152.44/39.76 YES 152.89/39.85 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 152.89/39.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 152.89/39.85 152.89/39.85 152.89/39.85 Termination w.r.t. Q of the given QTRS could be proven: 152.89/39.85 152.89/39.85 (0) QTRS 152.89/39.85 (1) QTRS Reverse [EQUIVALENT, 0 ms] 152.89/39.85 (2) QTRS 152.89/39.85 (3) FlatCCProof [EQUIVALENT, 0 ms] 152.89/39.85 (4) QTRS 152.89/39.85 (5) RootLabelingProof [EQUIVALENT, 0 ms] 152.89/39.85 (6) QTRS 152.89/39.85 (7) QTRSRRRProof [EQUIVALENT, 98 ms] 152.89/39.85 (8) QTRS 152.89/39.85 (9) DependencyPairsProof [EQUIVALENT, 161 ms] 152.89/39.85 (10) QDP 152.89/39.85 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 152.89/39.85 (12) QDP 152.89/39.85 (13) QDPOrderProof [EQUIVALENT, 782 ms] 152.89/39.85 (14) QDP 152.89/39.85 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 152.89/39.85 (16) AND 152.89/39.85 (17) QDP 152.89/39.85 (18) UsableRulesProof [EQUIVALENT, 0 ms] 152.89/39.85 (19) QDP 152.89/39.85 (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] 152.89/39.85 (21) YES 152.89/39.85 (22) QDP 152.89/39.85 (23) QDPOrderProof [EQUIVALENT, 52 ms] 152.89/39.85 (24) QDP 152.89/39.85 (25) QDPOrderProof [EQUIVALENT, 496 ms] 152.89/39.85 (26) QDP 152.89/39.85 (27) QDPOrderProof [EQUIVALENT, 6963 ms] 152.89/39.85 (28) QDP 152.89/39.85 (29) QDPOrderProof [EQUIVALENT, 39 ms] 152.89/39.85 (30) QDP 152.89/39.85 (31) QDPOrderProof [EQUIVALENT, 31 ms] 152.89/39.85 (32) QDP 152.89/39.85 (33) PisEmptyProof [EQUIVALENT, 0 ms] 152.89/39.85 (34) YES 152.89/39.85 152.89/39.85 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (0) 152.89/39.85 Obligation: 152.89/39.85 Q restricted rewrite system: 152.89/39.85 The TRS R consists of the following rules: 152.89/39.85 152.89/39.85 a(c(x1)) -> a(x1) 152.89/39.85 d(a(x1)) -> a(c(b(c(d(x1))))) 152.89/39.85 a(c(b(c(x1)))) -> c(b(c(c(x1)))) 152.89/39.85 c(x1) -> b(a(a(x1))) 152.89/39.85 d(c(x1)) -> a(c(d(a(x1)))) 152.89/39.85 152.89/39.85 Q is empty. 152.89/39.85 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (1) QTRS Reverse (EQUIVALENT) 152.89/39.85 We applied the QTRS Reverse Processor [REVERSE]. 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (2) 152.89/39.85 Obligation: 152.89/39.85 Q restricted rewrite system: 152.89/39.85 The TRS R consists of the following rules: 152.89/39.85 152.89/39.85 c(a(x1)) -> a(x1) 152.89/39.85 a(d(x1)) -> d(c(b(c(a(x1))))) 152.89/39.85 c(b(c(a(x1)))) -> c(c(b(c(x1)))) 152.89/39.85 c(x1) -> a(a(b(x1))) 152.89/39.85 c(d(x1)) -> a(d(c(a(x1)))) 152.89/39.85 152.89/39.85 Q is empty. 152.89/39.85 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (3) FlatCCProof (EQUIVALENT) 152.89/39.85 We used flat context closure [ROOTLAB] 152.89/39.85 As Q is empty the flat context closure was sound AND complete. 152.89/39.85 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (4) 152.89/39.85 Obligation: 152.89/39.85 Q restricted rewrite system: 152.89/39.85 The TRS R consists of the following rules: 152.89/39.85 152.89/39.85 c(b(c(a(x1)))) -> c(c(b(c(x1)))) 152.89/39.85 c(c(a(x1))) -> c(a(x1)) 152.89/39.85 a(c(a(x1))) -> a(a(x1)) 152.89/39.85 d(c(a(x1))) -> d(a(x1)) 152.89/39.85 b(c(a(x1))) -> b(a(x1)) 152.89/39.85 c(a(d(x1))) -> c(d(c(b(c(a(x1)))))) 152.89/39.85 a(a(d(x1))) -> a(d(c(b(c(a(x1)))))) 152.89/39.85 d(a(d(x1))) -> d(d(c(b(c(a(x1)))))) 152.89/39.85 b(a(d(x1))) -> b(d(c(b(c(a(x1)))))) 152.89/39.85 c(c(x1)) -> c(a(a(b(x1)))) 152.89/39.85 a(c(x1)) -> a(a(a(b(x1)))) 152.89/39.85 d(c(x1)) -> d(a(a(b(x1)))) 152.89/39.85 b(c(x1)) -> b(a(a(b(x1)))) 152.89/39.85 c(c(d(x1))) -> c(a(d(c(a(x1))))) 152.89/39.85 a(c(d(x1))) -> a(a(d(c(a(x1))))) 152.89/39.85 d(c(d(x1))) -> d(a(d(c(a(x1))))) 152.89/39.85 b(c(d(x1))) -> b(a(d(c(a(x1))))) 152.89/39.85 152.89/39.85 Q is empty. 152.89/39.85 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (5) RootLabelingProof (EQUIVALENT) 152.89/39.85 We used plain root labeling [ROOTLAB] with the following heuristic: 152.89/39.85 LabelAll: All function symbols get labeled 152.89/39.85 152.89/39.85 As Q is empty the root labeling was sound AND complete. 152.89/39.85 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (6) 152.89/39.85 Obligation: 152.89/39.85 Q restricted rewrite system: 152.89/39.85 The TRS R consists of the following rules: 152.89/39.85 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.85 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.85 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.85 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.85 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.85 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.85 c_{a_1}(a_{d_1}(d_{b_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) 152.89/39.85 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.85 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.85 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.85 a_{a_1}(a_{d_1}(d_{b_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) 152.89/39.85 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.85 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.85 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.85 d_{a_1}(a_{d_1}(d_{b_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) 152.89/39.85 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.85 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.85 b_{a_1}(a_{d_1}(d_{c_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.85 b_{a_1}(a_{d_1}(d_{b_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) 152.89/39.85 b_{a_1}(a_{d_1}(d_{a_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.85 b_{a_1}(a_{d_1}(d_{d_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.85 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 c_{c_1}(c_{d_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) 152.89/39.85 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 a_{c_1}(c_{d_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) 152.89/39.85 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 d_{c_1}(c_{d_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) 152.89/39.85 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 b_{c_1}(c_{d_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) 152.89/39.85 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 c_{c_1}(c_{d_1}(d_{b_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) 152.89/39.85 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 a_{c_1}(c_{d_1}(d_{b_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) 152.89/39.85 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 d_{c_1}(c_{d_1}(d_{b_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) 152.89/39.85 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 b_{c_1}(c_{d_1}(d_{b_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) 152.89/39.85 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 152.89/39.85 Q is empty. 152.89/39.85 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (7) QTRSRRRProof (EQUIVALENT) 152.89/39.85 Used ordering: 152.89/39.85 Polynomial interpretation [POLO]: 152.89/39.85 152.89/39.85 POL(a_{a_1}(x_1)) = x_1 152.89/39.85 POL(a_{b_1}(x_1)) = x_1 152.89/39.85 POL(a_{c_1}(x_1)) = x_1 152.89/39.85 POL(a_{d_1}(x_1)) = 1 + x_1 152.89/39.85 POL(b_{a_1}(x_1)) = x_1 152.89/39.85 POL(b_{b_1}(x_1)) = x_1 152.89/39.85 POL(b_{c_1}(x_1)) = x_1 152.89/39.85 POL(b_{d_1}(x_1)) = x_1 152.89/39.85 POL(c_{a_1}(x_1)) = x_1 152.89/39.85 POL(c_{b_1}(x_1)) = x_1 152.89/39.85 POL(c_{c_1}(x_1)) = x_1 152.89/39.85 POL(c_{d_1}(x_1)) = 1 + x_1 152.89/39.85 POL(d_{a_1}(x_1)) = x_1 152.89/39.85 POL(d_{b_1}(x_1)) = 1 + x_1 152.89/39.85 POL(d_{c_1}(x_1)) = x_1 152.89/39.85 POL(d_{d_1}(x_1)) = 1 + x_1 152.89/39.85 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 152.89/39.85 152.89/39.85 c_{a_1}(a_{d_1}(d_{b_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) 152.89/39.85 a_{a_1}(a_{d_1}(d_{b_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) 152.89/39.85 d_{a_1}(a_{d_1}(d_{b_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) 152.89/39.85 b_{a_1}(a_{d_1}(d_{c_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.85 b_{a_1}(a_{d_1}(d_{b_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))))) 152.89/39.85 b_{a_1}(a_{d_1}(d_{a_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.85 b_{a_1}(a_{d_1}(d_{d_1}(x1))) -> b_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.85 c_{c_1}(c_{d_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) 152.89/39.85 a_{c_1}(c_{d_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) 152.89/39.85 d_{c_1}(c_{d_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) 152.89/39.85 b_{c_1}(c_{d_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{d_1}(x1)))) 152.89/39.85 c_{c_1}(c_{d_1}(d_{b_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) 152.89/39.85 a_{c_1}(c_{d_1}(d_{b_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) 152.89/39.85 d_{c_1}(c_{d_1}(d_{b_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) 152.89/39.85 b_{c_1}(c_{d_1}(d_{b_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{b_1}(x1))))) 152.89/39.85 152.89/39.85 152.89/39.85 152.89/39.85 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (8) 152.89/39.85 Obligation: 152.89/39.85 Q restricted rewrite system: 152.89/39.85 The TRS R consists of the following rules: 152.89/39.85 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.85 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.85 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.85 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.85 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.85 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.85 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.85 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.85 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.85 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.85 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.85 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.85 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.85 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.85 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 152.89/39.85 Q is empty. 152.89/39.85 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (9) DependencyPairsProof (EQUIVALENT) 152.89/39.85 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 152.89/39.85 ---------------------------------------- 152.89/39.85 152.89/39.85 (10) 152.89/39.85 Obligation: 152.89/39.85 Q DP problem: 152.89/39.85 The TRS P consists of the following rules: 152.89/39.85 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(c_{c_1}(x1)) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(x1) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{b_1}(x1)) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{a_1}(x1)) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(x1) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) 152.89/39.85 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> B_{C_1}(c_{d_1}(x1)) 152.89/39.85 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 152.89/39.85 A_{C_1}(c_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 152.89/39.85 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 152.89/39.85 A_{C_1}(c_{a_1}(a_{d_1}(x1))) -> A_{A_1}(a_{d_1}(x1)) 152.89/39.85 D_{C_1}(c_{a_1}(a_{c_1}(x1))) -> D_{A_1}(a_{c_1}(x1)) 152.89/39.85 D_{C_1}(c_{a_1}(a_{b_1}(x1))) -> D_{A_1}(a_{b_1}(x1)) 152.89/39.85 D_{C_1}(c_{a_1}(a_{a_1}(x1))) -> D_{A_1}(a_{a_1}(x1)) 152.89/39.85 D_{C_1}(c_{a_1}(a_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) 152.89/39.85 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) 152.89/39.85 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.85 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.85 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.85 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) 152.89/39.85 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.85 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.85 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.85 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) 152.89/39.85 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.85 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.85 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) 152.89/39.85 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.85 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.85 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.85 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) 152.89/39.85 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.85 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.85 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.85 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) 152.89/39.85 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.85 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.85 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) 152.89/39.85 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.85 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.85 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.85 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) 152.89/39.85 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.85 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.85 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.85 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) 152.89/39.85 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.85 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.85 C_{C_1}(c_{c_1}(x1)) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 C_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{b_1}(b_{c_1}(x1))) 152.89/39.85 C_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.85 C_{C_1}(c_{b_1}(x1)) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 C_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 152.89/39.85 C_{C_1}(c_{a_1}(x1)) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 C_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 152.89/39.85 A_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 A_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{b_1}(b_{c_1}(x1))) 152.89/39.85 A_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.85 A_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 A_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 152.89/39.85 A_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 A_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 152.89/39.85 D_{C_1}(c_{c_1}(x1)) -> D_{A_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.85 D_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{b_1}(b_{c_1}(x1))) 152.89/39.85 D_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.85 D_{C_1}(c_{b_1}(x1)) -> D_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.85 D_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 152.89/39.85 D_{C_1}(c_{a_1}(x1)) -> D_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.85 D_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 152.89/39.85 B_{C_1}(c_{c_1}(x1)) -> A_{A_1}(a_{b_1}(b_{c_1}(x1))) 152.89/39.85 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.85 B_{C_1}(c_{b_1}(x1)) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 152.89/39.85 B_{C_1}(c_{a_1}(x1)) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 152.89/39.85 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.85 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.85 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.85 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.85 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.85 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.85 C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.85 C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.85 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.85 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.85 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.85 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.85 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.85 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.85 A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.85 A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.85 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.85 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.85 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.85 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.85 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.85 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.85 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.85 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.85 D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.85 D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.85 D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.85 B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.85 B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.85 B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.85 B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.85 B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.85 B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.85 B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.85 B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.85 152.89/39.85 The TRS R consists of the following rules: 152.89/39.85 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.85 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.85 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.85 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.85 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.85 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.85 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.85 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.85 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.85 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.85 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.85 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.85 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (11) DependencyGraphProof (EQUIVALENT) 152.89/39.88 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 23 less nodes. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (12) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 The TRS P consists of the following rules: 152.89/39.88 152.89/39.88 C_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 D_{C_1}(c_{a_1}(a_{c_1}(x1))) -> D_{A_1}(a_{c_1}(x1)) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 D_{C_1}(c_{a_1}(a_{a_1}(x1))) -> D_{A_1}(a_{a_1}(x1)) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 D_{C_1}(c_{a_1}(a_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(c_{c_1}(x1)) 152.89/39.88 B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 D_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(x1) 152.89/39.88 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{a_1}(a_{d_1}(x1))) -> A_{A_1}(a_{d_1}(x1)) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{b_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{a_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(x1) 152.89/39.88 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> B_{C_1}(c_{d_1}(x1)) 152.89/39.88 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 152.89/39.88 The TRS R consists of the following rules: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (13) QDPOrderProof (EQUIVALENT) 152.89/39.88 We use the reduction pair processor [LPAR04,JAR06]. 152.89/39.88 152.89/39.88 152.89/39.88 The following pairs can be oriented strictly and are deleted. 152.89/39.88 152.89/39.88 B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 B_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{C_1}(c_{a_1}(a_{c_1}(x1))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) 152.89/39.88 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 C_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 A_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) 152.89/39.88 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 A_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 D_{A_1}(a_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> B_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 D_{A_1}(a_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{c_1}(x1)) 152.89/39.88 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{C_1}(c_{a_1}(a_{a_1}(x1))) 152.89/39.88 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{a_1}(x1)) 152.89/39.88 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(x1) 152.89/39.88 A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{C_1}(c_{a_1}(a_{d_1}(x1))) 152.89/39.88 A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(x1)) 152.89/39.88 The remaining pairs can at least be oriented weakly. 152.89/39.88 Used ordering: Polynomial interpretation [POLO]: 152.89/39.88 152.89/39.88 POL(A_{A_1}(x_1)) = x_1 152.89/39.88 POL(A_{C_1}(x_1)) = x_1 152.89/39.88 POL(B_{C_1}(x_1)) = x_1 152.89/39.88 POL(C_{A_1}(x_1)) = x_1 152.89/39.88 POL(C_{B_1}(x_1)) = x_1 152.89/39.88 POL(C_{C_1}(x_1)) = x_1 152.89/39.88 POL(D_{A_1}(x_1)) = x_1 152.89/39.88 POL(D_{C_1}(x_1)) = x_1 152.89/39.88 POL(a_{a_1}(x_1)) = x_1 152.89/39.88 POL(a_{b_1}(x_1)) = x_1 152.89/39.88 POL(a_{c_1}(x_1)) = x_1 152.89/39.88 POL(a_{d_1}(x_1)) = 1 + x_1 152.89/39.88 POL(b_{a_1}(x_1)) = 0 152.89/39.88 POL(b_{b_1}(x_1)) = 0 152.89/39.88 POL(b_{c_1}(x_1)) = x_1 152.89/39.88 POL(c_{a_1}(x_1)) = x_1 152.89/39.88 POL(c_{b_1}(x_1)) = 0 152.89/39.88 POL(c_{c_1}(x_1)) = x_1 152.89/39.88 POL(c_{d_1}(x_1)) = 1 + x_1 152.89/39.88 POL(d_{a_1}(x_1)) = x_1 152.89/39.88 POL(d_{c_1}(x_1)) = x_1 152.89/39.88 POL(d_{d_1}(x_1)) = 1 + x_1 152.89/39.88 152.89/39.88 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 152.89/39.88 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 152.89/39.88 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (14) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 The TRS P consists of the following rules: 152.89/39.88 152.89/39.88 C_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 D_{C_1}(c_{a_1}(a_{c_1}(x1))) -> D_{A_1}(a_{c_1}(x1)) 152.89/39.88 D_{C_1}(c_{a_1}(a_{a_1}(x1))) -> D_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 C_{C_1}(c_{d_1}(d_{c_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 D_{C_1}(c_{a_1}(a_{d_1}(x1))) -> D_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> B_{C_1}(c_{c_1}(x1)) 152.89/39.88 A_{C_1}(c_{a_1}(a_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 152.89/39.88 D_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 D_{C_1}(c_{d_1}(d_{c_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 A_{C_1}(c_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{a_1}(a_{d_1}(x1))) -> A_{A_1}(a_{d_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{a_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 A_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 A_{C_1}(c_{d_1}(d_{c_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> B_{C_1}(c_{b_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 D_{C_1}(c_{d_1}(d_{d_1}(x1))) -> D_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> B_{C_1}(c_{a_1}(x1)) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{A_1}(x1) 152.89/39.88 C_{C_1}(c_{d_1}(d_{a_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> B_{C_1}(c_{d_1}(x1)) 152.89/39.88 C_{C_1}(c_{d_1}(d_{d_1}(x1))) -> C_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 A_{C_1}(c_{d_1}(d_{a_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 A_{C_1}(c_{d_1}(d_{d_1}(x1))) -> A_{A_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 The TRS R consists of the following rules: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (15) DependencyGraphProof (EQUIVALENT) 152.89/39.88 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 27 less nodes. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (16) 152.89/39.88 Complex Obligation (AND) 152.89/39.88 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (17) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 The TRS P consists of the following rules: 152.89/39.88 152.89/39.88 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 152.89/39.88 The TRS R consists of the following rules: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (18) UsableRulesProof (EQUIVALENT) 152.89/39.88 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. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (19) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 The TRS P consists of the following rules: 152.89/39.88 152.89/39.88 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 152.89/39.88 R is empty. 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (20) QDPSizeChangeProof (EQUIVALENT) 152.89/39.88 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 152.89/39.88 152.89/39.88 From the DPs we obtained the following set of size-change graphs: 152.89/39.88 *B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 152.89/39.88 The graph contains the following edges 1 > 1 152.89/39.88 152.89/39.88 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (21) 152.89/39.88 YES 152.89/39.88 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (22) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 The TRS P consists of the following rules: 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) 152.89/39.88 152.89/39.88 The TRS R consists of the following rules: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (23) QDPOrderProof (EQUIVALENT) 152.89/39.88 We use the reduction pair processor [LPAR04,JAR06]. 152.89/39.88 152.89/39.88 152.89/39.88 The following pairs can be oriented strictly and are deleted. 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{c_1}(x1))) 152.89/39.88 The remaining pairs can at least be oriented weakly. 152.89/39.88 Used ordering: Polynomial interpretation [POLO]: 152.89/39.88 152.89/39.88 POL(C_{B_1}(x_1)) = x_1 152.89/39.88 POL(a_{a_1}(x_1)) = x_1 152.89/39.88 POL(a_{b_1}(x_1)) = x_1 152.89/39.88 POL(a_{c_1}(x_1)) = 1 + x_1 152.89/39.88 POL(a_{d_1}(x_1)) = 0 152.89/39.88 POL(b_{a_1}(x_1)) = 0 152.89/39.88 POL(b_{b_1}(x_1)) = 0 152.89/39.88 POL(b_{c_1}(x_1)) = x_1 152.89/39.88 POL(c_{a_1}(x_1)) = x_1 152.89/39.88 POL(c_{b_1}(x_1)) = 0 152.89/39.88 POL(c_{c_1}(x_1)) = x_1 152.89/39.88 POL(c_{d_1}(x_1)) = 0 152.89/39.88 POL(d_{a_1}(x_1)) = 0 152.89/39.88 POL(d_{c_1}(x_1)) = 0 152.89/39.88 POL(d_{d_1}(x_1)) = 0 152.89/39.88 152.89/39.88 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 152.89/39.88 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (24) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 The TRS P consists of the following rules: 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) 152.89/39.88 152.89/39.88 The TRS R consists of the following rules: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (25) QDPOrderProof (EQUIVALENT) 152.89/39.88 We use the reduction pair processor [LPAR04,JAR06]. 152.89/39.88 152.89/39.88 152.89/39.88 The following pairs can be oriented strictly and are deleted. 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{d_1}(x1))) 152.89/39.88 The remaining pairs can at least be oriented weakly. 152.89/39.88 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 152.89/39.88 152.89/39.88 POL( C_{B_1}_1(x_1) ) = 2x_1 152.89/39.88 POL( b_{c_1}_1(x_1) ) = x_1 152.89/39.88 POL( a_{d_1}_1(x_1) ) = 1 152.89/39.88 POL( b_{a_1}_1(x_1) ) = max{0, -2} 152.89/39.88 POL( d_{c_1}_1(x_1) ) = max{0, -2} 152.89/39.88 POL( c_{b_1}_1(x_1) ) = 0 152.89/39.88 POL( c_{a_1}_1(x_1) ) = 2x_1 152.89/39.88 POL( a_{c_1}_1(x_1) ) = 2x_1 152.89/39.88 POL( c_{c_1}_1(x_1) ) = 2x_1 152.89/39.88 POL( a_{b_1}_1(x_1) ) = x_1 152.89/39.88 POL( a_{a_1}_1(x_1) ) = 2x_1 152.89/39.88 POL( c_{d_1}_1(x_1) ) = 1 152.89/39.88 POL( b_{b_1}_1(x_1) ) = 0 152.89/39.88 POL( d_{a_1}_1(x_1) ) = max{0, x_1 - 2} 152.89/39.88 POL( d_{d_1}_1(x_1) ) = max{0, x_1 - 2} 152.89/39.88 152.89/39.88 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 152.89/39.88 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (26) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 The TRS P consists of the following rules: 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) 152.89/39.88 152.89/39.88 The TRS R consists of the following rules: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (27) QDPOrderProof (EQUIVALENT) 152.89/39.88 We use the reduction pair processor [LPAR04,JAR06]. 152.89/39.88 152.89/39.88 152.89/39.88 The following pairs can be oriented strictly and are deleted. 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{a_1}(x1))) 152.89/39.88 The remaining pairs can at least be oriented weakly. 152.89/39.88 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(C_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(b_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(c_{a_1}(x_1)) = [[0A], [1A], [1A]] + [[0A, 0A, 0A], [-I, 0A, 0A], [1A, 0A, 0A]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(a_{b_1}(x_1)) = [[0A], [-I], [1A]] + [[0A, -I, -I], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(c_{b_1}(x_1)) = [[0A], [1A], [1A]] + [[0A, -I, -I], [0A, 0A, 0A], [1A, 0A, 0A]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(a_{a_1}(x_1)) = [[1A], [-I], [0A]] + [[1A, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(a_{c_1}(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, 1A], [0A, 1A, 1A], [1A, 0A, 0A]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(c_{c_1}(x_1)) = [[0A], [1A], [1A]] + [[0A, 0A, -I], [1A, 0A, 0A], [1A, 1A, 0A]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(a_{d_1}(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(c_{d_1}(x_1)) = [[0A], [1A], [1A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(b_{a_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(b_{b_1}(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [0A, -I, 0A], [0A, -I, -I]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(d_{c_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(d_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 1A], [0A, 1A, 0A], [0A, 1A, -I]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 <<< 152.89/39.88 POL(d_{d_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [-I, 0A, 0A], [-I, -I, -I]] * x_1 152.89/39.88 >>> 152.89/39.88 152.89/39.88 152.89/39.88 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 152.89/39.88 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (28) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 The TRS P consists of the following rules: 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) 152.89/39.88 152.89/39.88 The TRS R consists of the following rules: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (29) QDPOrderProof (EQUIVALENT) 152.89/39.88 We use the reduction pair processor [LPAR04,JAR06]. 152.89/39.88 152.89/39.88 152.89/39.88 The following pairs can be oriented strictly and are deleted. 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(x1) 152.89/39.88 The remaining pairs can at least be oriented weakly. 152.89/39.88 Used ordering: Polynomial interpretation [POLO]: 152.89/39.88 152.89/39.88 POL(C_{B_1}(x_1)) = x_1 152.89/39.88 POL(a_{a_1}(x_1)) = 0 152.89/39.88 POL(a_{b_1}(x_1)) = x_1 152.89/39.88 POL(a_{c_1}(x_1)) = 0 152.89/39.88 POL(a_{d_1}(x_1)) = 0 152.89/39.88 POL(b_{a_1}(x_1)) = 0 152.89/39.88 POL(b_{b_1}(x_1)) = x_1 152.89/39.88 POL(b_{c_1}(x_1)) = 1 + x_1 152.89/39.88 POL(c_{a_1}(x_1)) = x_1 152.89/39.88 POL(c_{b_1}(x_1)) = 0 152.89/39.88 POL(c_{c_1}(x_1)) = x_1 152.89/39.88 POL(c_{d_1}(x_1)) = 0 152.89/39.88 POL(d_{a_1}(x_1)) = 0 152.89/39.88 POL(d_{c_1}(x_1)) = 0 152.89/39.88 POL(d_{d_1}(x_1)) = 0 152.89/39.88 152.89/39.88 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 152.89/39.88 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (30) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 The TRS P consists of the following rules: 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) 152.89/39.88 152.89/39.88 The TRS R consists of the following rules: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (31) QDPOrderProof (EQUIVALENT) 152.89/39.88 We use the reduction pair processor [LPAR04,JAR06]. 152.89/39.88 152.89/39.88 152.89/39.88 The following pairs can be oriented strictly and are deleted. 152.89/39.88 152.89/39.88 C_{B_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{c_1}(c_{b_1}(x1))) 152.89/39.88 The remaining pairs can at least be oriented weakly. 152.89/39.88 Used ordering: Polynomial interpretation [POLO]: 152.89/39.88 152.89/39.88 POL(C_{B_1}(x_1)) = x_1 152.89/39.88 POL(a_{a_1}(x_1)) = 0 152.89/39.88 POL(a_{b_1}(x_1)) = 1 152.89/39.88 POL(a_{c_1}(x_1)) = 0 152.89/39.88 POL(a_{d_1}(x_1)) = 0 152.89/39.88 POL(b_{a_1}(x_1)) = 0 152.89/39.88 POL(b_{b_1}(x_1)) = x_1 152.89/39.88 POL(b_{c_1}(x_1)) = x_1 152.89/39.88 POL(c_{a_1}(x_1)) = x_1 152.89/39.88 POL(c_{b_1}(x_1)) = 0 152.89/39.88 POL(c_{c_1}(x_1)) = x_1 152.89/39.88 POL(c_{d_1}(x_1)) = 0 152.89/39.88 POL(d_{a_1}(x_1)) = 0 152.89/39.88 POL(d_{c_1}(x_1)) = 0 152.89/39.88 POL(d_{d_1}(x_1)) = 0 152.89/39.88 152.89/39.88 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 152.89/39.88 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (32) 152.89/39.88 Obligation: 152.89/39.88 Q DP problem: 152.89/39.88 P is empty. 152.89/39.88 The TRS R consists of the following rules: 152.89/39.88 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 152.89/39.88 c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{d_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 152.89/39.88 c_{c_1}(c_{a_1}(a_{d_1}(x1))) -> c_{a_1}(a_{d_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 152.89/39.88 a_{c_1}(c_{a_1}(a_{d_1}(x1))) -> a_{a_1}(a_{d_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{c_1}(x1))) -> d_{a_1}(a_{c_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{b_1}(x1))) -> d_{a_1}(a_{b_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{a_1}(x1))) -> d_{a_1}(a_{a_1}(x1)) 152.89/39.88 d_{c_1}(c_{a_1}(a_{d_1}(x1))) -> d_{a_1}(a_{d_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 152.89/39.88 b_{c_1}(c_{a_1}(a_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 152.89/39.88 c_{a_1}(a_{d_1}(d_{c_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{a_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 c_{a_1}(a_{d_1}(d_{d_1}(x1))) -> c_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{c_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{a_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 a_{a_1}(a_{d_1}(d_{d_1}(x1))) -> a_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{c_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{a_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))))) 152.89/39.88 d_{a_1}(a_{d_1}(d_{d_1}(x1))) -> d_{d_1}(d_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(a_{d_1}(x1)))))) 152.89/39.88 c_{c_1}(c_{c_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 c_{c_1}(c_{b_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 a_{c_1}(c_{c_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 a_{c_1}(c_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 d_{c_1}(c_{c_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 d_{c_1}(c_{b_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 d_{c_1}(c_{a_1}(x1)) -> d_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 152.89/39.88 b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 152.89/39.88 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{c_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{a_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 c_{c_1}(c_{d_1}(d_{d_1}(x1))) -> c_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{c_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{a_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 a_{c_1}(c_{d_1}(d_{d_1}(x1))) -> a_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{c_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{a_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 d_{c_1}(c_{d_1}(d_{d_1}(x1))) -> d_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{c_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{c_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{a_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{a_1}(x1))))) 152.89/39.88 b_{c_1}(c_{d_1}(d_{d_1}(x1))) -> b_{a_1}(a_{d_1}(d_{c_1}(c_{a_1}(a_{d_1}(x1))))) 152.89/39.88 152.89/39.88 Q is empty. 152.89/39.88 We have to consider all minimal (P,Q,R)-chains. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (33) PisEmptyProof (EQUIVALENT) 152.89/39.88 The TRS P is empty. Hence, there is no (P,Q,R) chain. 152.89/39.88 ---------------------------------------- 152.89/39.88 152.89/39.88 (34) 152.89/39.88 YES 153.30/40.01 EOF