/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS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, 92 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 221 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 779 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) QDPOrderProof [EQUIVALENT, 131 ms] (15) QDP (16) DependencyGraphProof [EQUIVALENT, 0 ms] (17) TRUE (18) QDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) QDP (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] (22) YES (23) QDP (24) QDPOrderProof [EQUIVALENT, 106 ms] (25) QDP (26) PisEmptyProof [EQUIVALENT, 0 ms] (27) YES (28) QDP (29) QDPOrderProof [EQUIVALENT, 105 ms] (30) QDP (31) PisEmptyProof [EQUIVALENT, 0 ms] (32) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: s(b(x1)) -> b(s(s(s(x1)))) s(b(s(x1))) -> b(t(x1)) t(b(x1)) -> b(s(x1)) t(b(s(x1))) -> u(t(b(x1))) b(u(x1)) -> b(s(x1)) t(s(x1)) -> t(t(x1)) t(u(x1)) -> u(t(x1)) s(u(x1)) -> s(s(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: b(u(x1)) -> b(s(x1)) t(s(x1)) -> t(t(x1)) s(u(x1)) -> s(s(x1)) s(s(b(x1))) -> s(b(s(s(s(x1))))) b(s(b(x1))) -> b(b(s(s(s(x1))))) t(s(b(x1))) -> t(b(s(s(s(x1))))) u(s(b(x1))) -> u(b(s(s(s(x1))))) s(s(b(s(x1)))) -> s(b(t(x1))) b(s(b(s(x1)))) -> b(b(t(x1))) t(s(b(s(x1)))) -> t(b(t(x1))) u(s(b(s(x1)))) -> u(b(t(x1))) s(t(b(x1))) -> s(b(s(x1))) b(t(b(x1))) -> b(b(s(x1))) t(t(b(x1))) -> t(b(s(x1))) u(t(b(x1))) -> u(b(s(x1))) s(t(b(s(x1)))) -> s(u(t(b(x1)))) b(t(b(s(x1)))) -> b(u(t(b(x1)))) t(t(b(s(x1)))) -> t(u(t(b(x1)))) u(t(b(s(x1)))) -> u(u(t(b(x1)))) s(t(u(x1))) -> s(u(t(x1))) b(t(u(x1))) -> b(u(t(x1))) t(t(u(x1))) -> t(u(t(x1))) u(t(u(x1))) -> u(u(t(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: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{s_1}(x1)) -> b_{s_1}(s_{s_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{s_1}(x1)) -> s_{s_1}(s_{s_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) u_{s_1}(s_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) u_{s_1}(s_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) u_{s_1}(s_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) u_{s_1}(s_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) u_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{b_1}(b_{t_1}(t_{b_1}(x1))) u_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{b_1}(b_{t_1}(t_{u_1}(x1))) u_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{b_1}(b_{t_1}(t_{s_1}(x1))) u_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{s_1}(x1))) -> s_{u_1}(u_{t_1}(t_{s_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{s_1}(x1))) -> b_{u_1}(u_{t_1}(t_{s_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{s_1}(x1))) -> t_{u_1}(u_{t_1}(t_{s_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{s_1}(x1))) -> u_{u_1}(u_{t_1}(t_{s_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(b_{b_1}(x_1)) = x_1 POL(b_{s_1}(x_1)) = x_1 POL(b_{t_1}(x_1)) = x_1 POL(b_{u_1}(x_1)) = x_1 POL(s_{b_1}(x_1)) = x_1 POL(s_{s_1}(x_1)) = x_1 POL(s_{t_1}(x_1)) = x_1 POL(s_{u_1}(x_1)) = x_1 POL(t_{b_1}(x_1)) = x_1 POL(t_{s_1}(x_1)) = x_1 POL(t_{t_1}(x_1)) = x_1 POL(t_{u_1}(x_1)) = x_1 POL(u_{b_1}(x_1)) = x_1 POL(u_{s_1}(x_1)) = 1 + x_1 POL(u_{t_1}(x_1)) = x_1 POL(u_{u_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: b_{u_1}(u_{s_1}(x1)) -> b_{s_1}(s_{s_1}(x1)) s_{u_1}(u_{s_1}(x1)) -> s_{s_1}(s_{s_1}(x1)) u_{s_1}(s_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) u_{s_1}(s_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) u_{s_1}(s_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) u_{s_1}(s_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) u_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{b_1}(b_{t_1}(t_{b_1}(x1))) u_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{b_1}(b_{t_1}(t_{u_1}(x1))) u_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{b_1}(b_{t_1}(t_{s_1}(x1))) u_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{u_1}(u_{s_1}(x1))) -> s_{u_1}(u_{t_1}(t_{s_1}(x1))) b_{t_1}(t_{u_1}(u_{s_1}(x1))) -> b_{u_1}(u_{t_1}(t_{s_1}(x1))) t_{t_1}(t_{u_1}(u_{s_1}(x1))) -> t_{u_1}(u_{t_1}(t_{s_1}(x1))) u_{t_1}(t_{u_1}(u_{s_1}(x1))) -> u_{u_1}(u_{t_1}(t_{s_1}(x1))) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_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: B_{U_1}(u_{b_1}(x1)) -> B_{S_1}(s_{b_1}(x1)) B_{U_1}(u_{u_1}(x1)) -> B_{S_1}(s_{u_1}(x1)) B_{U_1}(u_{u_1}(x1)) -> S_{U_1}(x1) B_{U_1}(u_{t_1}(x1)) -> B_{S_1}(s_{t_1}(x1)) B_{U_1}(u_{t_1}(x1)) -> S_{T_1}(x1) T_{S_1}(s_{b_1}(x1)) -> T_{T_1}(t_{b_1}(x1)) T_{S_1}(s_{u_1}(x1)) -> T_{T_1}(t_{u_1}(x1)) T_{S_1}(s_{s_1}(x1)) -> T_{T_1}(t_{s_1}(x1)) T_{S_1}(s_{s_1}(x1)) -> T_{S_1}(x1) T_{S_1}(s_{t_1}(x1)) -> T_{T_1}(t_{t_1}(x1)) T_{S_1}(s_{t_1}(x1)) -> T_{T_1}(x1) S_{U_1}(u_{b_1}(x1)) -> S_{S_1}(s_{b_1}(x1)) S_{U_1}(u_{u_1}(x1)) -> S_{S_1}(s_{u_1}(x1)) S_{U_1}(u_{u_1}(x1)) -> S_{U_1}(x1) S_{U_1}(u_{t_1}(x1)) -> S_{S_1}(s_{t_1}(x1)) S_{U_1}(u_{t_1}(x1)) -> S_{T_1}(x1) S_{S_1}(s_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1)))) S_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{s_1}(s_{b_1}(x1))) S_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{b_1}(x1)) S_{S_1}(s_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1)))) S_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{s_1}(s_{u_1}(x1))) S_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{u_1}(x1)) S_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) S_{S_1}(s_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1)))) S_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(s_{s_1}(x1))) S_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) S_{S_1}(s_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1)))) S_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{s_1}(s_{t_1}(x1))) S_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{t_1}(x1)) S_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) B_{S_1}(s_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1)))) B_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{s_1}(s_{b_1}(x1))) B_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{b_1}(x1)) B_{S_1}(s_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1)))) B_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{s_1}(s_{u_1}(x1))) B_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{u_1}(x1)) B_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) B_{S_1}(s_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1)))) B_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(s_{s_1}(x1))) B_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) B_{S_1}(s_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1)))) B_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{s_1}(s_{t_1}(x1))) B_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{t_1}(x1)) B_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) T_{S_1}(s_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1)))) T_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{s_1}(s_{b_1}(x1))) T_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{b_1}(x1)) T_{S_1}(s_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1)))) T_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{s_1}(s_{u_1}(x1))) T_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{u_1}(x1)) T_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) T_{S_1}(s_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1)))) T_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(s_{s_1}(x1))) T_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) T_{S_1}(s_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1)))) T_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{s_1}(s_{t_1}(x1))) T_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{t_1}(x1)) T_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) S_{S_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> B_{T_1}(t_{b_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{T_1}(t_{u_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{T_1}(t_{s_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> T_{S_1}(x1) S_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(t_{t_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> T_{T_1}(x1) B_{S_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> B_{T_1}(t_{b_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{T_1}(t_{u_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{T_1}(t_{s_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> T_{S_1}(x1) B_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(t_{t_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> T_{T_1}(x1) T_{S_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> B_{T_1}(t_{b_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{T_1}(t_{u_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{T_1}(t_{s_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> T_{S_1}(x1) T_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(t_{t_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> T_{T_1}(x1) S_{T_1}(t_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{b_1}(x1)) S_{T_1}(t_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{u_1}(x1)) S_{T_1}(t_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(x1)) S_{T_1}(t_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) S_{T_1}(t_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{t_1}(x1)) S_{T_1}(t_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) B_{T_1}(t_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{b_1}(x1)) B_{T_1}(t_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{u_1}(x1)) B_{T_1}(t_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) B_{T_1}(t_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(x1)) B_{T_1}(t_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) B_{T_1}(t_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{t_1}(x1)) B_{T_1}(t_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) T_{T_1}(t_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{b_1}(x1)) T_{T_1}(t_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{u_1}(x1)) T_{T_1}(t_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) T_{T_1}(t_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(x1)) T_{T_1}(t_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) T_{T_1}(t_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{t_1}(x1)) T_{T_1}(t_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) U_{T_1}(t_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{b_1}(x1)) U_{T_1}(t_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{u_1}(x1)) U_{T_1}(t_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) U_{T_1}(t_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(x1)) U_{T_1}(t_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) U_{T_1}(t_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{t_1}(x1)) U_{T_1}(t_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{b_1}(x1))) S_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) S_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) S_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{S_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) S_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(x1) B_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> B_{U_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) B_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{b_1}(x1))) B_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) B_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) B_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(x1) B_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{U_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) B_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) B_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{S_1}(x1) B_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{U_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) B_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) B_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(x1) T_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{b_1}(x1))) T_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) T_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(x1) T_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) T_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{S_1}(x1) T_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) T_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(x1) U_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{b_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(x1) U_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{S_1}(x1) U_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(x1) S_{T_1}(t_{u_1}(u_{b_1}(x1))) -> S_{U_1}(u_{t_1}(t_{b_1}(x1))) S_{T_1}(t_{u_1}(u_{b_1}(x1))) -> U_{T_1}(t_{b_1}(x1)) S_{T_1}(t_{u_1}(u_{u_1}(x1))) -> S_{U_1}(u_{t_1}(t_{u_1}(x1))) S_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) S_{T_1}(t_{u_1}(u_{t_1}(x1))) -> S_{U_1}(u_{t_1}(t_{t_1}(x1))) S_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) S_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) B_{T_1}(t_{u_1}(u_{b_1}(x1))) -> B_{U_1}(u_{t_1}(t_{b_1}(x1))) B_{T_1}(t_{u_1}(u_{b_1}(x1))) -> U_{T_1}(t_{b_1}(x1)) B_{T_1}(t_{u_1}(u_{u_1}(x1))) -> B_{U_1}(u_{t_1}(t_{u_1}(x1))) B_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) B_{T_1}(t_{u_1}(u_{t_1}(x1))) -> B_{U_1}(u_{t_1}(t_{t_1}(x1))) B_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) B_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) T_{T_1}(t_{u_1}(u_{b_1}(x1))) -> U_{T_1}(t_{b_1}(x1)) T_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) T_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) T_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) U_{T_1}(t_{u_1}(u_{b_1}(x1))) -> U_{T_1}(t_{b_1}(x1)) U_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) U_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) U_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{U_1}(u_{b_1}(x1)) -> B_{S_1}(s_{b_1}(x1)) B_{U_1}(u_{u_1}(x1)) -> B_{S_1}(s_{u_1}(x1)) B_{U_1}(u_{t_1}(x1)) -> B_{S_1}(s_{t_1}(x1)) T_{S_1}(s_{b_1}(x1)) -> T_{T_1}(t_{b_1}(x1)) T_{S_1}(s_{u_1}(x1)) -> T_{T_1}(t_{u_1}(x1)) T_{S_1}(s_{s_1}(x1)) -> T_{T_1}(t_{s_1}(x1)) T_{S_1}(s_{t_1}(x1)) -> T_{T_1}(t_{t_1}(x1)) T_{S_1}(s_{t_1}(x1)) -> T_{T_1}(x1) S_{U_1}(u_{b_1}(x1)) -> S_{S_1}(s_{b_1}(x1)) S_{U_1}(u_{u_1}(x1)) -> S_{S_1}(s_{u_1}(x1)) S_{U_1}(u_{t_1}(x1)) -> S_{S_1}(s_{t_1}(x1)) S_{S_1}(s_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1)))) S_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{s_1}(s_{b_1}(x1))) S_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{b_1}(x1)) S_{S_1}(s_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1)))) S_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{s_1}(s_{u_1}(x1))) S_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{u_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1)))) S_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(s_{s_1}(x1))) S_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) S_{S_1}(s_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1)))) S_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{s_1}(s_{t_1}(x1))) S_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{t_1}(x1)) B_{S_1}(s_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1)))) B_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{s_1}(s_{b_1}(x1))) B_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{b_1}(x1)) B_{S_1}(s_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1)))) B_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{s_1}(s_{u_1}(x1))) B_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{u_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1)))) B_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(s_{s_1}(x1))) B_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) B_{S_1}(s_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1)))) B_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{s_1}(s_{t_1}(x1))) B_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{t_1}(x1)) T_{S_1}(s_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1)))) T_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{s_1}(s_{b_1}(x1))) T_{S_1}(s_{b_1}(b_{b_1}(x1))) -> S_{S_1}(s_{b_1}(x1)) T_{S_1}(s_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1)))) T_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{s_1}(s_{u_1}(x1))) T_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{S_1}(s_{u_1}(x1)) T_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) T_{S_1}(s_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1)))) T_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(s_{s_1}(x1))) T_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(s_{s_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) T_{S_1}(s_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1)))) T_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{s_1}(s_{t_1}(x1))) T_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{S_1}(s_{t_1}(x1)) T_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) S_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> T_{T_1}(x1) B_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> T_{T_1}(x1) T_{S_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> B_{T_1}(t_{b_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{T_1}(t_{u_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{T_1}(t_{s_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> T_{S_1}(x1) T_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(t_{t_1}(x1)) T_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> T_{T_1}(x1) S_{T_1}(t_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{b_1}(x1)) S_{T_1}(t_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{u_1}(x1)) S_{T_1}(t_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(x1)) S_{T_1}(t_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) S_{T_1}(t_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{t_1}(x1)) S_{T_1}(t_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) B_{T_1}(t_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{b_1}(x1)) B_{T_1}(t_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{u_1}(x1)) B_{T_1}(t_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) B_{T_1}(t_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(x1)) B_{T_1}(t_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) B_{T_1}(t_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{t_1}(x1)) B_{T_1}(t_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) T_{T_1}(t_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{b_1}(x1)) T_{T_1}(t_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{u_1}(x1)) T_{T_1}(t_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(x1)) T_{T_1}(t_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) T_{T_1}(t_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{t_1}(x1)) U_{T_1}(t_{b_1}(b_{b_1}(x1))) -> B_{S_1}(s_{b_1}(x1)) U_{T_1}(t_{b_1}(b_{u_1}(x1))) -> B_{S_1}(s_{u_1}(x1)) U_{T_1}(t_{b_1}(b_{s_1}(x1))) -> B_{S_1}(s_{s_1}(x1)) U_{T_1}(t_{b_1}(b_{s_1}(x1))) -> S_{S_1}(x1) U_{T_1}(t_{b_1}(b_{t_1}(x1))) -> B_{S_1}(s_{t_1}(x1)) S_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{b_1}(x1))) S_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) S_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) S_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{S_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) S_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(x1) B_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{b_1}(x1))) B_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) B_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(x1) B_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) B_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{S_1}(x1) B_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) B_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(x1) T_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{S_1}(x1) U_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{S_1}(x1) S_{T_1}(t_{u_1}(u_{b_1}(x1))) -> U_{T_1}(t_{b_1}(x1)) S_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) S_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) S_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) B_{T_1}(t_{u_1}(u_{b_1}(x1))) -> U_{T_1}(t_{b_1}(x1)) B_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) B_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) B_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(B_{S_1}(x_1)) = x_1 POL(B_{T_1}(x_1)) = 1 + x_1 POL(B_{U_1}(x_1)) = 1 + x_1 POL(S_{S_1}(x_1)) = x_1 POL(S_{T_1}(x_1)) = 1 + x_1 POL(S_{U_1}(x_1)) = 1 + x_1 POL(T_{S_1}(x_1)) = 1 + x_1 POL(T_{T_1}(x_1)) = x_1 POL(U_{T_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 1 + x_1 POL(b_{s_1}(x_1)) = x_1 POL(b_{t_1}(x_1)) = x_1 POL(b_{u_1}(x_1)) = x_1 POL(s_{b_1}(x_1)) = 1 + x_1 POL(s_{s_1}(x_1)) = x_1 POL(s_{t_1}(x_1)) = x_1 POL(s_{u_1}(x_1)) = x_1 POL(t_{b_1}(x_1)) = 1 + x_1 POL(t_{s_1}(x_1)) = x_1 POL(t_{t_1}(x_1)) = x_1 POL(t_{u_1}(x_1)) = x_1 POL(u_{b_1}(x_1)) = 1 + x_1 POL(u_{t_1}(x_1)) = x_1 POL(u_{u_1}(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: B_{U_1}(u_{u_1}(x1)) -> S_{U_1}(x1) B_{U_1}(u_{t_1}(x1)) -> S_{T_1}(x1) T_{S_1}(s_{s_1}(x1)) -> T_{S_1}(x1) S_{U_1}(u_{u_1}(x1)) -> S_{U_1}(x1) S_{U_1}(u_{t_1}(x1)) -> S_{T_1}(x1) S_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) S_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) B_{S_1}(s_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) B_{S_1}(s_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) S_{S_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> B_{T_1}(t_{b_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{T_1}(t_{u_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{T_1}(t_{s_1}(x1)) S_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> T_{S_1}(x1) S_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(t_{t_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> B_{T_1}(t_{b_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{T_1}(t_{u_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{T_1}(t_{s_1}(x1)) B_{S_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> T_{S_1}(x1) B_{S_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(t_{t_1}(x1)) T_{T_1}(t_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) T_{T_1}(t_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) U_{T_1}(t_{b_1}(b_{u_1}(x1))) -> S_{U_1}(x1) U_{T_1}(t_{b_1}(b_{t_1}(x1))) -> S_{T_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) B_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> B_{U_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) B_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) B_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> B_{U_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) B_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{U_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) T_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{b_1}(x1))) T_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) T_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(x1) T_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) T_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) T_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(x1) U_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{b_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> B_{U_1}(x1) U_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> B_{T_1}(x1) S_{T_1}(t_{u_1}(u_{b_1}(x1))) -> S_{U_1}(u_{t_1}(t_{b_1}(x1))) S_{T_1}(t_{u_1}(u_{u_1}(x1))) -> S_{U_1}(u_{t_1}(t_{u_1}(x1))) S_{T_1}(t_{u_1}(u_{t_1}(x1))) -> S_{U_1}(u_{t_1}(t_{t_1}(x1))) B_{T_1}(t_{u_1}(u_{b_1}(x1))) -> B_{U_1}(u_{t_1}(t_{b_1}(x1))) B_{T_1}(t_{u_1}(u_{u_1}(x1))) -> B_{U_1}(u_{t_1}(t_{u_1}(x1))) B_{T_1}(t_{u_1}(u_{t_1}(x1))) -> B_{U_1}(u_{t_1}(t_{t_1}(x1))) T_{T_1}(t_{u_1}(u_{b_1}(x1))) -> U_{T_1}(t_{b_1}(x1)) T_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) T_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) T_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) U_{T_1}(t_{u_1}(u_{b_1}(x1))) -> U_{T_1}(t_{b_1}(x1)) U_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) U_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) U_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 38 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: S_{U_1}(u_{t_1}(x1)) -> S_{T_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) S_{U_1}(u_{u_1}(x1)) -> S_{U_1}(x1) S_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) S_{T_1}(t_{u_1}(u_{b_1}(x1))) -> S_{U_1}(u_{t_1}(t_{b_1}(x1))) S_{T_1}(t_{u_1}(u_{u_1}(x1))) -> S_{U_1}(u_{t_1}(t_{u_1}(x1))) S_{T_1}(t_{u_1}(u_{t_1}(x1))) -> S_{U_1}(u_{t_1}(t_{t_1}(x1))) The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. S_{U_1}(u_{t_1}(x1)) -> S_{T_1}(x1) S_{U_1}(u_{u_1}(x1)) -> S_{U_1}(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(S_{T_1}(x_1)) = 1 + x_1 POL(S_{U_1}(x_1)) = 1 + x_1 POL(b_{b_1}(x_1)) = 0 POL(b_{s_1}(x_1)) = 1 + x_1 POL(b_{t_1}(x_1)) = 1 + x_1 POL(b_{u_1}(x_1)) = 1 + x_1 POL(s_{b_1}(x_1)) = 0 POL(s_{s_1}(x_1)) = 1 + x_1 POL(s_{t_1}(x_1)) = 1 + x_1 POL(s_{u_1}(x_1)) = 1 + x_1 POL(t_{b_1}(x_1)) = 1 + x_1 POL(t_{s_1}(x_1)) = 0 POL(t_{t_1}(x_1)) = 1 + x_1 POL(t_{u_1}(x_1)) = 1 + x_1 POL(u_{b_1}(x_1)) = 1 + x_1 POL(u_{t_1}(x_1)) = 1 + x_1 POL(u_{u_1}(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: S_{T_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) S_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> S_{U_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) S_{T_1}(t_{u_1}(u_{b_1}(x1))) -> S_{U_1}(u_{t_1}(t_{b_1}(x1))) S_{T_1}(t_{u_1}(u_{u_1}(x1))) -> S_{U_1}(u_{t_1}(t_{u_1}(x1))) S_{T_1}(t_{u_1}(u_{t_1}(x1))) -> S_{U_1}(u_{t_1}(t_{t_1}(x1))) The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 7 less nodes. ---------------------------------------- (17) TRUE ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: T_{S_1}(s_{s_1}(x1)) -> T_{S_1}(x1) The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: T_{S_1}(s_{s_1}(x1)) -> T_{S_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *T_{S_1}(s_{s_1}(x1)) -> T_{S_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (22) YES ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: U_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U_{T_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{s_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{u_1}(x1))) U_{T_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> U_{T_1}(t_{b_1}(b_{t_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(U_{T_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 0 POL(b_{s_1}(x_1)) = x_1 POL(b_{t_1}(x_1)) = x_1 POL(b_{u_1}(x_1)) = x_1 POL(s_{b_1}(x_1)) = 1 POL(s_{s_1}(x_1)) = 1 + x_1 POL(s_{t_1}(x_1)) = 1 + x_1 POL(s_{u_1}(x_1)) = 1 + x_1 POL(t_{b_1}(x_1)) = 1 + x_1 POL(t_{s_1}(x_1)) = 0 POL(t_{t_1}(x_1)) = 1 + x_1 POL(t_{u_1}(x_1)) = 1 + x_1 POL(u_{b_1}(x_1)) = 1 + x_1 POL(u_{t_1}(x_1)) = 1 + x_1 POL(u_{u_1}(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) ---------------------------------------- (25) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (27) YES ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: T_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) U_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) U_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) U_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) T_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) T_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. T_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) U_{T_1}(t_{u_1}(u_{u_1}(x1))) -> U_{T_1}(t_{u_1}(x1)) U_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) U_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) T_{T_1}(t_{u_1}(u_{t_1}(x1))) -> U_{T_1}(t_{t_1}(x1)) T_{T_1}(t_{u_1}(u_{t_1}(x1))) -> T_{T_1}(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(T_{T_1}(x_1)) = x_1 POL(U_{T_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 0 POL(b_{s_1}(x_1)) = 1 + x_1 POL(b_{t_1}(x_1)) = 1 + x_1 POL(b_{u_1}(x_1)) = 1 + x_1 POL(s_{b_1}(x_1)) = 0 POL(s_{s_1}(x_1)) = 1 + x_1 POL(s_{t_1}(x_1)) = 1 + x_1 POL(s_{u_1}(x_1)) = 1 + x_1 POL(t_{b_1}(x_1)) = x_1 POL(t_{s_1}(x_1)) = 0 POL(t_{t_1}(x_1)) = 1 + x_1 POL(t_{u_1}(x_1)) = 1 + x_1 POL(u_{b_1}(x_1)) = x_1 POL(u_{t_1}(x_1)) = 1 + x_1 POL(u_{u_1}(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) ---------------------------------------- (30) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b_{u_1}(u_{b_1}(x1)) -> b_{s_1}(s_{b_1}(x1)) b_{u_1}(u_{u_1}(x1)) -> b_{s_1}(s_{u_1}(x1)) b_{u_1}(u_{t_1}(x1)) -> b_{s_1}(s_{t_1}(x1)) t_{s_1}(s_{b_1}(x1)) -> t_{t_1}(t_{b_1}(x1)) t_{s_1}(s_{u_1}(x1)) -> t_{t_1}(t_{u_1}(x1)) t_{s_1}(s_{s_1}(x1)) -> t_{t_1}(t_{s_1}(x1)) t_{s_1}(s_{t_1}(x1)) -> t_{t_1}(t_{t_1}(x1)) s_{u_1}(u_{b_1}(x1)) -> s_{s_1}(s_{b_1}(x1)) s_{u_1}(u_{u_1}(x1)) -> s_{s_1}(s_{u_1}(x1)) s_{u_1}(u_{t_1}(x1)) -> s_{s_1}(s_{t_1}(x1)) s_{s_1}(s_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) s_{s_1}(s_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) s_{s_1}(s_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) b_{s_1}(s_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) b_{s_1}(s_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) b_{s_1}(s_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) b_{s_1}(s_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) t_{s_1}(s_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{b_1}(x1))))) t_{s_1}(s_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{u_1}(x1))))) t_{s_1}(s_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{s_1}(x1))))) t_{s_1}(s_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(s_{s_1}(s_{t_1}(x1))))) s_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{b_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{u_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{s_1}(x1))) s_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{b_1}(b_{t_1}(t_{t_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{b_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{u_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{s_1}(x1))) b_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{b_1}(b_{t_1}(t_{t_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{b_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{u_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{s_1}(x1))) t_{s_1}(s_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{b_1}(b_{t_1}(t_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{b_1}(x1))) -> s_{b_1}(b_{s_1}(s_{b_1}(x1))) s_{t_1}(t_{b_1}(b_{u_1}(x1))) -> s_{b_1}(b_{s_1}(s_{u_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(x1))) -> s_{b_1}(b_{s_1}(s_{s_1}(x1))) s_{t_1}(t_{b_1}(b_{t_1}(x1))) -> s_{b_1}(b_{s_1}(s_{t_1}(x1))) b_{t_1}(t_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{s_1}(s_{b_1}(x1))) b_{t_1}(t_{b_1}(b_{u_1}(x1))) -> b_{b_1}(b_{s_1}(s_{u_1}(x1))) b_{t_1}(t_{b_1}(b_{s_1}(x1))) -> b_{b_1}(b_{s_1}(s_{s_1}(x1))) b_{t_1}(t_{b_1}(b_{t_1}(x1))) -> b_{b_1}(b_{s_1}(s_{t_1}(x1))) t_{t_1}(t_{b_1}(b_{b_1}(x1))) -> t_{b_1}(b_{s_1}(s_{b_1}(x1))) t_{t_1}(t_{b_1}(b_{u_1}(x1))) -> t_{b_1}(b_{s_1}(s_{u_1}(x1))) t_{t_1}(t_{b_1}(b_{s_1}(x1))) -> t_{b_1}(b_{s_1}(s_{s_1}(x1))) t_{t_1}(t_{b_1}(b_{t_1}(x1))) -> t_{b_1}(b_{s_1}(s_{t_1}(x1))) u_{t_1}(t_{b_1}(b_{b_1}(x1))) -> u_{b_1}(b_{s_1}(s_{b_1}(x1))) u_{t_1}(t_{b_1}(b_{u_1}(x1))) -> u_{b_1}(b_{s_1}(s_{u_1}(x1))) u_{t_1}(t_{b_1}(b_{s_1}(x1))) -> u_{b_1}(b_{s_1}(s_{s_1}(x1))) u_{t_1}(t_{b_1}(b_{t_1}(x1))) -> u_{b_1}(b_{s_1}(s_{t_1}(x1))) s_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) s_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> s_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) b_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> b_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) t_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> t_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{b_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{b_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{u_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{u_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{s_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{s_1}(x1)))) u_{t_1}(t_{b_1}(b_{s_1}(s_{t_1}(x1)))) -> u_{u_1}(u_{t_1}(t_{b_1}(b_{t_1}(x1)))) s_{t_1}(t_{u_1}(u_{b_1}(x1))) -> s_{u_1}(u_{t_1}(t_{b_1}(x1))) s_{t_1}(t_{u_1}(u_{u_1}(x1))) -> s_{u_1}(u_{t_1}(t_{u_1}(x1))) s_{t_1}(t_{u_1}(u_{t_1}(x1))) -> s_{u_1}(u_{t_1}(t_{t_1}(x1))) b_{t_1}(t_{u_1}(u_{b_1}(x1))) -> b_{u_1}(u_{t_1}(t_{b_1}(x1))) b_{t_1}(t_{u_1}(u_{u_1}(x1))) -> b_{u_1}(u_{t_1}(t_{u_1}(x1))) b_{t_1}(t_{u_1}(u_{t_1}(x1))) -> b_{u_1}(u_{t_1}(t_{t_1}(x1))) t_{t_1}(t_{u_1}(u_{b_1}(x1))) -> t_{u_1}(u_{t_1}(t_{b_1}(x1))) t_{t_1}(t_{u_1}(u_{u_1}(x1))) -> t_{u_1}(u_{t_1}(t_{u_1}(x1))) t_{t_1}(t_{u_1}(u_{t_1}(x1))) -> t_{u_1}(u_{t_1}(t_{t_1}(x1))) u_{t_1}(t_{u_1}(u_{b_1}(x1))) -> u_{u_1}(u_{t_1}(t_{b_1}(x1))) u_{t_1}(t_{u_1}(u_{u_1}(x1))) -> u_{u_1}(u_{t_1}(t_{u_1}(x1))) u_{t_1}(t_{u_1}(u_{t_1}(x1))) -> u_{u_1}(u_{t_1}(t_{t_1}(x1))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (32) YES