11.20/3.92 YES 11.79/4.06 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 11.79/4.06 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.79/4.06 11.79/4.06 11.79/4.06 Termination w.r.t. Q of the given QTRS could be proven: 11.79/4.06 11.79/4.06 (0) QTRS 11.79/4.06 (1) FlatCCProof [EQUIVALENT, 0 ms] 11.79/4.06 (2) QTRS 11.79/4.06 (3) RootLabelingProof [EQUIVALENT, 0 ms] 11.79/4.06 (4) QTRS 11.79/4.06 (5) QTRSRRRProof [EQUIVALENT, 91 ms] 11.79/4.06 (6) QTRS 11.79/4.06 (7) DependencyPairsProof [EQUIVALENT, 10 ms] 11.79/4.06 (8) QDP 11.79/4.06 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 11.79/4.06 (10) TRUE 11.79/4.06 11.79/4.06 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (0) 11.79/4.06 Obligation: 11.79/4.06 Q restricted rewrite system: 11.79/4.06 The TRS R consists of the following rules: 11.79/4.06 11.79/4.06 a(a(x1)) -> b(x1) 11.79/4.06 a(a(a(x1))) -> a(b(a(x1))) 11.79/4.06 a(b(a(x1))) -> b(b(b(x1))) 11.79/4.06 a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) 11.79/4.06 a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) 11.79/4.06 a(b(a(a(x1)))) -> b(a(b(b(a(x1))))) 11.79/4.06 a(b(b(a(x1)))) -> b(b(b(b(b(x1))))) 11.79/4.06 a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) 11.79/4.06 a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) 11.79/4.06 a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) 11.79/4.06 a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) 11.79/4.06 a(b(a(a(a(x1))))) -> b(a(a(b(b(a(a(x1))))))) 11.79/4.06 a(b(a(b(a(x1))))) -> b(a(b(b(b(a(b(x1))))))) 11.79/4.06 a(b(b(a(a(x1))))) -> b(b(a(b(b(b(a(x1))))))) 11.79/4.06 a(b(b(b(a(x1))))) -> b(b(b(b(b(b(b(x1))))))) 11.79/4.06 11.79/4.06 Q is empty. 11.79/4.06 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (1) FlatCCProof (EQUIVALENT) 11.79/4.06 We used flat context closure [ROOTLAB] 11.79/4.06 As Q is empty the flat context closure was sound AND complete. 11.79/4.06 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (2) 11.79/4.06 Obligation: 11.79/4.06 Q restricted rewrite system: 11.79/4.06 The TRS R consists of the following rules: 11.79/4.06 11.79/4.06 a(a(a(x1))) -> a(b(a(x1))) 11.79/4.06 a(a(a(a(x1)))) -> a(a(b(a(a(x1))))) 11.79/4.06 a(a(b(a(x1)))) -> a(b(b(a(b(x1))))) 11.79/4.06 a(a(a(a(a(x1))))) -> a(a(a(b(a(a(a(x1))))))) 11.79/4.06 a(a(a(b(a(x1))))) -> a(a(b(b(a(a(b(x1))))))) 11.79/4.06 a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(x1))))))) 11.79/4.06 a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(x1))))))) 11.79/4.06 a(a(a(x1))) -> a(b(x1)) 11.79/4.06 b(a(a(x1))) -> b(b(x1)) 11.79/4.06 a(a(b(a(x1)))) -> a(b(b(b(x1)))) 11.79/4.06 b(a(b(a(x1)))) -> b(b(b(b(x1)))) 11.79/4.06 a(a(b(a(a(x1))))) -> a(b(a(b(b(a(x1)))))) 11.79/4.06 b(a(b(a(a(x1))))) -> b(b(a(b(b(a(x1)))))) 11.79/4.06 a(a(b(b(a(x1))))) -> a(b(b(b(b(b(x1)))))) 11.79/4.06 b(a(b(b(a(x1))))) -> b(b(b(b(b(b(x1)))))) 11.79/4.06 a(a(b(a(a(a(x1)))))) -> a(b(a(a(b(b(a(a(x1)))))))) 11.79/4.06 b(a(b(a(a(a(x1)))))) -> b(b(a(a(b(b(a(a(x1)))))))) 11.79/4.06 a(a(b(a(b(a(x1)))))) -> a(b(a(b(b(b(a(b(x1)))))))) 11.79/4.06 b(a(b(a(b(a(x1)))))) -> b(b(a(b(b(b(a(b(x1)))))))) 11.79/4.06 a(a(b(b(a(a(x1)))))) -> a(b(b(a(b(b(b(a(x1)))))))) 11.79/4.06 b(a(b(b(a(a(x1)))))) -> b(b(b(a(b(b(b(a(x1)))))))) 11.79/4.06 a(a(b(b(b(a(x1)))))) -> a(b(b(b(b(b(b(b(x1)))))))) 11.79/4.06 b(a(b(b(b(a(x1)))))) -> b(b(b(b(b(b(b(b(x1)))))))) 11.79/4.06 11.79/4.06 Q is empty. 11.79/4.06 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (3) RootLabelingProof (EQUIVALENT) 11.79/4.06 We used plain root labeling [ROOTLAB] with the following heuristic: 11.79/4.06 LabelAll: All function symbols get labeled 11.79/4.06 11.79/4.06 As Q is empty the root labeling was sound AND complete. 11.79/4.06 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (4) 11.79/4.06 Obligation: 11.79/4.06 Q restricted rewrite system: 11.79/4.06 The TRS R consists of the following rules: 11.79/4.06 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{b_1}(b_{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_{a_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{b_1}(b_{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}(x1))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1)) 11.79/4.06 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 11.79/4.06 b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1)) 11.79/4.06 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(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}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 11.79/4.06 Q is empty. 11.79/4.06 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (5) QTRSRRRProof (EQUIVALENT) 11.79/4.06 Used ordering: 11.79/4.06 Polynomial interpretation [POLO]: 11.79/4.06 11.79/4.06 POL(a_{a_1}(x_1)) = 1 + x_1 11.79/4.06 POL(a_{b_1}(x_1)) = x_1 11.79/4.06 POL(b_{a_1}(x_1)) = x_1 11.79/4.06 POL(b_{b_1}(x_1)) = x_1 11.79/4.06 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 11.79/4.06 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{b_1}(b_{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_{a_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1)) 11.79/4.06 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 11.79/4.06 b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1)) 11.79/4.06 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(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}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 11.79/4.06 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 11.79/4.06 11.79/4.06 11.79/4.06 11.79/4.06 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (6) 11.79/4.06 Obligation: 11.79/4.06 Q restricted rewrite system: 11.79/4.06 The TRS R consists of the following rules: 11.79/4.06 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{b_1}(b_{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}(x1))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 11.79/4.06 Q is empty. 11.79/4.06 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (7) DependencyPairsProof (EQUIVALENT) 11.79/4.06 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (8) 11.79/4.06 Obligation: 11.79/4.06 Q DP problem: 11.79/4.06 The TRS P consists of the following rules: 11.79/4.06 11.79/4.06 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 11.79/4.06 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 11.79/4.06 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 11.79/4.06 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 11.79/4.06 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 11.79/4.06 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 11.79/4.06 A_{A_1}(a_{a_1}(a_{b_1}(b_{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}(x1))))))) 11.79/4.06 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 11.79/4.06 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 11.79/4.06 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 11.79/4.06 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 11.79/4.06 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1))) 11.79/4.06 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 11.79/4.06 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 11.79/4.06 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1))) 11.79/4.06 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 11.79/4.06 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1))) 11.79/4.06 11.79/4.06 The TRS R consists of the following rules: 11.79/4.06 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 11.79/4.06 a_{a_1}(a_{a_1}(a_{b_1}(b_{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}(x1))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 11.79/4.06 11.79/4.06 Q is empty. 11.79/4.06 We have to consider all minimal (P,Q,R)-chains. 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (9) DependencyGraphProof (EQUIVALENT) 11.79/4.06 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 17 less nodes. 11.79/4.06 ---------------------------------------- 11.79/4.06 11.79/4.06 (10) 11.79/4.06 TRUE 12.11/4.11 EOF