25.32/7.39 YES 25.57/7.40 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 25.57/7.40 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.57/7.40 25.57/7.40 25.57/7.40 Termination w.r.t. Q of the given QTRS could be proven: 25.57/7.40 25.57/7.40 (0) QTRS 25.57/7.40 (1) QTRS Reverse [EQUIVALENT, 0 ms] 25.57/7.40 (2) QTRS 25.57/7.40 (3) FlatCCProof [EQUIVALENT, 0 ms] 25.57/7.40 (4) QTRS 25.57/7.40 (5) RootLabelingProof [EQUIVALENT, 0 ms] 25.57/7.40 (6) QTRS 25.57/7.40 (7) QTRSRRRProof [EQUIVALENT, 68 ms] 25.57/7.40 (8) QTRS 25.57/7.40 (9) DependencyPairsProof [EQUIVALENT, 85 ms] 25.57/7.40 (10) QDP 25.57/7.40 (11) QDPOrderProof [EQUIVALENT, 119 ms] 25.57/7.40 (12) QDP 25.57/7.40 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 25.57/7.40 (14) AND 25.57/7.40 (15) QDP 25.57/7.40 (16) UsableRulesProof [EQUIVALENT, 0 ms] 25.57/7.40 (17) QDP 25.57/7.40 (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] 25.57/7.40 (19) YES 25.57/7.40 (20) QDP 25.57/7.40 (21) UsableRulesProof [EQUIVALENT, 0 ms] 25.57/7.40 (22) QDP 25.57/7.40 (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] 25.57/7.40 (24) YES 25.57/7.40 25.57/7.40 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (0) 25.57/7.40 Obligation: 25.57/7.40 Q restricted rewrite system: 25.57/7.40 The TRS R consists of the following rules: 25.57/7.40 25.57/7.40 a(x1) -> x1 25.57/7.40 a(b(x1)) -> b(a(c(x1))) 25.57/7.40 b(x1) -> x1 25.57/7.40 c(c(x1)) -> c(b(a(x1))) 25.57/7.40 25.57/7.40 Q is empty. 25.57/7.40 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (1) QTRS Reverse (EQUIVALENT) 25.57/7.40 We applied the QTRS Reverse Processor [REVERSE]. 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (2) 25.57/7.40 Obligation: 25.57/7.40 Q restricted rewrite system: 25.57/7.40 The TRS R consists of the following rules: 25.57/7.40 25.57/7.40 a(x1) -> x1 25.57/7.40 b(a(x1)) -> c(a(b(x1))) 25.57/7.40 b(x1) -> x1 25.57/7.40 c(c(x1)) -> a(b(c(x1))) 25.57/7.40 25.57/7.40 Q is empty. 25.57/7.40 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (3) FlatCCProof (EQUIVALENT) 25.57/7.40 We used flat context closure [ROOTLAB] 25.57/7.40 As Q is empty the flat context closure was sound AND complete. 25.57/7.40 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (4) 25.57/7.40 Obligation: 25.57/7.40 Q restricted rewrite system: 25.57/7.40 The TRS R consists of the following rules: 25.57/7.40 25.57/7.40 a(a(x1)) -> a(x1) 25.57/7.40 b(a(x1)) -> b(x1) 25.57/7.40 c(a(x1)) -> c(x1) 25.57/7.40 a(b(a(x1))) -> a(c(a(b(x1)))) 25.57/7.40 b(b(a(x1))) -> b(c(a(b(x1)))) 25.57/7.40 c(b(a(x1))) -> c(c(a(b(x1)))) 25.57/7.40 a(b(x1)) -> a(x1) 25.57/7.40 b(b(x1)) -> b(x1) 25.57/7.40 c(b(x1)) -> c(x1) 25.57/7.40 a(c(c(x1))) -> a(a(b(c(x1)))) 25.57/7.40 b(c(c(x1))) -> b(a(b(c(x1)))) 25.57/7.40 c(c(c(x1))) -> c(a(b(c(x1)))) 25.57/7.40 25.57/7.40 Q is empty. 25.57/7.40 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (5) RootLabelingProof (EQUIVALENT) 25.57/7.40 We used plain root labeling [ROOTLAB] with the following heuristic: 25.57/7.40 LabelAll: All function symbols get labeled 25.57/7.40 25.57/7.40 As Q is empty the root labeling was sound AND complete. 25.57/7.40 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (6) 25.57/7.40 Obligation: 25.57/7.40 Q restricted rewrite system: 25.57/7.40 The TRS R consists of the following rules: 25.57/7.40 25.57/7.40 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 25.57/7.40 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 25.57/7.40 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 25.57/7.40 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 25.57/7.40 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.57/7.40 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 25.57/7.40 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 25.57/7.40 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.40 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.40 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.40 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.40 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.40 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.40 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.40 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.40 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.40 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.40 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.40 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 25.57/7.40 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 25.57/7.40 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 25.57/7.40 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 25.57/7.40 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 25.57/7.40 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 25.57/7.40 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) 25.57/7.40 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.40 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.40 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.40 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.40 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.40 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.40 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.40 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.40 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.40 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.40 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.40 25.57/7.40 Q is empty. 25.57/7.40 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (7) QTRSRRRProof (EQUIVALENT) 25.57/7.40 Used ordering: 25.57/7.40 Polynomial interpretation [POLO]: 25.57/7.40 25.57/7.40 POL(a_{a_1}(x_1)) = 1 + x_1 25.57/7.40 POL(a_{b_1}(x_1)) = x_1 25.57/7.40 POL(a_{c_1}(x_1)) = x_1 25.57/7.40 POL(b_{a_1}(x_1)) = 1 + x_1 25.57/7.40 POL(b_{b_1}(x_1)) = x_1 25.57/7.40 POL(b_{c_1}(x_1)) = x_1 25.57/7.40 POL(c_{a_1}(x_1)) = 1 + x_1 25.57/7.40 POL(c_{b_1}(x_1)) = 1 + x_1 25.57/7.40 POL(c_{c_1}(x_1)) = 1 + x_1 25.57/7.40 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 25.57/7.40 25.57/7.40 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 25.57/7.40 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 25.57/7.40 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 25.57/7.40 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 25.57/7.40 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.57/7.40 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 25.57/7.40 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 25.57/7.40 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) 25.57/7.40 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.40 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.40 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.40 25.57/7.40 25.57/7.40 25.57/7.40 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (8) 25.57/7.40 Obligation: 25.57/7.40 Q restricted rewrite system: 25.57/7.40 The TRS R consists of the following rules: 25.57/7.40 25.57/7.40 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.40 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.40 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.40 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.40 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.40 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.40 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.40 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.40 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.40 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.40 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.40 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 25.57/7.40 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 25.57/7.40 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 25.57/7.40 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 25.57/7.40 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 25.57/7.40 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 25.57/7.40 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.40 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.40 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.40 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.40 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.40 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.40 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.40 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.40 25.57/7.40 Q is empty. 25.57/7.40 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (9) DependencyPairsProof (EQUIVALENT) 25.57/7.40 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.57/7.40 ---------------------------------------- 25.57/7.40 25.57/7.40 (10) 25.57/7.40 Obligation: 25.57/7.40 Q DP problem: 25.57/7.40 The TRS P consists of the following rules: 25.57/7.40 25.57/7.40 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(x1) 25.57/7.40 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.40 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 25.57/7.40 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 25.57/7.40 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.40 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 25.57/7.40 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 25.57/7.40 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 25.57/7.40 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.40 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 25.57/7.40 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 25.57/7.41 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 25.57/7.41 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 25.57/7.41 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 25.57/7.41 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 25.57/7.41 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 25.57/7.41 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 25.57/7.41 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 25.57/7.41 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 25.57/7.41 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 25.57/7.41 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 25.57/7.41 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 25.57/7.41 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 25.57/7.41 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 25.57/7.41 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 25.57/7.41 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 25.57/7.41 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1) 25.57/7.41 C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 25.57/7.41 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.57/7.41 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.57/7.41 A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.57/7.41 A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.57/7.41 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.57/7.41 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.57/7.41 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.57/7.41 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.57/7.41 B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.57/7.41 B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.57/7.41 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.57/7.41 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.57/7.41 25.57/7.41 The TRS R consists of the following rules: 25.57/7.41 25.57/7.41 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 25.57/7.41 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 25.57/7.41 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 25.57/7.41 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 25.57/7.41 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 25.57/7.41 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 25.57/7.41 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 25.57/7.41 Q is empty. 25.57/7.41 We have to consider all minimal (P,Q,R)-chains. 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (11) QDPOrderProof (EQUIVALENT) 25.57/7.41 We use the reduction pair processor [LPAR04,JAR06]. 25.57/7.41 25.57/7.41 25.57/7.41 The following pairs can be oriented strictly and are deleted. 25.57/7.41 25.57/7.41 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(x1) 25.57/7.41 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 25.57/7.41 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 25.57/7.41 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 25.57/7.41 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 25.57/7.41 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 25.57/7.41 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 25.57/7.41 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 25.57/7.41 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 25.57/7.41 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 25.57/7.41 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 25.57/7.41 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 25.57/7.41 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 25.57/7.41 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 25.57/7.41 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 25.57/7.41 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 25.57/7.41 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.57/7.41 A_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.57/7.41 A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.57/7.41 A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.57/7.41 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.57/7.41 A_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.57/7.41 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.57/7.41 B_{C_1}(c_{c_1}(c_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.57/7.41 B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.57/7.41 B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.57/7.41 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.57/7.41 B_{C_1}(c_{c_1}(c_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.57/7.41 The remaining pairs can at least be oriented weakly. 25.57/7.41 Used ordering: Polynomial interpretation [POLO]: 25.57/7.41 25.57/7.41 POL(A_{B_1}(x_1)) = x_1 25.57/7.41 POL(A_{C_1}(x_1)) = x_1 25.57/7.41 POL(B_{B_1}(x_1)) = x_1 25.57/7.41 POL(B_{C_1}(x_1)) = x_1 25.57/7.41 POL(C_{A_1}(x_1)) = x_1 25.57/7.41 POL(C_{B_1}(x_1)) = x_1 25.57/7.41 POL(a_{a_1}(x_1)) = 1 + x_1 25.57/7.41 POL(a_{b_1}(x_1)) = 1 + x_1 25.57/7.41 POL(a_{c_1}(x_1)) = 1 + x_1 25.57/7.41 POL(b_{a_1}(x_1)) = x_1 25.57/7.41 POL(b_{b_1}(x_1)) = x_1 25.57/7.41 POL(b_{c_1}(x_1)) = x_1 25.57/7.41 POL(c_{a_1}(x_1)) = x_1 25.57/7.41 POL(c_{b_1}(x_1)) = 1 + x_1 25.57/7.41 POL(c_{c_1}(x_1)) = 1 + x_1 25.57/7.41 25.57/7.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.57/7.41 25.57/7.41 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 25.57/7.41 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 25.57/7.41 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 25.57/7.41 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 25.57/7.41 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 25.57/7.41 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 25.57/7.41 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 25.57/7.41 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (12) 25.57/7.41 Obligation: 25.57/7.41 Q DP problem: 25.57/7.41 The TRS P consists of the following rules: 25.57/7.41 25.57/7.41 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 25.57/7.41 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 25.57/7.41 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 25.57/7.41 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 25.57/7.41 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 25.57/7.41 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 25.57/7.41 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 25.57/7.41 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1) 25.57/7.41 C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 25.57/7.41 25.57/7.41 The TRS R consists of the following rules: 25.57/7.41 25.57/7.41 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 25.57/7.41 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 25.57/7.41 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 25.57/7.41 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 25.57/7.41 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 25.57/7.41 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 25.57/7.41 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 25.57/7.41 Q is empty. 25.57/7.41 We have to consider all minimal (P,Q,R)-chains. 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (13) DependencyGraphProof (EQUIVALENT) 25.57/7.41 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 16 less nodes. 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (14) 25.57/7.41 Complex Obligation (AND) 25.57/7.41 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (15) 25.57/7.41 Obligation: 25.57/7.41 Q DP problem: 25.57/7.41 The TRS P consists of the following rules: 25.57/7.41 25.57/7.41 C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 25.57/7.41 25.57/7.41 The TRS R consists of the following rules: 25.57/7.41 25.57/7.41 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 25.57/7.41 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 25.57/7.41 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 25.57/7.41 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 25.57/7.41 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 25.57/7.41 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 25.57/7.41 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 25.57/7.41 Q is empty. 25.57/7.41 We have to consider all minimal (P,Q,R)-chains. 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (16) UsableRulesProof (EQUIVALENT) 25.57/7.41 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (17) 25.57/7.41 Obligation: 25.57/7.41 Q DP problem: 25.57/7.41 The TRS P consists of the following rules: 25.57/7.41 25.57/7.41 C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 25.57/7.41 25.57/7.41 R is empty. 25.57/7.41 Q is empty. 25.57/7.41 We have to consider all minimal (P,Q,R)-chains. 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (18) QDPSizeChangeProof (EQUIVALENT) 25.57/7.41 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 25.57/7.41 25.57/7.41 From the DPs we obtained the following set of size-change graphs: 25.57/7.41 *C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 25.57/7.41 The graph contains the following edges 1 > 1 25.57/7.41 25.57/7.41 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (19) 25.57/7.41 YES 25.57/7.41 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (20) 25.57/7.41 Obligation: 25.57/7.41 Q DP problem: 25.57/7.41 The TRS P consists of the following rules: 25.57/7.41 25.57/7.41 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 25.57/7.41 25.57/7.41 The TRS R consists of the following rules: 25.57/7.41 25.57/7.41 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 25.57/7.41 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 25.57/7.41 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 25.57/7.41 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 25.57/7.41 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 25.57/7.41 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 25.57/7.41 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 25.57/7.41 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 25.57/7.41 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 25.57/7.41 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 25.57/7.41 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.57/7.41 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.57/7.41 25.57/7.41 Q is empty. 25.57/7.41 We have to consider all minimal (P,Q,R)-chains. 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (21) UsableRulesProof (EQUIVALENT) 25.57/7.41 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (22) 25.57/7.41 Obligation: 25.57/7.41 Q DP problem: 25.57/7.41 The TRS P consists of the following rules: 25.57/7.41 25.57/7.41 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 25.57/7.41 25.57/7.41 R is empty. 25.57/7.41 Q is empty. 25.57/7.41 We have to consider all minimal (P,Q,R)-chains. 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (23) QDPSizeChangeProof (EQUIVALENT) 25.57/7.41 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 25.57/7.41 25.57/7.41 From the DPs we obtained the following set of size-change graphs: 25.57/7.41 *A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 25.57/7.41 The graph contains the following edges 1 > 1 25.57/7.41 25.57/7.41 25.57/7.41 ---------------------------------------- 25.57/7.41 25.57/7.41 (24) 25.57/7.41 YES 25.77/7.50 EOF