YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 79 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 0 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(x1)) -> b(a(b(x1))) a(a(a(x1))) -> a(b(a(b(a(x1))))) a(b(a(x1))) -> b(b(a(b(b(x1))))) a(a(a(a(x1)))) -> a(a(b(a(b(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(b(a(b(a(b(x1))))))) a(b(a(a(x1)))) -> b(a(b(a(b(b(a(x1))))))) a(b(b(a(x1)))) -> b(b(b(a(b(b(b(x1))))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(b(a(a(a(x1))))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(b(a(a(b(x1))))))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(b(a(x1))))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(a(b(b(x1))))))))) a(b(a(a(a(x1))))) -> b(a(a(b(a(b(b(a(a(x1))))))))) a(b(a(b(a(x1))))) -> b(a(b(b(a(b(b(a(b(x1))))))))) a(b(b(a(a(x1))))) -> b(b(a(b(a(b(b(b(a(x1))))))))) a(b(b(b(a(x1))))) -> b(b(b(b(a(b(b(b(b(x1))))))))) Q is empty. ---------------------------------------- (1) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(a(x1))) -> a(b(a(b(a(x1))))) a(a(a(a(x1)))) -> a(a(b(a(b(a(a(x1))))))) a(a(b(a(x1)))) -> a(b(b(a(b(a(b(x1))))))) a(a(a(a(a(x1))))) -> a(a(a(b(a(b(a(a(a(x1))))))))) a(a(a(b(a(x1))))) -> a(a(b(b(a(b(a(a(b(x1))))))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(a(b(a(x1))))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(a(b(b(x1))))))))) a(a(a(x1))) -> a(b(a(b(x1)))) b(a(a(x1))) -> b(b(a(b(x1)))) a(a(b(a(x1)))) -> a(b(b(a(b(b(x1)))))) b(a(b(a(x1)))) -> b(b(b(a(b(b(x1)))))) a(a(b(a(a(x1))))) -> a(b(a(b(a(b(b(a(x1)))))))) b(a(b(a(a(x1))))) -> b(b(a(b(a(b(b(a(x1)))))))) a(a(b(b(a(x1))))) -> a(b(b(b(a(b(b(b(x1)))))))) b(a(b(b(a(x1))))) -> b(b(b(b(a(b(b(b(x1)))))))) a(a(b(a(a(a(x1)))))) -> a(b(a(a(b(a(b(b(a(a(x1)))))))))) b(a(b(a(a(a(x1)))))) -> b(b(a(a(b(a(b(b(a(a(x1)))))))))) a(a(b(a(b(a(x1)))))) -> a(b(a(b(b(a(b(b(a(b(x1)))))))))) b(a(b(a(b(a(x1)))))) -> b(b(a(b(b(a(b(b(a(b(x1)))))))))) a(a(b(b(a(a(x1)))))) -> a(b(b(a(b(a(b(b(b(a(x1)))))))))) b(a(b(b(a(a(x1)))))) -> b(b(b(a(b(a(b(b(b(a(x1)))))))))) a(a(b(b(b(a(x1)))))) -> a(b(b(b(b(a(b(b(b(b(x1)))))))))) b(a(b(b(b(a(x1)))))) -> b(b(b(b(b(a(b(b(b(b(x1)))))))))) Q is empty. ---------------------------------------- (3) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled As Q is empty the root labeling was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 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}(a_{b_1}(b_{a_1}(x1))))))) 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_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 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_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 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_{b_1}(b_{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}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))) 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_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))) 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_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) 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}(b_{a_1}(a_{b_1}(x1))))))))) 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_{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_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 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_{b_1}(b_{a_1}(x1)))))) 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}(b_{b_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 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_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 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_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 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_{b_1}(b_{a_1}(x1)))))))) 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}(b_{b_1}(x1)))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 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_{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_{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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))))) 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_{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_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{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}(x1)))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))) a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))) 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}(a_{b_1}(b_{a_1}(x1))))))) 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_{a_1}(a_{b_1}(b_{b_1}(x1))))))) 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_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))))) 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_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) 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}(b_{a_1}(a_{b_1}(x1))))))))) 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_{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_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))))) a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 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_{b_1}(b_{a_1}(x1)))))) 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}(b_{b_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) 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_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 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_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))) 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_{b_1}(b_{a_1}(x1)))))))) 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}(b_{b_1}(x1)))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) 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_{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_{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_{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_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{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}(x1)))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))))) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 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_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 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_{b_1}(b_{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_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))) Q is empty. ---------------------------------------- (7) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 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_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 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_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 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)))) 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_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 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_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 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)))) 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_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))) A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 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)))) 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))) 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}(x1)))) B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) 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))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) 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))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(x1)))))) 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))) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) The TRS R consists of the following rules: 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_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))))) 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_{b_1}(b_{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_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))))) 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_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 26 less nodes. ---------------------------------------- (10) TRUE