232.54/59.98 YES 232.97/60.08 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 232.97/60.08 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 232.97/60.08 232.97/60.08 232.97/60.08 Termination w.r.t. Q of the given QTRS could be proven: 232.97/60.08 232.97/60.08 (0) QTRS 232.97/60.08 (1) FlatCCProof [EQUIVALENT, 0 ms] 232.97/60.08 (2) QTRS 232.97/60.08 (3) RootLabelingProof [EQUIVALENT, 1 ms] 232.97/60.08 (4) QTRS 232.97/60.08 (5) QTRSRRRProof [EQUIVALENT, 85 ms] 232.97/60.08 (6) QTRS 232.97/60.08 (7) DependencyPairsProof [EQUIVALENT, 206 ms] 232.97/60.08 (8) QDP 232.97/60.08 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 232.97/60.08 (10) QDP 232.97/60.08 (11) UsableRulesProof [EQUIVALENT, 166 ms] 232.97/60.08 (12) QDP 232.97/60.08 (13) QDPOrderProof [EQUIVALENT, 2749 ms] 232.97/60.08 (14) QDP 232.97/60.08 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 232.97/60.08 (16) AND 232.97/60.08 (17) QDP 232.97/60.08 (18) QDPOrderProof [EQUIVALENT, 1476 ms] 232.97/60.08 (19) QDP 232.97/60.08 (20) QDPOrderProof [EQUIVALENT, 1911 ms] 232.97/60.08 (21) QDP 232.97/60.08 (22) QDPOrderProof [EQUIVALENT, 18 ms] 232.97/60.08 (23) QDP 232.97/60.08 (24) QDPOrderProof [EQUIVALENT, 2473 ms] 232.97/60.08 (25) QDP 232.97/60.08 (26) QDPOrderProof [EQUIVALENT, 787 ms] 232.97/60.08 (27) QDP 232.97/60.08 (28) QDPOrderProof [EQUIVALENT, 1824 ms] 232.97/60.08 (29) QDP 232.97/60.08 (30) PisEmptyProof [EQUIVALENT, 0 ms] 232.97/60.08 (31) YES 232.97/60.08 (32) QDP 232.97/60.08 (33) QDPOrderProof [EQUIVALENT, 1408 ms] 232.97/60.08 (34) QDP 232.97/60.08 (35) QDPOrderProof [EQUIVALENT, 1748 ms] 232.97/60.08 (36) QDP 232.97/60.08 (37) QDPOrderProof [EQUIVALENT, 1223 ms] 232.97/60.08 (38) QDP 232.97/60.08 (39) QDPOrderProof [EQUIVALENT, 1930 ms] 232.97/60.08 (40) QDP 232.97/60.08 (41) DependencyGraphProof [EQUIVALENT, 0 ms] 232.97/60.08 (42) QDP 232.97/60.08 (43) QDPOrderProof [EQUIVALENT, 2809 ms] 232.97/60.08 (44) QDP 232.97/60.08 (45) DependencyGraphProof [EQUIVALENT, 0 ms] 232.97/60.08 (46) TRUE 232.97/60.08 232.97/60.08 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (0) 232.97/60.08 Obligation: 232.97/60.08 Q restricted rewrite system: 232.97/60.08 The TRS R consists of the following rules: 232.97/60.08 232.97/60.08 c(c(x1)) -> a(a(a(b(b(b(x1)))))) 232.97/60.08 b(b(b(a(x1)))) -> b(b(b(b(b(b(b(b(x1)))))))) 232.97/60.08 b(b(c(c(x1)))) -> c(c(c(a(a(a(a(x1))))))) 232.97/60.08 232.97/60.08 Q is empty. 232.97/60.08 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (1) FlatCCProof (EQUIVALENT) 232.97/60.08 We used flat context closure [ROOTLAB] 232.97/60.08 As Q is empty the flat context closure was sound AND complete. 232.97/60.08 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (2) 232.97/60.08 Obligation: 232.97/60.08 Q restricted rewrite system: 232.97/60.08 The TRS R consists of the following rules: 232.97/60.08 232.97/60.08 b(b(b(a(x1)))) -> b(b(b(b(b(b(b(b(x1)))))))) 232.97/60.08 c(c(c(x1))) -> c(a(a(a(b(b(b(x1))))))) 232.97/60.08 a(c(c(x1))) -> a(a(a(a(b(b(b(x1))))))) 232.97/60.08 b(c(c(x1))) -> b(a(a(a(b(b(b(x1))))))) 232.97/60.08 c(b(b(c(c(x1))))) -> c(c(c(c(a(a(a(a(x1)))))))) 232.97/60.08 a(b(b(c(c(x1))))) -> a(c(c(c(a(a(a(a(x1)))))))) 232.97/60.08 b(b(b(c(c(x1))))) -> b(c(c(c(a(a(a(a(x1)))))))) 232.97/60.08 232.97/60.08 Q is empty. 232.97/60.08 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (3) RootLabelingProof (EQUIVALENT) 232.97/60.08 We used plain root labeling [ROOTLAB] with the following heuristic: 232.97/60.08 LabelAll: All function symbols get labeled 232.97/60.08 232.97/60.08 As Q is empty the root labeling was sound AND complete. 232.97/60.08 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (4) 232.97/60.08 Obligation: 232.97/60.08 Q restricted rewrite system: 232.97/60.08 The TRS R consists of the following rules: 232.97/60.08 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 232.97/60.08 Q is empty. 232.97/60.08 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (5) QTRSRRRProof (EQUIVALENT) 232.97/60.08 Used ordering: 232.97/60.08 Polynomial interpretation [POLO]: 232.97/60.08 232.97/60.08 POL(a_{a_1}(x_1)) = x_1 232.97/60.08 POL(a_{b_1}(x_1)) = x_1 232.97/60.08 POL(a_{c_1}(x_1)) = x_1 232.97/60.08 POL(b_{a_1}(x_1)) = x_1 232.97/60.08 POL(b_{b_1}(x_1)) = x_1 232.97/60.08 POL(b_{c_1}(x_1)) = x_1 232.97/60.08 POL(c_{a_1}(x_1)) = x_1 232.97/60.08 POL(c_{b_1}(x_1)) = 1 + x_1 232.97/60.08 POL(c_{c_1}(x_1)) = x_1 232.97/60.08 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 232.97/60.08 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 232.97/60.08 232.97/60.08 232.97/60.08 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (6) 232.97/60.08 Obligation: 232.97/60.08 Q restricted rewrite system: 232.97/60.08 The TRS R consists of the following rules: 232.97/60.08 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 232.97/60.08 Q is empty. 232.97/60.08 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (7) DependencyPairsProof (EQUIVALENT) 232.97/60.08 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (8) 232.97/60.08 Obligation: 232.97/60.08 Q DP problem: 232.97/60.08 The TRS P consists of the following rules: 232.97/60.08 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))) 232.97/60.08 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))))) 232.97/60.08 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)))) 232.97/60.08 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 232.97/60.08 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 232.97/60.08 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))) 232.97/60.08 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))))) 232.97/60.08 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)))) 232.97/60.08 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.08 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))) 232.97/60.08 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))))) 232.97/60.08 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)))) 232.97/60.08 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.08 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.08 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(x1) 232.97/60.08 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.08 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.08 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 232.97/60.08 C_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.08 C_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.08 C_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.08 C_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) 232.97/60.08 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.08 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.08 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 232.97/60.08 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.08 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.08 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.08 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) 232.97/60.08 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.08 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.08 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 232.97/60.08 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.08 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.08 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.08 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))) 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))) 232.97/60.08 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))) 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))) 232.97/60.08 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.08 232.97/60.08 The TRS R consists of the following rules: 232.97/60.08 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.08 232.97/60.08 Q is empty. 232.97/60.08 We have to consider all minimal (P,Q,R)-chains. 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (9) DependencyGraphProof (EQUIVALENT) 232.97/60.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes. 232.97/60.08 ---------------------------------------- 232.97/60.08 232.97/60.08 (10) 232.97/60.08 Obligation: 232.97/60.08 Q DP problem: 232.97/60.08 The TRS P consists of the following rules: 232.97/60.08 232.97/60.08 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)))))))) 232.97/60.08 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))))))) 232.97/60.08 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)))))) 232.97/60.08 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))))) 232.97/60.08 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)))) 232.97/60.08 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.08 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)))))))) 232.97/60.08 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(x1) 232.97/60.10 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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)))))))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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))))))) 232.97/60.10 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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)))))))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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)))))) 232.97/60.10 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))))))) 232.97/60.10 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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))))) 232.97/60.10 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)))))))) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.10 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))))))) 232.97/60.10 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 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))))))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (11) UsableRulesProof (EQUIVALENT) 232.97/60.10 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. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (12) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(x1) 232.97/60.10 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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)))))))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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))))))) 232.97/60.10 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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)))))))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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)))))) 232.97/60.10 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))))))) 232.97/60.10 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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))))) 232.97/60.10 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)))))))) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.10 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))))))) 232.97/60.10 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 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))))))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (13) QDPOrderProof (EQUIVALENT) 232.97/60.10 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.10 232.97/60.10 232.97/60.10 The following pairs can be oriented strictly and are deleted. 232.97/60.10 232.97/60.10 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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))))))) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.10 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))))))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) 232.97/60.10 The remaining pairs can at least be oriented weakly. 232.97/60.10 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(B_{B_1}(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{a_1}(x_1)) = [[-I], [0A], [-I]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{a_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{b_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, -I], [1A, 0A, 0A], [-I, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{c_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(B_{C_1}(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{c_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 1A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(A_{B_1}(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(A_{C_1}(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(C_{C_1}(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 232.97/60.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (14) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(x1) 232.97/60.10 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)))))))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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))))))) 232.97/60.10 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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)))))))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))) 232.97/60.10 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(x1) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (15) DependencyGraphProof (EQUIVALENT) 232.97/60.10 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (16) 232.97/60.10 Complex Obligation (AND) 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (17) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (18) QDPOrderProof (EQUIVALENT) 232.97/60.10 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.10 232.97/60.10 232.97/60.10 The following pairs can be oriented strictly and are deleted. 232.97/60.10 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 232.97/60.10 The remaining pairs can at least be oriented weakly. 232.97/60.10 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(B_{B_1}(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{a_1}(x_1)) = [[1A], [0A], [-I]] + [[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[1A, 0A, 1A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{c_1}(x_1)) = [[-I], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{c_1}(x_1)) = [[-I], [1A], [0A]] + [[0A, 1A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{a_1}(x_1)) = [[0A], [1A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 232.97/60.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (19) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (20) QDPOrderProof (EQUIVALENT) 232.97/60.10 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.10 232.97/60.10 232.97/60.10 The following pairs can be oriented strictly and are deleted. 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 232.97/60.10 The remaining pairs can at least be oriented weakly. 232.97/60.10 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(B_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{b_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{b_1}(x_1)) = [[-I], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 1A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{c_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{c_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{c_1}(x_1)) = [[-I], [1A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [1A, 0A, 1A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 232.97/60.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (21) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (22) QDPOrderProof (EQUIVALENT) 232.97/60.10 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.10 232.97/60.10 232.97/60.10 The following pairs can be oriented strictly and are deleted. 232.97/60.10 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{c_1}(x1)) 232.97/60.10 The remaining pairs can at least be oriented weakly. 232.97/60.10 Used ordering: Polynomial interpretation [POLO]: 232.97/60.10 232.97/60.10 POL(B_{B_1}(x_1)) = x_1 232.97/60.10 POL(a_{a_1}(x_1)) = 0 232.97/60.10 POL(a_{b_1}(x_1)) = 0 232.97/60.10 POL(a_{c_1}(x_1)) = 0 232.97/60.10 POL(b_{a_1}(x_1)) = 0 232.97/60.10 POL(b_{b_1}(x_1)) = 1 232.97/60.10 POL(b_{c_1}(x_1)) = 0 232.97/60.10 POL(c_{a_1}(x_1)) = 0 232.97/60.10 POL(c_{c_1}(x_1)) = 0 232.97/60.10 232.97/60.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (23) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (24) QDPOrderProof (EQUIVALENT) 232.97/60.10 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.10 232.97/60.10 232.97/60.10 The following pairs can be oriented strictly and are deleted. 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{c_1}(x1))) 232.97/60.10 The remaining pairs can at least be oriented weakly. 232.97/60.10 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(B_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{b_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{c_1}(x_1)) = [[-I], [-I], [1A]] + [[0A, -I, 0A], [0A, -I, 0A], [1A, -I, 1A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{c_1}(x_1)) = [[0A], [1A], [0A]] + [[-I, 0A, 0A], [1A, -I, 1A], [-I, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{a_1}(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, -I], [1A, 1A, 1A], [-I, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 232.97/60.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (25) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (26) QDPOrderProof (EQUIVALENT) 232.97/60.10 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.10 232.97/60.10 232.97/60.10 The following pairs can be oriented strictly and are deleted. 232.97/60.10 232.97/60.10 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 232.97/60.10 The remaining pairs can at least be oriented weakly. 232.97/60.10 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(B_{B_1}(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [0A, -I, -I], [-I, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{a_1}(x_1)) = [[0A], [1A], [-I]] + [[0A, 0A, 0A], [0A, 1A, 0A], [0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{b_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [1A, -I, -I], [0A, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[1A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{c_1}(x_1)) = [[0A], [1A], [0A]] + [[-I, 1A, 0A], [1A, -I, -I], [0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{a_1}(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, -I], [1A, 1A, 1A], [-I, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 232.97/60.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (27) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (28) QDPOrderProof (EQUIVALENT) 232.97/60.10 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.10 232.97/60.10 232.97/60.10 The following pairs can be oriented strictly and are deleted. 232.97/60.10 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))) 232.97/60.10 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))))) 232.97/60.10 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)))) 232.97/60.10 The remaining pairs can at least be oriented weakly. 232.97/60.10 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(B_{B_1}(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{b_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{b_1}(x_1)) = [[-I], [0A], [-I]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{c_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 1A], [-I, -I, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{c_1}(x_1)) = [[0A], [-I], [1A]] + [[0A, -I, 0A], [0A, 0A, 1A], [0A, 1A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{a_1}(x_1)) = [[0A], [-I], [-I]] + [[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 232.97/60.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (29) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 P is empty. 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (30) PisEmptyProof (EQUIVALENT) 232.97/60.10 The TRS P is empty. Hence, there is no (P,Q,R) chain. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (31) 232.97/60.10 YES 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (32) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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)))))))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.10 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))))))) 232.97/60.10 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (33) QDPOrderProof (EQUIVALENT) 232.97/60.10 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.10 232.97/60.10 232.97/60.10 The following pairs can be oriented strictly and are deleted. 232.97/60.10 232.97/60.10 A_{B_1}(b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))) -> A_{C_1}(x1) 232.97/60.10 The remaining pairs can at least be oriented weakly. 232.97/60.10 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(A_{C_1}(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{c_1}(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, -I], [-I, -I, 0A], [0A, 1A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [-I, -I, -I], [0A, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(A_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{a_1}(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(C_{C_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{c_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, 1A, 0A], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 232.97/60.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (34) 232.97/60.10 Obligation: 232.97/60.10 Q DP problem: 232.97/60.10 The TRS P consists of the following rules: 232.97/60.10 232.97/60.10 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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)))))))) 232.97/60.10 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.10 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))))))) 232.97/60.10 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 232.97/60.10 The TRS R consists of the following rules: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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))))))) 232.97/60.10 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)))))))) 232.97/60.10 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)))))))) 232.97/60.10 232.97/60.10 Q is empty. 232.97/60.10 We have to consider all minimal (P,Q,R)-chains. 232.97/60.10 ---------------------------------------- 232.97/60.10 232.97/60.10 (35) QDPOrderProof (EQUIVALENT) 232.97/60.10 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.10 232.97/60.10 232.97/60.10 The following pairs can be oriented strictly and are deleted. 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.10 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))))))) 232.97/60.10 The remaining pairs can at least be oriented weakly. 232.97/60.10 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(A_{C_1}(x_1)) = [[0A]] + [[-I, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [-I, -I, 0A], [-I, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(c_{a_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [-I, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(A_{B_1}(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, -I], [-I, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{a_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(C_{C_1}(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{c_1}(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 1A], [-I, 0A, -I]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 <<< 232.97/60.10 POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.10 >>> 232.97/60.10 232.97/60.10 232.97/60.10 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.10 232.97/60.10 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 232.97/60.11 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (36) 232.97/60.11 Obligation: 232.97/60.11 Q DP problem: 232.97/60.11 The TRS P consists of the following rules: 232.97/60.11 232.97/60.11 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.11 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.11 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 232.97/60.11 The TRS R consists of the following rules: 232.97/60.11 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 232.97/60.11 Q is empty. 232.97/60.11 We have to consider all minimal (P,Q,R)-chains. 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (37) QDPOrderProof (EQUIVALENT) 232.97/60.11 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.11 232.97/60.11 232.97/60.11 The following pairs can be oriented strictly and are deleted. 232.97/60.11 232.97/60.11 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.11 The remaining pairs can at least be oriented weakly. 232.97/60.11 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(A_{C_1}(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(c_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, 0A, -I], [0A, 0A, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(c_{a_1}(x_1)) = [[1A], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(A_{B_1}(x_1)) = [[0A]] + [[-I, -I, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(b_{b_1}(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, -I], [-I, 0A, 0A], [-I, 0A, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(b_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(b_{c_1}(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, 0A], [-I, -I, -I], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(C_{C_1}(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(a_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 232.97/60.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.11 232.97/60.11 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))))))) 232.97/60.11 232.97/60.11 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (38) 232.97/60.11 Obligation: 232.97/60.11 Q DP problem: 232.97/60.11 The TRS P consists of the following rules: 232.97/60.11 232.97/60.11 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.11 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 232.97/60.11 The TRS R consists of the following rules: 232.97/60.11 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 232.97/60.11 Q is empty. 232.97/60.11 We have to consider all minimal (P,Q,R)-chains. 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (39) QDPOrderProof (EQUIVALENT) 232.97/60.11 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.11 232.97/60.11 232.97/60.11 The following pairs can be oriented strictly and are deleted. 232.97/60.11 232.97/60.11 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) 232.97/60.11 The remaining pairs can at least be oriented weakly. 232.97/60.11 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(A_{C_1}(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(c_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(A_{B_1}(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(b_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(b_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(C_{C_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(c_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [1A, 0A, -I], [0A, 0A, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(a_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(a_{c_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 232.97/60.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.11 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 232.97/60.11 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (40) 232.97/60.11 Obligation: 232.97/60.11 Q DP problem: 232.97/60.11 The TRS P consists of the following rules: 232.97/60.11 232.97/60.11 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 232.97/60.11 The TRS R consists of the following rules: 232.97/60.11 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 232.97/60.11 Q is empty. 232.97/60.11 We have to consider all minimal (P,Q,R)-chains. 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (41) DependencyGraphProof (EQUIVALENT) 232.97/60.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (42) 232.97/60.11 Obligation: 232.97/60.11 Q DP problem: 232.97/60.11 The TRS P consists of the following rules: 232.97/60.11 232.97/60.11 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))))))) 232.97/60.11 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.11 232.97/60.11 The TRS R consists of the following rules: 232.97/60.11 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 232.97/60.11 Q is empty. 232.97/60.11 We have to consider all minimal (P,Q,R)-chains. 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (43) QDPOrderProof (EQUIVALENT) 232.97/60.11 We use the reduction pair processor [LPAR04,JAR06]. 232.97/60.11 232.97/60.11 232.97/60.11 The following pairs can be oriented strictly and are deleted. 232.97/60.11 232.97/60.11 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))))))) 232.97/60.11 The remaining pairs can at least be oriented weakly. 232.97/60.11 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(A_{B_1}(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(b_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(b_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, -I], [0A, -I, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(c_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [-I, 1A, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(C_{C_1}(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(c_{a_1}(x_1)) = [[-I], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(a_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(a_{c_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(b_{a_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 <<< 232.97/60.11 POL(a_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 232.97/60.11 >>> 232.97/60.11 232.97/60.11 232.97/60.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 232.97/60.11 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 232.97/60.11 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (44) 232.97/60.11 Obligation: 232.97/60.11 Q DP problem: 232.97/60.11 The TRS P consists of the following rules: 232.97/60.11 232.97/60.11 C_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 232.97/60.11 232.97/60.11 The TRS R consists of the following rules: 232.97/60.11 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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))))))) 232.97/60.11 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)))))))) 232.97/60.11 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)))))))) 232.97/60.11 232.97/60.11 Q is empty. 232.97/60.11 We have to consider all minimal (P,Q,R)-chains. 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (45) DependencyGraphProof (EQUIVALENT) 232.97/60.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 232.97/60.11 ---------------------------------------- 232.97/60.11 232.97/60.11 (46) 232.97/60.11 TRUE 233.22/60.20 EOF