YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 88 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 107 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) QDP (11) UsableRulesProof [EQUIVALENT, 45 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 2752 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) AND (17) QDP (18) QDPOrderProof [EQUIVALENT, 2146 ms] (19) QDP (20) QDPOrderProof [EQUIVALENT, 1765 ms] (21) QDP (22) QDPOrderProof [EQUIVALENT, 22 ms] (23) QDP (24) QDPOrderProof [EQUIVALENT, 2373 ms] (25) QDP (26) QDPOrderProof [EQUIVALENT, 1489 ms] (27) QDP (28) QDPOrderProof [EQUIVALENT, 1863 ms] (29) QDP (30) PisEmptyProof [EQUIVALENT, 0 ms] (31) YES (32) QDP (33) QDPOrderProof [EQUIVALENT, 2168 ms] (34) QDP (35) QDPOrderProof [EQUIVALENT, 1775 ms] (36) QDP (37) QDPOrderProof [EQUIVALENT, 690 ms] (38) QDP (39) QDPOrderProof [EQUIVALENT, 1926 ms] (40) QDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) QDP (43) QDPOrderProof [EQUIVALENT, 2764 ms] (44) QDP (45) DependencyGraphProof [EQUIVALENT, 0 ms] (46) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c(c(x1)) -> a(a(a(b(b(b(x1)))))) b(b(b(a(x1)))) -> b(b(b(b(b(b(b(b(x1)))))))) b(b(c(c(x1)))) -> c(c(c(a(a(a(a(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(b(b(a(x1)))) -> b(b(b(b(b(b(b(b(x1)))))))) c(c(c(x1))) -> c(a(a(a(b(b(b(x1))))))) a(c(c(x1))) -> a(a(a(a(b(b(b(x1))))))) b(c(c(x1))) -> b(a(a(a(b(b(b(x1))))))) c(b(b(c(c(x1))))) -> c(c(c(c(a(a(a(a(x1)))))))) a(b(b(c(c(x1))))) -> a(c(c(c(a(a(a(a(x1)))))))) b(b(b(c(c(x1))))) -> b(c(c(c(a(a(a(a(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_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = x_1 POL(a_{c_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = x_1 POL(b_{c_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = 1 + x_1 POL(c_{c_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: c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_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_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(x1) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) C_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) C_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) C_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) C_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) 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. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(B_{B_1}(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[-I], [0A], [-I]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, -I], [1A, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(B_{C_1}(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 1A, 0A]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(A_{B_1}(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 >>> <<< POL(A_{C_1}(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 >>> <<< POL(C_{C_1}(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(x1) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> B_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes. ---------------------------------------- (16) Complex Obligation (AND) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(B_{B_1}(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[1A], [0A], [-I]] + [[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[1A, 0A, 1A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[-I], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[-I], [1A], [0A]] + [[0A, 1A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[0A], [1A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(B_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [0A, -I, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [0A, -I, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[-I], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 1A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[-I], [1A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 0A]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [1A, 0A, 1A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(B_{B_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = 0 POL(a_{b_1}(x_1)) = 0 POL(a_{c_1}(x_1)) = 0 POL(b_{a_1}(x_1)) = 0 POL(b_{b_1}(x_1)) = 1 POL(b_{c_1}(x_1)) = 0 POL(c_{a_1}(x_1)) = 0 POL(c_{c_1}(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_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. B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(B_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[-I], [-I], [1A]] + [[0A, -I, 0A], [0A, -I, 0A], [1A, -I, 1A]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[0A], [1A], [0A]] + [[-I, 0A, 0A], [1A, -I, 1A], [-I, -I, 0A]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, -I], [1A, 1A, 1A], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(B_{B_1}(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [0A, -I, -I], [-I, 0A, 0A]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0A], [1A], [-I]] + [[0A, 0A, 0A], [0A, 1A, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [1A, -I, -I], [0A, -I, -I]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[1A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[0A], [1A], [0A]] + [[-I, 1A, 0A], [1A, -I, -I], [0A, -I, 0A]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, -I], [1A, 1A, 1A], [-I, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(B_{B_1}(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[-I], [0A], [-I]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 1A], [-I, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[0A], [-I], [1A]] + [[0A, -I, 0A], [0A, 0A, 1A], [0A, 1A, 0A]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[0A], [-I], [-I]] + [[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) ---------------------------------------- (29) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (31) YES ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A_{C_1}(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, -I], [-I, -I, 0A], [0A, 1A, -I]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [-I, -I, -I], [0A, -I, -I]] * x_1 >>> <<< POL(A_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(C_{C_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, 1A, 0A], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A_{C_1}(x_1)) = [[0A]] + [[-I, -I, -I]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [-I, -I, 0A], [-I, 0A, -I]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(A_{B_1}(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(C_{C_1}(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 1A], [-I, 0A, -I]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A_{C_1}(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[1A], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(A_{B_1}(x_1)) = [[0A]] + [[-I, -I, -I]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, -I], [-I, 0A, 0A], [-I, 0A, -I]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, 0A], [-I, -I, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(C_{C_1}(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A_{C_1}(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(A_{B_1}(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 >>> <<< POL(C_{C_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [1A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> C_{C_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A_{B_1}(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, -I], [0A, -I, 0A]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [-I, 1A, -I]] * x_1 >>> <<< POL(C_{C_1}(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[-I], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) The TRS R consists of the following rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (46) TRUE