following rules: 29.89/8.62 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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) 29.89/8.62 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) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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) 29.89/8.62 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) 29.89/8.62 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) 29.89/8.62 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) 29.89/8.62 29.89/8.62 The TRS R consists of the following rules: 29.89/8.62 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 29.89/8.62 Q is empty. 29.89/8.62 We have to consider all minimal (P,Q,R)-chains. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (13) DependencyGraphProof (EQUIVALENT) 29.89/8.62 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (14) 29.89/8.62 Obligation: 29.89/8.62 Q DP problem: 29.89/8.62 The TRS P consists of the following rules: 29.89/8.62 29.89/8.62 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) 29.89/8.62 29.89/8.62 The TRS R consists of the following rules: 29.89/8.62 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 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))))))) 29.89/8.62 29.89/8.62 Q is empty. 29.89/8.62 We have to consider all minimal (P,Q,R)-chains. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (15) UsableRulesProof (EQUIVALENT) 29.89/8.62 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. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (16) 29.89/8.62 Obligation: 29.89/8.62 Q DP problem: 29.89/8.62 The TRS P consists of the following rules: 29.89/8.62 29.89/8.62 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) 29.89/8.62 29.89/8.62 R is empty. 29.89/8.62 Q is empty. 29.89/8.62 We have to consider all minimal (P,Q,R)-chains. 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (17) QDPSizeChangeProof (EQUIVALENT) 29.89/8.62 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. 29.89/8.62 29.89/8.62 From the DPs we obtained the following set of size-change graphs: 29.89/8.62 *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) 29.89/8.62 The graph contains the following edges 1 > 1 29.89/8.62 29.89/8.62 29.89/8.62 ---------------------------------------- 29.89/8.62 29.89/8.62 (18) 29.89/8.62 YES 30.31/8.68 EOF