34.97/10.01 YES 35.28/10.05 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 35.28/10.05 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 35.28/10.05 35.28/10.05 35.28/10.05 Termination w.r.t. Q of the given QTRS could be proven: 35.28/10.05 35.28/10.05 (0) QTRS 35.28/10.05 (1) QTRS Reverse [EQUIVALENT, 0 ms] 35.28/10.05 (2) QTRS 35.28/10.05 (3) FlatCCProof [EQUIVALENT, 0 ms] 35.28/10.05 (4) QTRS 35.28/10.05 (5) RootLabelingProof [EQUIVALENT, 0 ms] 35.28/10.05 (6) QTRS 35.28/10.05 (7) QTRSRRRProof [EQUIVALENT, 50 ms] 35.28/10.05 (8) QTRS 35.28/10.05 (9) QTRSRRRProof [EQUIVALENT, 9 ms] 35.28/10.05 (10) QTRS 35.28/10.05 (11) QTRSRRRProof [EQUIVALENT, 28 ms] 35.28/10.05 (12) QTRS 35.28/10.05 (13) QTRSRRRProof [EQUIVALENT, 5 ms] 35.28/10.05 (14) QTRS 35.28/10.05 (15) DependencyPairsProof [EQUIVALENT, 162 ms] 35.28/10.05 (16) QDP 35.28/10.05 (17) DependencyGraphProof [EQUIVALENT, 0 ms] 35.28/10.05 (18) QDP 35.28/10.05 (19) QDPOrderProof [EQUIVALENT, 298 ms] 35.28/10.05 (20) QDP 35.28/10.05 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 35.28/10.05 (22) QDP 35.28/10.05 (23) QDPOrderProof [EQUIVALENT, 52 ms] 35.28/10.05 (24) QDP 35.28/10.05 (25) QDPOrderProof [EQUIVALENT, 55 ms] 35.28/10.05 (26) QDP 35.28/10.05 (27) DependencyGraphProof [EQUIVALENT, 0 ms] 35.28/10.05 (28) TRUE 35.28/10.05 35.28/10.05 35.28/10.05 ---------------------------------------- 35.28/10.05 35.28/10.05 (0) 35.28/10.05 Obligation: 35.28/10.05 Q restricted rewrite system: 35.28/10.05 The TRS R consists of the following rules: 35.28/10.05 35.28/10.05 a(a(x1)) -> a(b(a(b(a(x1))))) 35.28/10.05 c(a(x1)) -> a(b(a(a(c(x1))))) 35.28/10.05 b(b(b(x1))) -> a(b(x1)) 35.28/10.05 c(b(x1)) -> a(a(c(x1))) 35.28/10.05 c(b(x1)) -> b(a(d(x1))) 35.28/10.05 d(d(x1)) -> d(b(d(b(d(x1))))) 35.28/10.05 c(c(x1)) -> c(d(c(x1))) 35.28/10.05 a(a(a(x1))) -> a(b(b(x1))) 35.28/10.05 35.28/10.05 Q is empty. 35.28/10.05 35.28/10.05 ---------------------------------------- 35.28/10.05 35.28/10.05 (1) QTRS Reverse (EQUIVALENT) 35.28/10.05 We applied the QTRS Reverse Processor [REVERSE]. 35.28/10.05 ---------------------------------------- 35.28/10.05 35.28/10.05 (2) 35.28/10.05 Obligation: 35.28/10.05 Q restricted rewrite system: 35.28/10.05 The TRS R consists of the following rules: 35.28/10.05 35.28/10.05 a(a(x1)) -> a(b(a(b(a(x1))))) 35.28/10.05 a(c(x1)) -> c(a(a(b(a(x1))))) 35.28/10.05 b(b(b(x1))) -> b(a(x1)) 35.28/10.05 b(c(x1)) -> c(a(a(x1))) 35.28/10.05 b(c(x1)) -> d(a(b(x1))) 35.28/10.05 d(d(x1)) -> d(b(d(b(d(x1))))) 35.28/10.05 c(c(x1)) -> c(d(c(x1))) 35.28/10.05 a(a(a(x1))) -> b(b(a(x1))) 35.28/10.05 35.28/10.05 Q is empty. 35.28/10.05 35.28/10.05 ---------------------------------------- 35.28/10.05 35.28/10.05 (3) FlatCCProof (EQUIVALENT) 35.28/10.05 We used flat context closure [ROOTLAB] 35.28/10.05 As Q is empty the flat context closure was sound AND complete. 35.28/10.05 35.28/10.05 ---------------------------------------- 35.28/10.05 35.28/10.05 (4) 35.28/10.05 Obligation: 35.28/10.05 Q restricted rewrite system: 35.28/10.05 The TRS R consists of the following rules: 35.28/10.05 35.28/10.05 a(a(x1)) -> a(b(a(b(a(x1))))) 35.28/10.05 b(b(b(x1))) -> b(a(x1)) 35.28/10.05 d(d(x1)) -> d(b(d(b(d(x1))))) 35.28/10.05 c(c(x1)) -> c(d(c(x1))) 35.28/10.05 a(a(c(x1))) -> a(c(a(a(b(a(x1)))))) 35.28/10.05 b(a(c(x1))) -> b(c(a(a(b(a(x1)))))) 35.28/10.05 c(a(c(x1))) -> c(c(a(a(b(a(x1)))))) 35.28/10.05 d(a(c(x1))) -> d(c(a(a(b(a(x1)))))) 35.28/10.05 a(b(c(x1))) -> a(c(a(a(x1)))) 35.28/10.05 b(b(c(x1))) -> b(c(a(a(x1)))) 35.28/10.05 c(b(c(x1))) -> c(c(a(a(x1)))) 35.28/10.05 d(b(c(x1))) -> d(c(a(a(x1)))) 35.28/10.05 a(b(c(x1))) -> a(d(a(b(x1)))) 35.28/10.05 b(b(c(x1))) -> b(d(a(b(x1)))) 35.28/10.05 c(b(c(x1))) -> c(d(a(b(x1)))) 35.28/10.05 d(b(c(x1))) -> d(d(a(b(x1)))) 35.28/10.05 a(a(a(a(x1)))) -> a(b(b(a(x1)))) 35.28/10.05 b(a(a(a(x1)))) -> b(b(b(a(x1)))) 35.28/10.05 c(a(a(a(x1)))) -> c(b(b(a(x1)))) 35.28/10.05 d(a(a(a(x1)))) -> d(b(b(a(x1)))) 35.28/10.05 35.28/10.05 Q is empty. 35.28/10.05 35.28/10.05 ---------------------------------------- 35.28/10.05 35.28/10.05 (5) RootLabelingProof (EQUIVALENT) 35.28/10.05 We used plain root labeling [ROOTLAB] with the following heuristic: 35.28/10.05 LabelAll: All function symbols get labeled 35.28/10.05 35.28/10.05 As Q is empty the root labeling was sound AND complete. 35.28/10.05 35.28/10.05 ---------------------------------------- 35.28/10.05 35.28/10.05 (6) 35.28/10.05 Obligation: 35.28/10.05 Q restricted rewrite system: 35.28/10.05 The TRS R consists of the following rules: 35.28/10.05 35.28/10.05 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.28/10.05 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.28/10.05 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.28/10.05 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.28/10.05 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.28/10.05 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.28/10.05 b_{b_1}(b_{b_1}(b_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 35.28/10.05 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.28/10.05 d_{d_1}(d_{a_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{a_1}(x1))))) 35.28/10.05 d_{d_1}(d_{b_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{b_1}(x1))))) 35.28/10.05 d_{d_1}(d_{d_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{d_1}(x1))))) 35.28/10.05 d_{d_1}(d_{c_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{c_1}(x1))))) 35.28/10.05 c_{c_1}(c_{a_1}(x1)) -> c_{d_1}(d_{c_1}(c_{a_1}(x1))) 35.28/10.05 c_{c_1}(c_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) 35.28/10.05 c_{c_1}(c_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) 35.28/10.05 c_{c_1}(c_{c_1}(x1)) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) 35.28/10.05 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.28/10.05 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.28/10.05 a_{a_1}(a_{c_1}(c_{d_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1)))))) 35.28/10.05 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.28/10.05 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.28/10.05 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.28/10.05 b_{a_1}(a_{c_1}(c_{d_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1)))))) 35.28/10.05 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.28/10.05 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.28/10.05 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.28/10.05 c_{a_1}(a_{c_1}(c_{d_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1)))))) 35.28/10.05 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.28/10.05 d_{a_1}(a_{c_1}(c_{a_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.28/10.05 d_{a_1}(a_{c_1}(c_{b_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.28/10.05 d_{a_1}(a_{c_1}(c_{d_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1)))))) 35.28/10.05 d_{a_1}(a_{c_1}(c_{c_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{d_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{d_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{d_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{a_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{b_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{d_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{c_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{a_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{c_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x1)))) 35.28/10.05 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.28/10.05 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.28/10.05 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.28/10.05 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.28/10.05 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.28/10.05 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.28/10.05 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.28/10.05 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.28/10.05 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.28/10.05 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.28/10.05 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.28/10.05 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.28/10.05 d_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.28/10.05 d_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.28/10.05 d_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.28/10.05 d_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.28/10.05 35.28/10.05 Q is empty. 35.28/10.05 35.28/10.05 ---------------------------------------- 35.28/10.05 35.28/10.05 (7) QTRSRRRProof (EQUIVALENT) 35.28/10.05 Used ordering: 35.28/10.05 Polynomial interpretation [POLO]: 35.28/10.05 35.28/10.05 POL(a_{a_1}(x_1)) = x_1 35.28/10.05 POL(a_{b_1}(x_1)) = x_1 35.28/10.05 POL(a_{c_1}(x_1)) = 1 + x_1 35.28/10.05 POL(a_{d_1}(x_1)) = x_1 35.28/10.05 POL(b_{a_1}(x_1)) = x_1 35.28/10.05 POL(b_{b_1}(x_1)) = x_1 35.28/10.05 POL(b_{c_1}(x_1)) = 1 + x_1 35.28/10.05 POL(b_{d_1}(x_1)) = 1 + x_1 35.28/10.05 POL(c_{a_1}(x_1)) = 2 + x_1 35.28/10.05 POL(c_{b_1}(x_1)) = 2 + x_1 35.28/10.05 POL(c_{c_1}(x_1)) = 3 + x_1 35.28/10.05 POL(c_{d_1}(x_1)) = 3 + x_1 35.28/10.05 POL(d_{a_1}(x_1)) = x_1 35.28/10.05 POL(d_{b_1}(x_1)) = x_1 35.28/10.05 POL(d_{c_1}(x_1)) = x_1 35.28/10.05 POL(d_{d_1}(x_1)) = 2 + x_1 35.28/10.05 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 35.28/10.05 35.28/10.05 b_{b_1}(b_{b_1}(b_{d_1}(x1))) -> b_{a_1}(a_{d_1}(x1)) 35.28/10.05 a_{a_1}(a_{c_1}(c_{d_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1)))))) 35.28/10.05 b_{a_1}(a_{c_1}(c_{d_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1)))))) 35.28/10.05 c_{a_1}(a_{c_1}(c_{d_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1)))))) 35.28/10.05 d_{a_1}(a_{c_1}(c_{a_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.28/10.05 d_{a_1}(a_{c_1}(c_{b_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.28/10.05 d_{a_1}(a_{c_1}(c_{d_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1)))))) 35.28/10.05 d_{a_1}(a_{c_1}(c_{c_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{d_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{d_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{d_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{a_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{b_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{d_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{d_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{c_1}(x1))) -> d_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{d_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{d_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{d_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{a_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(b_{a_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{b_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(b_{b_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{d_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(b_{d_1}(x1)))) 35.28/10.05 d_{b_1}(b_{c_1}(c_{c_1}(x1))) -> d_{d_1}(d_{a_1}(a_{b_1}(b_{c_1}(x1)))) 35.28/10.05 35.28/10.05 35.28/10.05 35.28/10.05 35.28/10.05 ---------------------------------------- 35.28/10.05 35.28/10.05 (8) 35.28/10.05 Obligation: 35.28/10.05 Q restricted rewrite system: 35.28/10.05 The TRS R consists of the following rules: 35.28/10.05 35.28/10.05 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.28/10.05 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.28/10.05 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.28/10.05 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.28/10.05 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.28/10.05 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.28/10.05 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.28/10.05 d_{d_1}(d_{a_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{a_1}(x1))))) 35.28/10.05 d_{d_1}(d_{b_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{b_1}(x1))))) 35.28/10.05 d_{d_1}(d_{d_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{d_1}(x1))))) 35.28/10.05 d_{d_1}(d_{c_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{c_1}(x1))))) 35.28/10.05 c_{c_1}(c_{a_1}(x1)) -> c_{d_1}(d_{c_1}(c_{a_1}(x1))) 35.28/10.05 c_{c_1}(c_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) 35.28/10.05 c_{c_1}(c_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) 35.28/10.05 c_{c_1}(c_{c_1}(x1)) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) 35.28/10.05 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.28/10.05 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.28/10.05 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.28/10.05 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.28/10.05 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.28/10.05 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.28/10.05 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.28/10.05 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.28/10.05 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.28/10.05 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.28/10.05 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.28/10.05 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.28/10.05 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.28/10.05 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.28/10.05 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.28/10.05 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.28/10.05 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.28/10.05 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.28/10.05 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.28/10.05 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.28/10.05 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.28/10.05 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.28/10.05 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.28/10.05 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.28/10.05 d_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.28/10.05 d_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 35.36/10.08 Q is empty. 35.36/10.08 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (9) QTRSRRRProof (EQUIVALENT) 35.36/10.08 Used ordering: 35.36/10.08 Polynomial interpretation [POLO]: 35.36/10.08 35.36/10.08 POL(a_{a_1}(x_1)) = x_1 35.36/10.08 POL(a_{b_1}(x_1)) = x_1 35.36/10.08 POL(a_{c_1}(x_1)) = x_1 35.36/10.08 POL(a_{d_1}(x_1)) = x_1 35.36/10.08 POL(b_{a_1}(x_1)) = x_1 35.36/10.08 POL(b_{b_1}(x_1)) = x_1 35.36/10.08 POL(b_{c_1}(x_1)) = x_1 35.36/10.08 POL(b_{d_1}(x_1)) = x_1 35.36/10.08 POL(c_{a_1}(x_1)) = x_1 35.36/10.08 POL(c_{b_1}(x_1)) = x_1 35.36/10.08 POL(c_{c_1}(x_1)) = x_1 35.36/10.08 POL(c_{d_1}(x_1)) = x_1 35.36/10.08 POL(d_{a_1}(x_1)) = x_1 35.36/10.08 POL(d_{b_1}(x_1)) = x_1 35.36/10.08 POL(d_{c_1}(x_1)) = x_1 35.36/10.08 POL(d_{d_1}(x_1)) = 1 + x_1 35.36/10.08 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 35.36/10.08 35.36/10.08 d_{d_1}(d_{a_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{a_1}(x1))))) 35.36/10.08 d_{d_1}(d_{b_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{b_1}(x1))))) 35.36/10.08 d_{d_1}(d_{d_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{d_1}(x1))))) 35.36/10.08 d_{d_1}(d_{c_1}(x1)) -> d_{b_1}(b_{d_1}(d_{b_1}(b_{d_1}(d_{c_1}(x1))))) 35.36/10.08 35.36/10.08 35.36/10.08 35.36/10.08 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (10) 35.36/10.08 Obligation: 35.36/10.08 Q restricted rewrite system: 35.36/10.08 The TRS R consists of the following rules: 35.36/10.08 35.36/10.08 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.08 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.08 c_{c_1}(c_{a_1}(x1)) -> c_{d_1}(d_{c_1}(c_{a_1}(x1))) 35.36/10.08 c_{c_1}(c_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) 35.36/10.08 c_{c_1}(c_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) 35.36/10.08 c_{c_1}(c_{c_1}(x1)) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 35.36/10.08 Q is empty. 35.36/10.08 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (11) QTRSRRRProof (EQUIVALENT) 35.36/10.08 Used ordering: 35.36/10.08 Polynomial interpretation [POLO]: 35.36/10.08 35.36/10.08 POL(a_{a_1}(x_1)) = x_1 35.36/10.08 POL(a_{b_1}(x_1)) = x_1 35.36/10.08 POL(a_{c_1}(x_1)) = x_1 35.36/10.08 POL(a_{d_1}(x_1)) = x_1 35.36/10.08 POL(b_{a_1}(x_1)) = x_1 35.36/10.08 POL(b_{b_1}(x_1)) = x_1 35.36/10.08 POL(b_{c_1}(x_1)) = x_1 35.36/10.08 POL(c_{a_1}(x_1)) = 1 + x_1 35.36/10.08 POL(c_{b_1}(x_1)) = 1 + x_1 35.36/10.08 POL(c_{c_1}(x_1)) = 1 + x_1 35.36/10.08 POL(c_{d_1}(x_1)) = x_1 35.36/10.08 POL(d_{a_1}(x_1)) = x_1 35.36/10.08 POL(d_{b_1}(x_1)) = x_1 35.36/10.08 POL(d_{c_1}(x_1)) = x_1 35.36/10.08 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 35.36/10.08 35.36/10.08 c_{c_1}(c_{a_1}(x1)) -> c_{d_1}(d_{c_1}(c_{a_1}(x1))) 35.36/10.08 c_{c_1}(c_{b_1}(x1)) -> c_{d_1}(d_{c_1}(c_{b_1}(x1))) 35.36/10.08 c_{c_1}(c_{d_1}(x1)) -> c_{d_1}(d_{c_1}(c_{d_1}(x1))) 35.36/10.08 c_{c_1}(c_{c_1}(x1)) -> c_{d_1}(d_{c_1}(c_{c_1}(x1))) 35.36/10.08 35.36/10.08 35.36/10.08 35.36/10.08 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (12) 35.36/10.08 Obligation: 35.36/10.08 Q restricted rewrite system: 35.36/10.08 The TRS R consists of the following rules: 35.36/10.08 35.36/10.08 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.08 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.08 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 35.36/10.08 Q is empty. 35.36/10.08 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (13) QTRSRRRProof (EQUIVALENT) 35.36/10.08 Used ordering: 35.36/10.08 Polynomial interpretation [POLO]: 35.36/10.08 35.36/10.08 POL(a_{a_1}(x_1)) = x_1 35.36/10.08 POL(a_{b_1}(x_1)) = x_1 35.36/10.08 POL(a_{c_1}(x_1)) = x_1 35.36/10.08 POL(a_{d_1}(x_1)) = x_1 35.36/10.08 POL(b_{a_1}(x_1)) = x_1 35.36/10.08 POL(b_{b_1}(x_1)) = x_1 35.36/10.08 POL(b_{c_1}(x_1)) = x_1 35.36/10.08 POL(c_{a_1}(x_1)) = x_1 35.36/10.08 POL(c_{b_1}(x_1)) = x_1 35.36/10.08 POL(c_{c_1}(x_1)) = x_1 35.36/10.08 POL(d_{a_1}(x_1)) = 1 + x_1 35.36/10.08 POL(d_{b_1}(x_1)) = x_1 35.36/10.08 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 35.36/10.08 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 d_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> d_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 35.36/10.08 35.36/10.08 35.36/10.08 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (14) 35.36/10.08 Obligation: 35.36/10.08 Q restricted rewrite system: 35.36/10.08 The TRS R consists of the following rules: 35.36/10.08 35.36/10.08 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.08 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.08 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 35.36/10.08 Q is empty. 35.36/10.08 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (15) DependencyPairsProof (EQUIVALENT) 35.36/10.08 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (16) 35.36/10.08 Obligation: 35.36/10.08 Q DP problem: 35.36/10.08 The TRS P consists of the following rules: 35.36/10.08 35.36/10.08 A_{A_1}(a_{a_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 A_{A_1}(a_{a_1}(x1)) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 A_{A_1}(a_{b_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 A_{A_1}(a_{b_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 A_{A_1}(a_{d_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.08 A_{A_1}(a_{d_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 A_{A_1}(a_{d_1}(x1)) -> A_{B_1}(b_{a_1}(a_{d_1}(x1))) 35.36/10.08 A_{A_1}(a_{d_1}(x1)) -> B_{A_1}(a_{d_1}(x1)) 35.36/10.08 A_{A_1}(a_{c_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 A_{A_1}(a_{c_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(x1)) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 A_{A_1}(a_{c_1}(x1)) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 B_{B_1}(b_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) 35.36/10.08 A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 35.36/10.08 A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) 35.36/10.08 A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 35.36/10.08 A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) 35.36/10.08 A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 35.36/10.08 B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) 35.36/10.08 B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) 35.36/10.08 B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) 35.36/10.08 B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 35.36/10.08 C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) 35.36/10.08 C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 35.36/10.08 C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) 35.36/10.08 C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 35.36/10.08 C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) 35.36/10.08 C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{d_1}(x1))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{A_1}(a_{d_1}(x1)) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{d_1}(x1))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{A_1}(a_{d_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{d_1}(x1))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{A_1}(a_{d_1}(x1)) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 35.36/10.08 The TRS R consists of the following rules: 35.36/10.08 35.36/10.08 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.08 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.08 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 35.36/10.08 Q is empty. 35.36/10.08 We have to consider all minimal (P,Q,R)-chains. 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (17) DependencyGraphProof (EQUIVALENT) 35.36/10.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes. 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (18) 35.36/10.08 Obligation: 35.36/10.08 Q DP problem: 35.36/10.08 The TRS P consists of the following rules: 35.36/10.08 35.36/10.08 A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 A_{A_1}(a_{b_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 35.36/10.08 A_{A_1}(a_{a_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 35.36/10.08 A_{A_1}(a_{b_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 A_{A_1}(a_{a_1}(x1)) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 35.36/10.08 A_{A_1}(a_{c_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 A_{A_1}(a_{c_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(x1)) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 A_{A_1}(a_{c_1}(x1)) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 B_{B_1}(b_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.08 B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) 35.36/10.08 B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) 35.36/10.08 B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 35.36/10.08 C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.08 C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) 35.36/10.08 C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 35.36/10.08 C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.08 C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) 35.36/10.08 C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.08 C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.08 35.36/10.08 The TRS R consists of the following rules: 35.36/10.08 35.36/10.08 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.08 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.08 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.08 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.08 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.08 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.08 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.08 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.08 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.08 35.36/10.08 Q is empty. 35.36/10.08 We have to consider all minimal (P,Q,R)-chains. 35.36/10.08 ---------------------------------------- 35.36/10.08 35.36/10.08 (19) QDPOrderProof (EQUIVALENT) 35.36/10.08 We use the reduction pair processor [LPAR04,JAR06]. 35.36/10.08 35.36/10.08 35.36/10.08 The following pairs can be oriented strictly and are deleted. 35.36/10.08 35.36/10.08 A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.09 A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 35.36/10.09 A_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.09 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.09 A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.09 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.09 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.09 A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 35.36/10.09 A_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.09 A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.09 A_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 35.36/10.09 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.09 B_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.09 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.09 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.09 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.09 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.09 B_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.09 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.09 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.09 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.09 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.09 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.09 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.09 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.09 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.09 B_{A_1}(a_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.09 A_{A_1}(a_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.09 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.09 C_{A_1}(a_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.09 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.09 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.09 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.09 A_{A_1}(a_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.09 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.09 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.09 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.09 A_{A_1}(a_{c_1}(c_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.09 B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) 35.36/10.09 C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> C_{A_1}(a_{a_1}(a_{a_1}(x1))) 35.36/10.09 B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 35.36/10.09 B_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.09 B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) 35.36/10.09 B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 35.36/10.09 B_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.09 B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) 35.36/10.09 B_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 35.36/10.09 C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(a_{a_1}(x1)) 35.36/10.09 C_{B_1}(b_{c_1}(c_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.09 C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{a_1}(a_{b_1}(x1))) 35.36/10.09 C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) 35.36/10.09 C_{B_1}(b_{c_1}(c_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.09 C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> C_{A_1}(a_{a_1}(a_{c_1}(x1))) 35.36/10.09 C_{B_1}(b_{c_1}(c_{c_1}(x1))) -> A_{A_1}(a_{c_1}(x1)) 35.36/10.09 The remaining pairs can at least be oriented weakly. 35.36/10.09 Used ordering: Polynomial interpretation [POLO]: 35.36/10.09 35.36/10.09 POL(A_{A_1}(x_1)) = x_1 35.36/10.09 POL(A_{B_1}(x_1)) = x_1 35.36/10.09 POL(B_{A_1}(x_1)) = x_1 35.36/10.09 POL(B_{B_1}(x_1)) = x_1 35.36/10.09 POL(C_{A_1}(x_1)) = x_1 35.36/10.09 POL(C_{B_1}(x_1)) = x_1 35.36/10.09 POL(a_{a_1}(x_1)) = x_1 35.36/10.09 POL(a_{b_1}(x_1)) = x_1 35.36/10.09 POL(a_{c_1}(x_1)) = x_1 35.36/10.09 POL(a_{d_1}(x_1)) = x_1 35.36/10.09 POL(b_{a_1}(x_1)) = x_1 35.36/10.09 POL(b_{b_1}(x_1)) = x_1 35.36/10.09 POL(b_{c_1}(x_1)) = x_1 35.36/10.09 POL(c_{a_1}(x_1)) = 1 + x_1 35.36/10.09 POL(c_{b_1}(x_1)) = 1 + x_1 35.36/10.09 POL(c_{c_1}(x_1)) = 1 + x_1 35.36/10.09 35.36/10.09 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.36/10.09 35.36/10.09 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.09 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.09 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.09 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.10 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.10 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.10 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.10 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.10 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.10 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.10 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.10 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.10 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.10 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.10 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.10 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.10 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.10 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.10 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.10 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.10 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.10 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.10 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.10 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.10 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.10 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.10 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.10 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.10 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.10 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.10 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.10 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.10 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.10 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.10 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.10 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.10 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.10 35.36/10.10 35.36/10.10 ---------------------------------------- 35.36/10.10 35.36/10.10 (20) 35.36/10.10 Obligation: 35.36/10.10 Q DP problem: 35.36/10.10 The TRS P consists of the following rules: 35.36/10.10 35.36/10.10 A_{A_1}(a_{b_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.10 A_{A_1}(a_{a_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.10 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.10 A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.10 A_{A_1}(a_{b_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.10 A_{A_1}(a_{a_1}(x1)) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.10 A_{A_1}(a_{c_1}(x1)) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.10 A_{A_1}(a_{c_1}(x1)) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.10 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.10 A_{A_1}(a_{b_1}(x1)) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.10 A_{A_1}(a_{c_1}(x1)) -> A_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.10 A_{A_1}(a_{c_1}(x1)) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.10 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.10 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.10 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.10 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.10 B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.10 B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1) 35.36/10.10 B_{B_1}(b_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.10 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.10 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.10 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.10 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.10 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.10 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.10 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.10 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.10 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.10 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.10 C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.10 C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.10 C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.10 C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 35.36/10.10 C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.10 C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.10 C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{A_1}(a_{c_1}(x1)) 35.36/10.10 35.36/10.10 The TRS R consists of the following rules: 35.36/10.10 35.36/10.10 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.10 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.10 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.10 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.10 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.10 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.10 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.10 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.10 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.10 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.10 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.10 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.10 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.10 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.10 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.10 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.10 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.10 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.10 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.10 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.10 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.10 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.10 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.10 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.10 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.10 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.10 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.10 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 35.36/10.11 Q is empty. 35.36/10.11 We have to consider all minimal (P,Q,R)-chains. 35.36/10.11 ---------------------------------------- 35.36/10.11 35.36/10.11 (21) DependencyGraphProof (EQUIVALENT) 35.36/10.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 30 less nodes. 35.36/10.11 ---------------------------------------- 35.36/10.11 35.36/10.11 (22) 35.36/10.11 Obligation: 35.36/10.11 Q DP problem: 35.36/10.11 The TRS P consists of the following rules: 35.36/10.11 35.36/10.11 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.11 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.11 35.36/10.11 The TRS R consists of the following rules: 35.36/10.11 35.36/10.11 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.11 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.11 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.11 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.11 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.11 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.11 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.11 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 35.36/10.11 Q is empty. 35.36/10.11 We have to consider all minimal (P,Q,R)-chains. 35.36/10.11 ---------------------------------------- 35.36/10.11 35.36/10.11 (23) QDPOrderProof (EQUIVALENT) 35.36/10.11 We use the reduction pair processor [LPAR04,JAR06]. 35.36/10.11 35.36/10.11 35.36/10.11 The following pairs can be oriented strictly and are deleted. 35.36/10.11 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{c_1}(x1))) 35.36/10.11 The remaining pairs can at least be oriented weakly. 35.36/10.11 Used ordering: Polynomial interpretation [POLO]: 35.36/10.11 35.36/10.11 POL(A_{A_1}(x_1)) = 1 35.36/10.11 POL(B_{A_1}(x_1)) = 1 35.36/10.11 POL(B_{B_1}(x_1)) = x_1 35.36/10.11 POL(a_{a_1}(x_1)) = 1 35.36/10.11 POL(a_{b_1}(x_1)) = 0 35.36/10.11 POL(a_{c_1}(x_1)) = 0 35.36/10.11 POL(a_{d_1}(x_1)) = x_1 35.36/10.11 POL(b_{a_1}(x_1)) = x_1 35.36/10.11 POL(b_{b_1}(x_1)) = 1 35.36/10.11 POL(b_{c_1}(x_1)) = 0 35.36/10.11 POL(c_{a_1}(x_1)) = x_1 35.36/10.11 POL(c_{b_1}(x_1)) = 0 35.36/10.11 POL(c_{c_1}(x_1)) = 0 35.36/10.11 35.36/10.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.36/10.11 35.36/10.11 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.11 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.11 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.11 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.11 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 35.36/10.11 35.36/10.11 ---------------------------------------- 35.36/10.11 35.36/10.11 (24) 35.36/10.11 Obligation: 35.36/10.11 Q DP problem: 35.36/10.11 The TRS P consists of the following rules: 35.36/10.11 35.36/10.11 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.11 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 35.36/10.11 The TRS R consists of the following rules: 35.36/10.11 35.36/10.11 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.11 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.11 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.11 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.11 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.11 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.11 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.11 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 35.36/10.11 Q is empty. 35.36/10.11 We have to consider all minimal (P,Q,R)-chains. 35.36/10.11 ---------------------------------------- 35.36/10.11 35.36/10.11 (25) QDPOrderProof (EQUIVALENT) 35.36/10.11 We use the reduction pair processor [LPAR04,JAR06]. 35.36/10.11 35.36/10.11 35.36/10.11 The following pairs can be oriented strictly and are deleted. 35.36/10.11 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.11 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 35.36/10.11 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 B_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 The remaining pairs can at least be oriented weakly. 35.36/10.11 Used ordering: Polynomial interpretation [POLO]: 35.36/10.11 35.36/10.11 POL(A_{A_1}(x_1)) = x_1 35.36/10.11 POL(B_{A_1}(x_1)) = x_1 35.36/10.11 POL(B_{B_1}(x_1)) = x_1 35.36/10.11 POL(a_{a_1}(x_1)) = 1 + x_1 35.36/10.11 POL(a_{b_1}(x_1)) = 1 35.36/10.11 POL(a_{c_1}(x_1)) = 1 35.36/10.11 POL(a_{d_1}(x_1)) = 1 + x_1 35.36/10.11 POL(b_{a_1}(x_1)) = x_1 35.36/10.11 POL(b_{b_1}(x_1)) = 1 + x_1 35.36/10.11 POL(b_{c_1}(x_1)) = 0 35.36/10.11 POL(c_{a_1}(x_1)) = 0 35.36/10.11 POL(c_{b_1}(x_1)) = 0 35.36/10.11 POL(c_{c_1}(x_1)) = 0 35.36/10.11 35.36/10.11 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 35.36/10.11 35.36/10.11 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.11 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.11 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.11 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.11 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 35.36/10.11 35.36/10.11 ---------------------------------------- 35.36/10.11 35.36/10.11 (26) 35.36/10.11 Obligation: 35.36/10.11 Q DP problem: 35.36/10.11 The TRS P consists of the following rules: 35.36/10.11 35.36/10.11 A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 35.36/10.11 35.36/10.11 The TRS R consists of the following rules: 35.36/10.11 35.36/10.11 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 35.36/10.11 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 35.36/10.11 a_{a_1}(a_{d_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{d_1}(x1))))) 35.36/10.11 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) 35.36/10.11 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 35.36/10.11 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 35.36/10.11 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 35.36/10.11 a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))) 35.36/10.11 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))) 35.36/10.11 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 c_{b_1}(b_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) 35.36/10.11 c_{b_1}(b_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) 35.36/10.11 c_{b_1}(b_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{d_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{d_1}(x1)))) 35.36/10.11 c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) 35.36/10.11 35.36/10.11 Q is empty. 35.36/10.11 We have to consider all minimal (P,Q,R)-chains. 35.36/10.11 ---------------------------------------- 35.36/10.11 35.36/10.11 (27) DependencyGraphProof (EQUIVALENT) 35.36/10.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 35.36/10.11 ---------------------------------------- 35.36/10.11 35.36/10.11 (28) 35.36/10.11 TRUE 35.78/10.20 EOF