38.81/10.95 YES 39.12/11.01 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 39.12/11.01 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 39.12/11.01 39.12/11.01 39.12/11.01 Termination w.r.t. Q of the given QTRS could be proven: 39.12/11.01 39.12/11.01 (0) QTRS 39.12/11.01 (1) QTRS Reverse [EQUIVALENT, 0 ms] 39.12/11.01 (2) QTRS 39.12/11.01 (3) FlatCCProof [EQUIVALENT, 1 ms] 39.12/11.01 (4) QTRS 39.12/11.01 (5) RootLabelingProof [EQUIVALENT, 0 ms] 39.12/11.01 (6) QTRS 39.12/11.01 (7) QTRSRRRProof [EQUIVALENT, 75 ms] 39.12/11.01 (8) QTRS 39.12/11.01 (9) DependencyPairsProof [EQUIVALENT, 16 ms] 39.12/11.01 (10) QDP 39.12/11.01 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 39.12/11.01 (12) QDP 39.12/11.01 (13) QDPOrderProof [EQUIVALENT, 324 ms] 39.12/11.01 (14) QDP 39.12/11.01 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 39.12/11.01 (16) TRUE 39.12/11.01 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (0) 39.12/11.01 Obligation: 39.12/11.01 Q restricted rewrite system: 39.12/11.01 The TRS R consists of the following rules: 39.12/11.01 39.12/11.01 a(a(s(s(x1)))) -> s(s(a(a(x1)))) 39.12/11.01 b(b(a(a(b(b(s(s(x1)))))))) -> a(a(b(b(s(s(a(a(x1)))))))) 39.12/11.01 b(b(a(a(b(b(b(b(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) 39.12/11.01 a(a(b(b(a(a(a(a(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) 39.12/11.01 39.12/11.01 Q is empty. 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (1) QTRS Reverse (EQUIVALENT) 39.12/11.01 We applied the QTRS Reverse Processor [REVERSE]. 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (2) 39.12/11.01 Obligation: 39.12/11.01 Q restricted rewrite system: 39.12/11.01 The TRS R consists of the following rules: 39.12/11.01 39.12/11.01 s(s(a(a(x1)))) -> a(a(s(s(x1)))) 39.12/11.01 s(s(b(b(a(a(b(b(x1)))))))) -> a(a(s(s(b(b(a(a(x1)))))))) 39.12/11.01 b(b(b(b(a(a(b(b(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) 39.12/11.01 a(a(a(a(b(b(a(a(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) 39.12/11.01 39.12/11.01 Q is empty. 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (3) FlatCCProof (EQUIVALENT) 39.12/11.01 We used flat context closure [ROOTLAB] 39.12/11.01 As Q is empty the flat context closure was sound AND complete. 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (4) 39.12/11.01 Obligation: 39.12/11.01 Q restricted rewrite system: 39.12/11.01 The TRS R consists of the following rules: 39.12/11.01 39.12/11.01 b(b(b(b(a(a(b(b(x1)))))))) -> b(b(a(a(b(b(a(a(x1)))))))) 39.12/11.01 a(a(a(a(b(b(a(a(x1)))))))) -> a(a(b(b(a(a(b(b(x1)))))))) 39.12/11.01 s(s(s(a(a(x1))))) -> s(a(a(s(s(x1))))) 39.12/11.01 a(s(s(a(a(x1))))) -> a(a(a(s(s(x1))))) 39.12/11.01 b(s(s(a(a(x1))))) -> b(a(a(s(s(x1))))) 39.12/11.01 s(s(s(b(b(a(a(b(b(x1))))))))) -> s(a(a(s(s(b(b(a(a(x1))))))))) 39.12/11.01 a(s(s(b(b(a(a(b(b(x1))))))))) -> a(a(a(s(s(b(b(a(a(x1))))))))) 39.12/11.01 b(s(s(b(b(a(a(b(b(x1))))))))) -> b(a(a(s(s(b(b(a(a(x1))))))))) 39.12/11.01 39.12/11.01 Q is empty. 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (5) RootLabelingProof (EQUIVALENT) 39.12/11.01 We used plain root labeling [ROOTLAB] with the following heuristic: 39.12/11.01 LabelAll: All function symbols get labeled 39.12/11.01 39.12/11.01 As Q is empty the root labeling was sound AND complete. 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (6) 39.12/11.01 Obligation: 39.12/11.01 Q restricted rewrite system: 39.12/11.01 The TRS R consists of the following rules: 39.12/11.01 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 39.12/11.01 Q is empty. 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (7) QTRSRRRProof (EQUIVALENT) 39.12/11.01 Used ordering: 39.12/11.01 Polynomial interpretation [POLO]: 39.12/11.01 39.12/11.01 POL(a_{a_1}(x_1)) = 1 + x_1 39.12/11.01 POL(a_{b_1}(x_1)) = 2 + x_1 39.12/11.01 POL(a_{s_1}(x_1)) = 1 + x_1 39.12/11.01 POL(b_{a_1}(x_1)) = x_1 39.12/11.01 POL(b_{b_1}(x_1)) = 1 + x_1 39.12/11.01 POL(b_{s_1}(x_1)) = x_1 39.12/11.01 POL(s_{a_1}(x_1)) = x_1 39.12/11.01 POL(s_{b_1}(x_1)) = x_1 39.12/11.01 POL(s_{s_1}(x_1)) = x_1 39.12/11.01 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 39.12/11.01 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(x1))))) 39.12/11.01 39.12/11.01 39.12/11.01 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (8) 39.12/11.01 Obligation: 39.12/11.01 Q restricted rewrite system: 39.12/11.01 The TRS R consists of the following rules: 39.12/11.01 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 39.12/11.01 Q is empty. 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (9) DependencyPairsProof (EQUIVALENT) 39.12/11.01 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (10) 39.12/11.01 Obligation: 39.12/11.01 Q DP problem: 39.12/11.01 The TRS P consists of the following rules: 39.12/11.01 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{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}(x1)))) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) -> A_{A_1}(a_{b_1}(x1)) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{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}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{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_{a_1}(x1)))) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) -> A_{A_1}(x1) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> A_{S_1}(x1) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(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}(x1)))) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) -> B_{B_1}(b_{b_1}(x1)) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) -> B_{B_1}(x1) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(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_{a_1}(x1)))) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) -> B_{B_1}(b_{a_1}(x1)) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> B_{B_1}(b_{s_1}(x1)) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> B_{S_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1)))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{a_1}(x1))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> S_{S_1}(s_{a_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1)))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{s_1}(x1))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(s_{s_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(x1) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1)))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{a_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> S_{S_1}(s_{a_1}(x1)) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1)))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{s_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(s_{s_1}(x1)) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1)))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{a_1}(x1))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> S_{S_1}(s_{a_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1)))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{s_1}(x1))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(s_{s_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> S_{S_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{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}(x1)))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{b_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> S_{S_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{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_{a_1}(x1)))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> S_{S_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(x1) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> S_{S_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{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}(x1)))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{b_1}(x1)) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> S_{S_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{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_{a_1}(x1)))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(x1) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> S_{S_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> S_{S_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{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}(x1)))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{b_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> S_{S_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{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_{a_1}(x1)))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> S_{S_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(x1) 39.12/11.01 39.12/11.01 The TRS R consists of the following rules: 39.12/11.01 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 39.12/11.01 Q is empty. 39.12/11.01 We have to consider all minimal (P,Q,R)-chains. 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (11) DependencyGraphProof (EQUIVALENT) 39.12/11.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 44 less nodes. 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (12) 39.12/11.01 Obligation: 39.12/11.01 Q DP problem: 39.12/11.01 The TRS P consists of the following rules: 39.12/11.01 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) -> B_{B_1}(b_{b_1}(x1)) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) -> A_{A_1}(x1) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) -> B_{B_1}(x1) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> B_{B_1}(b_{s_1}(x1)) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> B_{S_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1)))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{a_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1)))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{a_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1)))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{s_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(s_{s_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1)))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{a_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1)))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{s_1}(x1))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(s_{s_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(x1) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1)))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{s_1}(x1))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(s_{s_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(x1) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> A_{S_1}(x1) 39.12/11.01 39.12/11.01 The TRS R consists of the following rules: 39.12/11.01 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 39.12/11.01 Q is empty. 39.12/11.01 We have to consider all minimal (P,Q,R)-chains. 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (13) QDPOrderProof (EQUIVALENT) 39.12/11.01 We use the reduction pair processor [LPAR04,JAR06]. 39.12/11.01 39.12/11.01 39.12/11.01 The following pairs can be oriented strictly and are deleted. 39.12/11.01 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) -> B_{B_1}(b_{b_1}(x1)) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) -> A_{A_1}(x1) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) -> B_{B_1}(x1) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> B_{B_1}(b_{s_1}(x1)) 39.12/11.01 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> B_{S_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1)))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{a_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1)))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{a_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1)))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{s_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(s_{s_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1)))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{a_1}(x1))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1)))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{s_1}(x1))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(s_{s_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(x1) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(x1) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 S_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1)))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{S_1}(s_{s_1}(s_{s_1}(x1))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(s_{s_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> S_{S_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(x1) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{s_1}(x1)) 39.12/11.01 B_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{S_1}(x1) 39.12/11.01 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> A_{S_1}(x1) 39.12/11.01 The remaining pairs can at least be oriented weakly. 39.12/11.01 Used ordering: Polynomial interpretation [POLO]: 39.12/11.01 39.12/11.01 POL(A_{A_1}(x_1)) = x_1 39.12/11.01 POL(A_{S_1}(x_1)) = x_1 39.12/11.01 POL(B_{B_1}(x_1)) = x_1 39.12/11.01 POL(B_{S_1}(x_1)) = 1 + x_1 39.12/11.01 POL(S_{S_1}(x_1)) = x_1 39.12/11.01 POL(a_{a_1}(x_1)) = 1 + x_1 39.12/11.01 POL(a_{b_1}(x_1)) = 1 + x_1 39.12/11.01 POL(a_{s_1}(x_1)) = 1 + x_1 39.12/11.01 POL(b_{a_1}(x_1)) = 1 + x_1 39.12/11.01 POL(b_{b_1}(x_1)) = 1 + x_1 39.12/11.01 POL(b_{s_1}(x_1)) = 1 + x_1 39.12/11.01 POL(s_{a_1}(x_1)) = x_1 39.12/11.01 POL(s_{b_1}(x_1)) = x_1 39.12/11.01 POL(s_{s_1}(x_1)) = x_1 39.12/11.01 39.12/11.01 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 39.12/11.01 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 39.12/11.01 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (14) 39.12/11.01 Obligation: 39.12/11.01 Q DP problem: 39.12/11.01 The TRS P consists of the following rules: 39.12/11.01 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 A_{S_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> A_{A_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 39.12/11.01 The TRS R consists of the following rules: 39.12/11.01 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{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}(a_{a_1}(a_{a_1}(x1)))))))) 39.12/11.01 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 39.12/11.01 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1)))))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1)))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{a_1}(x1))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{a_1}(a_{a_1}(a_{s_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{s_1}(x1))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 s_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> s_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> a_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 39.12/11.01 b_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{s_1}(x1))))))))) -> b_{a_1}(a_{a_1}(a_{s_1}(s_{s_1}(s_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{s_1}(x1))))))))) 39.12/11.01 39.12/11.01 Q is empty. 39.12/11.01 We have to consider all minimal (P,Q,R)-chains. 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (15) DependencyGraphProof (EQUIVALENT) 39.12/11.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes. 39.12/11.01 ---------------------------------------- 39.12/11.01 39.12/11.01 (16) 39.12/11.01 TRUE 39.23/11.06 EOF