26.99/7.86 YES 28.34/8.14 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 28.34/8.14 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.34/8.14 28.34/8.14 28.34/8.14 Termination w.r.t. Q of the given QTRS could be proven: 28.34/8.14 28.34/8.14 (0) QTRS 28.34/8.14 (1) QTRS Reverse [EQUIVALENT, 0 ms] 28.34/8.14 (2) QTRS 28.34/8.14 (3) FlatCCProof [EQUIVALENT, 0 ms] 28.34/8.14 (4) QTRS 28.34/8.14 (5) RootLabelingProof [EQUIVALENT, 3 ms] 28.34/8.14 (6) QTRS 28.34/8.14 (7) QTRSRRRProof [EQUIVALENT, 31 ms] 28.34/8.14 (8) QTRS 28.34/8.14 (9) DependencyPairsProof [EQUIVALENT, 92 ms] 28.34/8.14 (10) QDP 28.34/8.14 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 28.34/8.14 (12) QDP 28.34/8.14 (13) QDPOrderProof [EQUIVALENT, 296 ms] 28.34/8.14 (14) QDP 28.34/8.14 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 28.34/8.14 (16) TRUE 28.34/8.14 28.34/8.14 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (0) 28.34/8.14 Obligation: 28.34/8.14 Q restricted rewrite system: 28.34/8.14 The TRS R consists of the following rules: 28.34/8.14 28.34/8.14 a(x1) -> x1 28.34/8.14 a(b(x1)) -> b(b(a(c(x1)))) 28.34/8.14 b(x1) -> a(c(x1)) 28.34/8.14 c(c(x1)) -> x1 28.34/8.14 28.34/8.14 Q is empty. 28.34/8.14 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (1) QTRS Reverse (EQUIVALENT) 28.34/8.14 We applied the QTRS Reverse Processor [REVERSE]. 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (2) 28.34/8.14 Obligation: 28.34/8.14 Q restricted rewrite system: 28.34/8.14 The TRS R consists of the following rules: 28.34/8.14 28.34/8.14 a(x1) -> x1 28.34/8.14 b(a(x1)) -> c(a(b(b(x1)))) 28.34/8.14 b(x1) -> c(a(x1)) 28.34/8.14 c(c(x1)) -> x1 28.34/8.14 28.34/8.14 Q is empty. 28.34/8.14 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (3) FlatCCProof (EQUIVALENT) 28.34/8.14 We used flat context closure [ROOTLAB] 28.34/8.14 As Q is empty the flat context closure was sound AND complete. 28.34/8.14 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (4) 28.34/8.14 Obligation: 28.34/8.14 Q restricted rewrite system: 28.34/8.14 The TRS R consists of the following rules: 28.34/8.14 28.34/8.14 a(a(x1)) -> a(x1) 28.34/8.14 b(a(x1)) -> b(x1) 28.34/8.14 c(a(x1)) -> c(x1) 28.34/8.14 a(b(a(x1))) -> a(c(a(b(b(x1))))) 28.34/8.14 b(b(a(x1))) -> b(c(a(b(b(x1))))) 28.34/8.14 c(b(a(x1))) -> c(c(a(b(b(x1))))) 28.34/8.14 a(b(x1)) -> a(c(a(x1))) 28.34/8.14 b(b(x1)) -> b(c(a(x1))) 28.34/8.14 c(b(x1)) -> c(c(a(x1))) 28.34/8.14 a(c(c(x1))) -> a(x1) 28.34/8.14 b(c(c(x1))) -> b(x1) 28.34/8.14 c(c(c(x1))) -> c(x1) 28.34/8.14 28.34/8.14 Q is empty. 28.34/8.14 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (5) RootLabelingProof (EQUIVALENT) 28.34/8.14 We used plain root labeling [ROOTLAB] with the following heuristic: 28.34/8.14 LabelAll: All function symbols get labeled 28.34/8.14 28.34/8.14 As Q is empty the root labeling was sound AND complete. 28.34/8.14 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (6) 28.34/8.14 Obligation: 28.34/8.14 Q restricted rewrite system: 28.34/8.14 The TRS R consists of the following rules: 28.34/8.14 28.34/8.14 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 28.34/8.14 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 28.34/8.14 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 28.34/8.14 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 28.34/8.14 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 28.34/8.14 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 28.34/8.14 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 28.34/8.14 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.34/8.14 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.34/8.14 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.14 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.14 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.14 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.14 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.14 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.14 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.14 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.14 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.14 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.14 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.14 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.14 b_{b_1}(b_{a_1}(x1)) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.14 b_{b_1}(b_{b_1}(x1)) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.14 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.14 c_{b_1}(b_{a_1}(x1)) -> c_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.14 c_{b_1}(b_{b_1}(x1)) -> c_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.14 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.14 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1) 28.34/8.14 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1) 28.34/8.14 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{c_1}(x1) 28.34/8.14 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1) 28.34/8.14 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1) 28.34/8.14 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{c_1}(x1) 28.34/8.14 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(x1) 28.34/8.14 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(x1) 28.34/8.14 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(x1) 28.34/8.14 28.34/8.14 Q is empty. 28.34/8.14 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (7) QTRSRRRProof (EQUIVALENT) 28.34/8.14 Used ordering: 28.34/8.14 Polynomial interpretation [POLO]: 28.34/8.14 28.34/8.14 POL(a_{a_1}(x_1)) = 2 + x_1 28.34/8.14 POL(a_{b_1}(x_1)) = 1 + x_1 28.34/8.14 POL(a_{c_1}(x_1)) = x_1 28.34/8.14 POL(b_{a_1}(x_1)) = 2 + x_1 28.34/8.14 POL(b_{b_1}(x_1)) = 1 + x_1 28.34/8.14 POL(b_{c_1}(x_1)) = x_1 28.34/8.14 POL(c_{a_1}(x_1)) = 1 + x_1 28.34/8.14 POL(c_{b_1}(x_1)) = 2 + x_1 28.34/8.14 POL(c_{c_1}(x_1)) = 1 + x_1 28.34/8.14 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 28.34/8.14 28.34/8.14 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 28.34/8.14 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 28.34/8.14 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 28.34/8.14 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 28.34/8.14 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 28.34/8.14 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 28.34/8.14 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 28.34/8.14 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(x1) 28.34/8.14 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{c_1}(x1) 28.34/8.14 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(x1) 28.34/8.14 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{c_1}(x1) 28.34/8.14 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(x1) 28.34/8.14 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(x1) 28.34/8.14 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(x1) 28.34/8.14 28.34/8.14 28.34/8.14 28.34/8.14 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (8) 28.34/8.14 Obligation: 28.34/8.14 Q restricted rewrite system: 28.34/8.14 The TRS R consists of the following rules: 28.34/8.14 28.34/8.14 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.34/8.14 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.34/8.14 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.14 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.14 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.14 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.14 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.14 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.14 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.14 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.14 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.14 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.14 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.14 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.14 b_{b_1}(b_{a_1}(x1)) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.14 b_{b_1}(b_{b_1}(x1)) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.14 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.14 c_{b_1}(b_{a_1}(x1)) -> c_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.14 c_{b_1}(b_{b_1}(x1)) -> c_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.14 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.14 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1) 28.34/8.14 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1) 28.34/8.14 28.34/8.14 Q is empty. 28.34/8.14 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (9) DependencyPairsProof (EQUIVALENT) 28.34/8.14 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 28.34/8.14 ---------------------------------------- 28.34/8.14 28.34/8.14 (10) 28.34/8.14 Obligation: 28.34/8.14 Q DP problem: 28.34/8.14 The TRS P consists of the following rules: 28.34/8.14 28.34/8.14 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(x1) 28.34/8.14 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.14 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 28.34/8.14 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 28.34/8.14 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 28.34/8.14 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.14 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 28.34/8.14 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 28.34/8.14 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) 28.34/8.14 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.34/8.14 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.14 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 28.34/8.14 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 28.34/8.14 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 28.34/8.15 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 28.34/8.15 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 28.34/8.15 A_{B_1}(b_{a_1}(x1)) -> A_{C_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(x1)) -> C_{A_1}(a_{a_1}(x1)) 28.34/8.15 A_{B_1}(b_{b_1}(x1)) -> A_{C_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 A_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 28.34/8.15 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.34/8.15 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 A_{B_1}(b_{c_1}(x1)) -> C_{A_1}(a_{c_1}(x1)) 28.34/8.15 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1) 28.34/8.15 B_{B_1}(b_{a_1}(x1)) -> B_{C_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 B_{B_1}(b_{a_1}(x1)) -> C_{A_1}(a_{a_1}(x1)) 28.34/8.15 B_{B_1}(b_{b_1}(x1)) -> B_{C_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 B_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 28.34/8.15 B_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.34/8.15 B_{B_1}(b_{c_1}(x1)) -> B_{C_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 B_{B_1}(b_{c_1}(x1)) -> C_{A_1}(a_{c_1}(x1)) 28.34/8.15 B_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1) 28.34/8.15 C_{B_1}(b_{a_1}(x1)) -> C_{A_1}(a_{a_1}(x1)) 28.34/8.15 C_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 28.34/8.15 C_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.34/8.15 C_{B_1}(b_{c_1}(x1)) -> C_{A_1}(a_{c_1}(x1)) 28.34/8.15 C_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1) 28.34/8.15 28.34/8.15 The TRS R consists of the following rules: 28.34/8.15 28.34/8.15 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.34/8.15 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.34/8.15 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 b_{b_1}(b_{a_1}(x1)) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 b_{b_1}(b_{b_1}(x1)) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 c_{b_1}(b_{a_1}(x1)) -> c_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 c_{b_1}(b_{b_1}(x1)) -> c_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1) 28.34/8.15 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1) 28.34/8.15 28.34/8.15 Q is empty. 28.34/8.15 We have to consider all minimal (P,Q,R)-chains. 28.34/8.15 ---------------------------------------- 28.34/8.15 28.34/8.15 (11) DependencyGraphProof (EQUIVALENT) 28.34/8.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 24 less nodes. 28.34/8.15 ---------------------------------------- 28.34/8.15 28.34/8.15 (12) 28.34/8.15 Obligation: 28.34/8.15 Q DP problem: 28.34/8.15 The TRS P consists of the following rules: 28.34/8.15 28.34/8.15 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 28.34/8.15 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(x1) 28.34/8.15 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 28.34/8.15 A_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 28.34/8.15 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.34/8.15 B_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 28.34/8.15 B_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.34/8.15 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 28.34/8.15 C_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 28.34/8.15 C_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.34/8.15 28.34/8.15 The TRS R consists of the following rules: 28.34/8.15 28.34/8.15 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.34/8.15 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.34/8.15 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 b_{b_1}(b_{a_1}(x1)) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 b_{b_1}(b_{b_1}(x1)) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 c_{b_1}(b_{a_1}(x1)) -> c_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 c_{b_1}(b_{b_1}(x1)) -> c_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1) 28.34/8.15 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1) 28.34/8.15 28.34/8.15 Q is empty. 28.34/8.15 We have to consider all minimal (P,Q,R)-chains. 28.34/8.15 ---------------------------------------- 28.34/8.15 28.34/8.15 (13) QDPOrderProof (EQUIVALENT) 28.34/8.15 We use the reduction pair processor [LPAR04,JAR06]. 28.34/8.15 28.34/8.15 28.34/8.15 The following pairs can be oriented strictly and are deleted. 28.34/8.15 28.34/8.15 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(x1) 28.34/8.15 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 28.34/8.15 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 28.34/8.15 A_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 28.34/8.15 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.34/8.15 B_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.34/8.15 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1)) 28.34/8.15 C_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.34/8.15 The remaining pairs can at least be oriented weakly. 28.34/8.15 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 28.34/8.15 28.34/8.15 POL( A_{B_1}_1(x_1) ) = 2x_1 + 1 28.34/8.15 POL( B_{B_1}_1(x_1) ) = 2x_1 28.34/8.15 POL( C_{A_1}_1(x_1) ) = 2x_1 28.34/8.15 POL( a_{b_1}_1(x_1) ) = x_1 + 1 28.34/8.15 POL( b_{b_1}_1(x_1) ) = x_1 + 1 28.34/8.15 POL( b_{a_1}_1(x_1) ) = x_1 + 2 28.34/8.15 POL( a_{a_1}_1(x_1) ) = x_1 + 2 28.34/8.15 POL( b_{c_1}_1(x_1) ) = x_1 + 1 28.34/8.15 POL( c_{a_1}_1(x_1) ) = x_1 28.34/8.15 POL( a_{c_1}_1(x_1) ) = x_1 + 1 28.34/8.15 POL( c_{c_1}_1(x_1) ) = x_1 + 1 28.34/8.15 POL( c_{b_1}_1(x_1) ) = x_1 + 1 28.34/8.15 POL( C_{B_1}_1(x_1) ) = 2x_1 28.34/8.15 28.34/8.15 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.34/8.15 28.34/8.15 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(x1)) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 b_{b_1}(b_{b_1}(x1)) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1) 28.34/8.15 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.34/8.15 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 c_{b_1}(b_{b_1}(x1)) -> c_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.34/8.15 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1) 28.34/8.15 c_{b_1}(b_{a_1}(x1)) -> c_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 28.34/8.15 28.34/8.15 ---------------------------------------- 28.34/8.15 28.34/8.15 (14) 28.34/8.15 Obligation: 28.34/8.15 Q DP problem: 28.34/8.15 The TRS P consists of the following rules: 28.34/8.15 28.34/8.15 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 28.34/8.15 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 28.34/8.15 B_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 28.34/8.15 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 28.34/8.15 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 28.34/8.15 C_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 28.34/8.15 28.34/8.15 The TRS R consists of the following rules: 28.34/8.15 28.34/8.15 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.34/8.15 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.34/8.15 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 28.34/8.15 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))) 28.34/8.15 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 b_{b_1}(b_{a_1}(x1)) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 b_{b_1}(b_{b_1}(x1)) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 c_{b_1}(b_{a_1}(x1)) -> c_{c_1}(c_{a_1}(a_{a_1}(x1))) 28.34/8.15 c_{b_1}(b_{b_1}(x1)) -> c_{c_1}(c_{a_1}(a_{b_1}(x1))) 28.34/8.15 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(c_{a_1}(a_{c_1}(x1))) 28.34/8.15 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(x1) 28.34/8.15 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(x1) 28.34/8.15 28.34/8.15 Q is empty. 28.34/8.15 We have to consider all minimal (P,Q,R)-chains. 28.34/8.15 ---------------------------------------- 28.34/8.15 28.34/8.15 (15) DependencyGraphProof (EQUIVALENT) 28.34/8.15 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 8 less nodes. 28.34/8.15 ---------------------------------------- 28.34/8.15 28.34/8.15 (16) 28.34/8.15 TRUE 28.47/8.21 EOF