29.56/8.47 YES 29.73/8.48 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 29.73/8.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.73/8.48 29.73/8.48 29.73/8.48 Termination w.r.t. Q of the given QTRS could be proven: 29.73/8.48 29.73/8.48 (0) QTRS 29.73/8.48 (1) FlatCCProof [EQUIVALENT, 0 ms] 29.73/8.48 (2) QTRS 29.73/8.48 (3) RootLabelingProof [EQUIVALENT, 0 ms] 29.73/8.48 (4) QTRS 29.73/8.48 (5) DependencyPairsProof [EQUIVALENT, 233 ms] 29.73/8.48 (6) QDP 29.73/8.48 (7) DependencyGraphProof [EQUIVALENT, 3 ms] 29.73/8.48 (8) AND 29.73/8.48 (9) QDP 29.73/8.48 (10) QDPOrderProof [EQUIVALENT, 162 ms] 29.73/8.48 (11) QDP 29.73/8.48 (12) DependencyGraphProof [EQUIVALENT, 0 ms] 29.73/8.48 (13) QDP 29.73/8.48 (14) QDPOrderProof [EQUIVALENT, 83 ms] 29.73/8.48 (15) QDP 29.73/8.48 (16) PisEmptyProof [EQUIVALENT, 0 ms] 29.73/8.48 (17) YES 29.73/8.48 (18) QDP 29.73/8.48 (19) QDPOrderProof [EQUIVALENT, 157 ms] 29.73/8.48 (20) QDP 29.73/8.48 (21) PisEmptyProof [EQUIVALENT, 0 ms] 29.73/8.48 (22) YES 29.73/8.48 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (0) 29.73/8.48 Obligation: 29.73/8.48 Q restricted rewrite system: 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a(x1) -> b(x1) 29.73/8.48 a(a(x1)) -> a(b(a(x1))) 29.73/8.48 a(b(x1)) -> b(b(b(x1))) 29.73/8.48 a(a(a(x1))) -> a(a(b(a(a(x1))))) 29.73/8.48 a(a(b(x1))) -> a(b(b(a(b(x1))))) 29.73/8.48 a(b(a(x1))) -> b(a(b(b(a(x1))))) 29.73/8.48 a(b(b(x1))) -> b(b(b(b(b(x1))))) 29.73/8.48 a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) 29.73/8.48 a(a(a(b(x1)))) -> a(a(b(b(a(a(b(x1))))))) 29.73/8.48 a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) 29.73/8.48 a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) 29.73/8.48 a(b(a(a(x1)))) -> b(a(a(b(b(a(a(x1))))))) 29.73/8.48 a(b(a(b(x1)))) -> b(a(b(b(b(a(b(x1))))))) 29.73/8.48 a(b(b(a(x1)))) -> b(b(a(b(b(b(a(x1))))))) 29.73/8.48 a(b(b(b(x1)))) -> b(b(b(b(b(b(b(x1))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (1) FlatCCProof (EQUIVALENT) 29.73/8.48 We used flat context closure [ROOTLAB] 29.73/8.48 As Q is empty the flat context closure was sound AND complete. 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (2) 29.73/8.48 Obligation: 29.73/8.48 Q restricted rewrite system: 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a(a(x1)) -> a(b(a(x1))) 29.73/8.48 a(a(a(x1))) -> a(a(b(a(a(x1))))) 29.73/8.48 a(a(b(x1))) -> a(b(b(a(b(x1))))) 29.73/8.48 a(a(a(a(x1)))) -> a(a(a(b(a(a(a(x1))))))) 29.73/8.48 a(a(a(b(x1)))) -> a(a(b(b(a(a(b(x1))))))) 29.73/8.48 a(a(b(a(x1)))) -> a(b(a(b(a(b(a(x1))))))) 29.73/8.48 a(a(b(b(x1)))) -> a(b(b(b(a(b(b(x1))))))) 29.73/8.48 a(a(x1)) -> a(b(x1)) 29.73/8.48 b(a(x1)) -> b(b(x1)) 29.73/8.48 a(a(b(x1))) -> a(b(b(b(x1)))) 29.73/8.48 b(a(b(x1))) -> b(b(b(b(x1)))) 29.73/8.48 a(a(b(a(x1)))) -> a(b(a(b(b(a(x1)))))) 29.73/8.48 b(a(b(a(x1)))) -> b(b(a(b(b(a(x1)))))) 29.73/8.48 a(a(b(b(x1)))) -> a(b(b(b(b(b(x1)))))) 29.73/8.48 b(a(b(b(x1)))) -> b(b(b(b(b(b(x1)))))) 29.73/8.48 a(a(b(a(a(x1))))) -> a(b(a(a(b(b(a(a(x1)))))))) 29.73/8.48 b(a(b(a(a(x1))))) -> b(b(a(a(b(b(a(a(x1)))))))) 29.73/8.48 a(a(b(a(b(x1))))) -> a(b(a(b(b(b(a(b(x1)))))))) 29.73/8.48 b(a(b(a(b(x1))))) -> b(b(a(b(b(b(a(b(x1)))))))) 29.73/8.48 a(a(b(b(a(x1))))) -> a(b(b(a(b(b(b(a(x1)))))))) 29.73/8.48 b(a(b(b(a(x1))))) -> b(b(b(a(b(b(b(a(x1)))))))) 29.73/8.48 a(a(b(b(b(x1))))) -> a(b(b(b(b(b(b(b(x1)))))))) 29.73/8.48 b(a(b(b(b(x1))))) -> b(b(b(b(b(b(b(b(x1)))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (3) RootLabelingProof (EQUIVALENT) 29.73/8.48 We used plain root labeling [ROOTLAB] with the following heuristic: 29.73/8.48 LabelAll: All function symbols get labeled 29.73/8.48 29.73/8.48 As Q is empty the root labeling was sound AND complete. 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (4) 29.73/8.48 Obligation: 29.73/8.48 Q restricted rewrite system: 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (5) DependencyPairsProof (EQUIVALENT) 29.73/8.48 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (6) 29.73/8.48 Obligation: 29.73/8.48 Q DP problem: 29.73/8.48 The TRS P consists of the following rules: 29.73/8.48 29.73/8.48 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{a_1}(x1)) 29.73/8.48 A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{b_1}(x1)) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 29.73/8.48 B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 We have to consider all minimal (P,Q,R)-chains. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (7) DependencyGraphProof (EQUIVALENT) 29.73/8.48 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 38 less nodes. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (8) 29.73/8.48 Complex Obligation (AND) 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (9) 29.73/8.48 Obligation: 29.73/8.48 Q DP problem: 29.73/8.48 The TRS P consists of the following rules: 29.73/8.48 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{b_1}(x1)) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 We have to consider all minimal (P,Q,R)-chains. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (10) QDPOrderProof (EQUIVALENT) 29.73/8.48 We use the reduction pair processor [LPAR04,JAR06]. 29.73/8.48 29.73/8.48 29.73/8.48 The following pairs can be oriented strictly and are deleted. 29.73/8.48 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 The remaining pairs can at least be oriented weakly. 29.73/8.48 Used ordering: Polynomial interpretation [POLO]: 29.73/8.48 29.73/8.48 POL(A_{A_1}(x_1)) = 1 + x_1 29.73/8.48 POL(B_{A_1}(x_1)) = 1 + x_1 29.73/8.48 POL(a_{a_1}(x_1)) = 1 + x_1 29.73/8.48 POL(a_{b_1}(x_1)) = x_1 29.73/8.48 POL(b_{a_1}(x_1)) = 1 + x_1 29.73/8.48 POL(b_{b_1}(x_1)) = 1 29.73/8.48 29.73/8.48 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.73/8.48 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (11) 29.73/8.48 Obligation: 29.73/8.48 Q DP problem: 29.73/8.48 The TRS P consists of the following rules: 29.73/8.48 29.73/8.48 A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{b_1}(x1)) 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 We have to consider all minimal (P,Q,R)-chains. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (12) DependencyGraphProof (EQUIVALENT) 29.73/8.48 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (13) 29.73/8.48 Obligation: 29.73/8.48 Q DP problem: 29.73/8.48 The TRS P consists of the following rules: 29.73/8.48 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 We have to consider all minimal (P,Q,R)-chains. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (14) QDPOrderProof (EQUIVALENT) 29.73/8.48 We use the reduction pair processor [LPAR04,JAR06]. 29.73/8.48 29.73/8.48 29.73/8.48 The following pairs can be oriented strictly and are deleted. 29.73/8.48 29.73/8.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 The remaining pairs can at least be oriented weakly. 29.73/8.48 Used ordering: Polynomial interpretation [POLO]: 29.73/8.48 29.73/8.48 POL(B_{A_1}(x_1)) = x_1 29.73/8.48 POL(a_{a_1}(x_1)) = 1 + x_1 29.73/8.48 POL(a_{b_1}(x_1)) = x_1 29.73/8.48 POL(b_{a_1}(x_1)) = 1 + x_1 29.73/8.48 POL(b_{b_1}(x_1)) = 0 29.73/8.48 29.73/8.48 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.73/8.48 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (15) 29.73/8.48 Obligation: 29.73/8.48 Q DP problem: 29.73/8.48 P is empty. 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 We have to consider all minimal (P,Q,R)-chains. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (16) PisEmptyProof (EQUIVALENT) 29.73/8.48 The TRS P is empty. Hence, there is no (P,Q,R) chain. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (17) 29.73/8.48 YES 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (18) 29.73/8.48 Obligation: 29.73/8.48 Q DP problem: 29.73/8.48 The TRS P consists of the following rules: 29.73/8.48 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 We have to consider all minimal (P,Q,R)-chains. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (19) QDPOrderProof (EQUIVALENT) 29.73/8.48 We use the reduction pair processor [LPAR04,JAR06]. 29.73/8.48 29.73/8.48 29.73/8.48 The following pairs can be oriented strictly and are deleted. 29.73/8.48 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 The remaining pairs can at least be oriented weakly. 29.73/8.48 Used ordering: Polynomial interpretation [POLO]: 29.73/8.48 29.73/8.48 POL(A_{A_1}(x_1)) = x_1 29.73/8.48 POL(a_{a_1}(x_1)) = 1 + x_1 29.73/8.48 POL(a_{b_1}(x_1)) = 0 29.73/8.48 POL(b_{a_1}(x_1)) = 0 29.73/8.48 POL(b_{b_1}(x_1)) = 0 29.73/8.48 29.73/8.48 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.73/8.48 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (20) 29.73/8.48 Obligation: 29.73/8.48 Q DP problem: 29.73/8.48 P is empty. 29.73/8.48 The TRS R consists of the following rules: 29.73/8.48 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 29.73/8.48 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 29.73/8.48 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 29.73/8.48 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 29.73/8.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 29.73/8.48 29.73/8.48 Q is empty. 29.73/8.48 We have to consider all minimal (P,Q,R)-chains. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (21) PisEmptyProof (EQUIVALENT) 29.73/8.48 The TRS P is empty. Hence, there is no (P,Q,R) chain. 29.73/8.48 ---------------------------------------- 29.73/8.48 29.73/8.48 (22) 29.73/8.48 YES 29.95/8.57 EOF