42.28/11.73 YES 42.42/11.78 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 42.42/11.78 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 42.42/11.78 42.42/11.78 42.42/11.78 Termination w.r.t. Q of the given QTRS could be proven: 42.42/11.78 42.42/11.78 (0) QTRS 42.42/11.78 (1) FlatCCProof [EQUIVALENT, 0 ms] 42.42/11.78 (2) QTRS 42.42/11.78 (3) RootLabelingProof [EQUIVALENT, 1 ms] 42.42/11.78 (4) QTRS 42.42/11.78 (5) QTRSRRRProof [EQUIVALENT, 84 ms] 42.42/11.78 (6) QTRS 42.42/11.78 (7) DependencyPairsProof [EQUIVALENT, 395 ms] 42.42/11.78 (8) QDP 42.42/11.78 (9) DependencyGraphProof [EQUIVALENT, 11 ms] 42.42/11.78 (10) QDP 42.42/11.78 (11) QDPOrderProof [EQUIVALENT, 342 ms] 42.42/11.78 (12) QDP 42.42/11.78 (13) DependencyGraphProof [EQUIVALENT, 4 ms] 42.42/11.78 (14) AND 42.42/11.78 (15) QDP 42.42/11.78 (16) QDPOrderProof [EQUIVALENT, 0 ms] 42.42/11.78 (17) QDP 42.42/11.78 (18) DependencyGraphProof [EQUIVALENT, 0 ms] 42.42/11.78 (19) TRUE 42.42/11.78 (20) QDP 42.42/11.78 (21) QDPOrderProof [EQUIVALENT, 90 ms] 42.42/11.78 (22) QDP 42.42/11.78 (23) PisEmptyProof [EQUIVALENT, 0 ms] 42.42/11.78 (24) YES 42.42/11.78 42.42/11.78 42.42/11.78 ---------------------------------------- 42.42/11.78 42.42/11.78 (0) 42.42/11.78 Obligation: 42.42/11.78 Q restricted rewrite system: 42.42/11.78 The TRS R consists of the following rules: 42.42/11.78 42.42/11.78 a(a(a(a(b(b(x1)))))) -> b(b(a(a(b(b(x1)))))) 42.42/11.78 b(b(a(a(x1)))) -> a(a(b(b(b(b(x1)))))) 42.42/11.78 b(b(c(c(a(a(x1)))))) -> c(c(c(c(a(a(a(a(b(b(x1)))))))))) 42.42/11.78 42.42/11.78 Q is empty. 42.42/11.78 42.42/11.78 ---------------------------------------- 42.42/11.78 42.42/11.78 (1) FlatCCProof (EQUIVALENT) 42.42/11.78 We used flat context closure [ROOTLAB] 42.42/11.78 As Q is empty the flat context closure was sound AND complete. 42.42/11.78 42.42/11.78 ---------------------------------------- 42.42/11.78 42.42/11.78 (2) 42.42/11.78 Obligation: 42.42/11.78 Q restricted rewrite system: 42.42/11.78 The TRS R consists of the following rules: 42.42/11.78 42.42/11.78 a(a(a(a(a(b(b(x1))))))) -> a(b(b(a(a(b(b(x1))))))) 42.42/11.78 b(a(a(a(a(b(b(x1))))))) -> b(b(b(a(a(b(b(x1))))))) 42.42/11.78 c(a(a(a(a(b(b(x1))))))) -> c(b(b(a(a(b(b(x1))))))) 42.42/11.78 a(b(b(a(a(x1))))) -> a(a(a(b(b(b(b(x1))))))) 42.42/11.78 b(b(b(a(a(x1))))) -> b(a(a(b(b(b(b(x1))))))) 42.42/11.78 c(b(b(a(a(x1))))) -> c(a(a(b(b(b(b(x1))))))) 42.42/11.78 a(b(b(c(c(a(a(x1))))))) -> a(c(c(c(c(a(a(a(a(b(b(x1))))))))))) 42.42/11.78 b(b(b(c(c(a(a(x1))))))) -> b(c(c(c(c(a(a(a(a(b(b(x1))))))))))) 42.42/11.78 c(b(b(c(c(a(a(x1))))))) -> c(c(c(c(c(a(a(a(a(b(b(x1))))))))))) 42.42/11.78 42.42/11.78 Q is empty. 42.42/11.78 42.42/11.78 ---------------------------------------- 42.42/11.78 42.42/11.78 (3) RootLabelingProof (EQUIVALENT) 42.42/11.78 We used plain root labeling [ROOTLAB] with the following heuristic: 42.42/11.78 LabelAll: All function symbols get labeled 42.42/11.78 42.42/11.78 As Q is empty the root labeling was sound AND complete. 42.42/11.78 42.42/11.78 ---------------------------------------- 42.42/11.78 42.42/11.78 (4) 42.42/11.78 Obligation: 42.42/11.78 Q restricted rewrite system: 42.42/11.78 The TRS R consists of the following rules: 42.42/11.78 42.42/11.78 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.78 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.78 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.78 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.78 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.78 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.78 c_{b_1}(b_{b_1}(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_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.78 c_{b_1}(b_{b_1}(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_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.78 c_{b_1}(b_{b_1}(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}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.78 42.42/11.78 Q is empty. 42.42/11.78 42.42/11.78 ---------------------------------------- 42.42/11.78 42.42/11.78 (5) QTRSRRRProof (EQUIVALENT) 42.42/11.78 Used ordering: 42.42/11.78 Polynomial interpretation [POLO]: 42.42/11.78 42.42/11.78 POL(a_{a_1}(x_1)) = x_1 42.42/11.78 POL(a_{b_1}(x_1)) = x_1 42.42/11.78 POL(a_{c_1}(x_1)) = 1 + x_1 42.42/11.78 POL(b_{a_1}(x_1)) = x_1 42.42/11.78 POL(b_{b_1}(x_1)) = x_1 42.42/11.78 POL(b_{c_1}(x_1)) = 1 + x_1 42.42/11.78 POL(c_{a_1}(x_1)) = x_1 42.42/11.78 POL(c_{b_1}(x_1)) = x_1 42.42/11.78 POL(c_{c_1}(x_1)) = x_1 42.42/11.78 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 42.42/11.78 42.42/11.78 c_{b_1}(b_{b_1}(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_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.78 c_{b_1}(b_{b_1}(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_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.78 c_{b_1}(b_{b_1}(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}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.78 42.42/11.78 42.42/11.78 42.42/11.78 42.42/11.78 ---------------------------------------- 42.42/11.78 42.42/11.78 (6) 42.42/11.78 Obligation: 42.42/11.78 Q restricted rewrite system: 42.42/11.78 The TRS R consists of the following rules: 42.42/11.78 42.42/11.78 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.78 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.78 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.78 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.78 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.78 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.78 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.78 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.78 42.42/11.78 Q is empty. 42.42/11.78 42.42/11.78 ---------------------------------------- 42.42/11.78 42.42/11.78 (7) DependencyPairsProof (EQUIVALENT) 42.42/11.78 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 42.42/11.78 ---------------------------------------- 42.42/11.78 42.42/11.78 (8) 42.42/11.78 Obligation: 42.42/11.78 Q DP problem: 42.42/11.78 The TRS P consists of the following rules: 42.42/11.78 42.42/11.78 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.78 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.79 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.79 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.79 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.79 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.79 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.79 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.79 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.79 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.79 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.79 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.79 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.79 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.79 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.79 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.79 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.79 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.79 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.79 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.79 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.79 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.79 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.79 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.79 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.79 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.79 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.79 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{c_1}(x1)) 42.42/11.81 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 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}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.81 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{c_1}(x1)) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{c_1}(x1)) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> B_{B_1}(b_{c_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> B_{B_1}(b_{c_1}(x1)) 42.42/11.81 42.42/11.81 The TRS R consists of the following rules: 42.42/11.81 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.81 42.42/11.81 Q is empty. 42.42/11.81 We have to consider all minimal (P,Q,R)-chains. 42.42/11.81 ---------------------------------------- 42.42/11.81 42.42/11.81 (9) DependencyGraphProof (EQUIVALENT) 42.42/11.81 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. 42.42/11.81 ---------------------------------------- 42.42/11.81 42.42/11.81 (10) 42.42/11.81 Obligation: 42.42/11.81 Q DP problem: 42.42/11.81 The TRS P consists of the following rules: 42.42/11.81 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 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}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 42.42/11.81 The TRS R consists of the following rules: 42.42/11.81 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.81 42.42/11.81 Q is empty. 42.42/11.81 We have to consider all minimal (P,Q,R)-chains. 42.42/11.81 ---------------------------------------- 42.42/11.81 42.42/11.81 (11) QDPOrderProof (EQUIVALENT) 42.42/11.81 We use the reduction pair processor [LPAR04,JAR06]. 42.42/11.81 42.42/11.81 42.42/11.81 The following pairs can be oriented strictly and are deleted. 42.42/11.81 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> B_{A_1}(x1) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> B_{B_1}(x1) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 The remaining pairs can at least be oriented weakly. 42.42/11.81 Used ordering: Polynomial interpretation [POLO]: 42.42/11.81 42.42/11.81 POL(A_{A_1}(x_1)) = x_1 42.42/11.81 POL(A_{B_1}(x_1)) = x_1 42.42/11.81 POL(B_{A_1}(x_1)) = x_1 42.42/11.81 POL(B_{B_1}(x_1)) = x_1 42.42/11.81 POL(C_{A_1}(x_1)) = 1 + x_1 42.42/11.81 POL(C_{B_1}(x_1)) = 1 + x_1 42.42/11.81 POL(a_{a_1}(x_1)) = x_1 42.42/11.81 POL(a_{b_1}(x_1)) = x_1 42.42/11.81 POL(a_{c_1}(x_1)) = x_1 42.42/11.81 POL(b_{a_1}(x_1)) = x_1 42.42/11.81 POL(b_{b_1}(x_1)) = x_1 42.42/11.81 POL(b_{c_1}(x_1)) = x_1 42.42/11.81 POL(c_{a_1}(x_1)) = 1 + x_1 42.42/11.81 POL(c_{b_1}(x_1)) = 1 + x_1 42.42/11.81 POL(c_{c_1}(x_1)) = x_1 42.42/11.81 42.42/11.81 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.42/11.81 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 42.42/11.81 42.42/11.81 ---------------------------------------- 42.42/11.81 42.42/11.81 (12) 42.42/11.81 Obligation: 42.42/11.81 Q DP problem: 42.42/11.81 The TRS P consists of the following rules: 42.42/11.81 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.81 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}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.81 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 42.42/11.81 The TRS R consists of the following rules: 42.42/11.81 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.81 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.81 42.42/11.81 Q is empty. 42.42/11.81 We have to consider all minimal (P,Q,R)-chains. 42.42/11.81 ---------------------------------------- 42.42/11.81 42.42/11.81 (13) DependencyGraphProof (EQUIVALENT) 42.42/11.81 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes. 42.42/11.81 ---------------------------------------- 42.42/11.81 42.42/11.81 (14) 42.42/11.81 Complex Obligation (AND) 42.42/11.81 42.42/11.81 ---------------------------------------- 42.42/11.81 42.42/11.81 (15) 42.42/11.81 Obligation: 42.42/11.81 Q DP problem: 42.42/11.81 The TRS P consists of the following rules: 42.42/11.81 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 42.42/11.81 The TRS R consists of the following rules: 42.42/11.81 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.81 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 42.42/11.82 Q is empty. 42.42/11.82 We have to consider all minimal (P,Q,R)-chains. 42.42/11.82 ---------------------------------------- 42.42/11.82 42.42/11.82 (16) QDPOrderProof (EQUIVALENT) 42.42/11.82 We use the reduction pair processor [LPAR04,JAR06]. 42.42/11.82 42.42/11.82 42.42/11.82 The following pairs can be oriented strictly and are deleted. 42.42/11.82 42.42/11.82 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 The remaining pairs can at least be oriented weakly. 42.42/11.82 Used ordering: Polynomial interpretation [POLO]: 42.42/11.82 42.42/11.82 POL(C_{A_1}(x_1)) = x_1 42.42/11.82 POL(C_{B_1}(x_1)) = 1 + x_1 42.42/11.82 POL(a_{a_1}(x_1)) = 1 + x_1 42.42/11.82 POL(a_{b_1}(x_1)) = x_1 42.42/11.82 POL(a_{c_1}(x_1)) = 0 42.42/11.82 POL(b_{a_1}(x_1)) = 1 + x_1 42.42/11.82 POL(b_{b_1}(x_1)) = x_1 42.42/11.82 POL(b_{c_1}(x_1)) = 0 42.42/11.82 POL(c_{a_1}(x_1)) = 0 42.42/11.82 POL(c_{b_1}(x_1)) = 0 42.42/11.82 POL(c_{c_1}(x_1)) = 0 42.42/11.82 42.42/11.82 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.42/11.82 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 42.42/11.82 42.42/11.82 ---------------------------------------- 42.42/11.82 42.42/11.82 (17) 42.42/11.82 Obligation: 42.42/11.82 Q DP problem: 42.42/11.82 The TRS P consists of the following rules: 42.42/11.82 42.42/11.82 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 42.42/11.82 The TRS R consists of the following rules: 42.42/11.82 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 42.42/11.82 Q is empty. 42.42/11.82 We have to consider all minimal (P,Q,R)-chains. 42.42/11.82 ---------------------------------------- 42.42/11.82 42.42/11.82 (18) DependencyGraphProof (EQUIVALENT) 42.42/11.82 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 42.42/11.82 ---------------------------------------- 42.42/11.82 42.42/11.82 (19) 42.42/11.82 TRUE 42.42/11.82 42.42/11.82 ---------------------------------------- 42.42/11.82 42.42/11.82 (20) 42.42/11.82 Obligation: 42.42/11.82 Q DP problem: 42.42/11.82 The TRS P consists of the following rules: 42.42/11.82 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.82 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}(x1))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.82 42.42/11.82 The TRS R consists of the following rules: 42.42/11.82 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 42.42/11.82 Q is empty. 42.42/11.82 We have to consider all minimal (P,Q,R)-chains. 42.42/11.82 ---------------------------------------- 42.42/11.82 42.42/11.82 (21) QDPOrderProof (EQUIVALENT) 42.42/11.82 We use the reduction pair processor [LPAR04,JAR06]. 42.42/11.82 42.42/11.82 42.42/11.82 The following pairs can be oriented strictly and are deleted. 42.42/11.82 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 42.42/11.82 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}(x1))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 B_{B_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 42.42/11.82 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 42.42/11.82 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.82 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) 42.42/11.82 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 42.42/11.82 The remaining pairs can at least be oriented weakly. 42.42/11.82 Used ordering: Polynomial interpretation [POLO]: 42.42/11.82 42.42/11.82 POL(A_{A_1}(x_1)) = 1 + x_1 42.42/11.82 POL(A_{B_1}(x_1)) = 1 + x_1 42.42/11.82 POL(B_{A_1}(x_1)) = x_1 42.42/11.82 POL(B_{B_1}(x_1)) = x_1 42.42/11.82 POL(a_{a_1}(x_1)) = 1 + x_1 42.42/11.82 POL(a_{b_1}(x_1)) = x_1 42.42/11.82 POL(a_{c_1}(x_1)) = 0 42.42/11.82 POL(b_{a_1}(x_1)) = 1 + x_1 42.42/11.82 POL(b_{b_1}(x_1)) = x_1 42.42/11.82 POL(b_{c_1}(x_1)) = 0 42.42/11.82 POL(c_{a_1}(x_1)) = 0 42.42/11.82 POL(c_{b_1}(x_1)) = 1 42.42/11.82 POL(c_{c_1}(x_1)) = 0 42.42/11.82 42.42/11.82 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.42/11.82 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 42.42/11.82 42.42/11.82 ---------------------------------------- 42.42/11.82 42.42/11.82 (22) 42.42/11.82 Obligation: 42.42/11.82 Q DP problem: 42.42/11.82 P is empty. 42.42/11.82 The TRS R consists of the following rules: 42.42/11.82 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{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_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 42.42/11.82 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 42.42/11.82 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 a_{b_1}(b_{b_1}(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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))))) 42.42/11.82 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))))))))) 42.42/11.82 42.42/11.82 Q is empty. 42.42/11.82 We have to consider all minimal (P,Q,R)-chains. 42.42/11.82 ---------------------------------------- 42.42/11.82 42.42/11.82 (23) PisEmptyProof (EQUIVALENT) 42.42/11.82 The TRS P is empty. Hence, there is no (P,Q,R) chain. 42.42/11.82 ---------------------------------------- 42.42/11.82 42.42/11.82 (24) 42.42/11.82 YES 42.78/11.88 EOF