25.88/7.48 YES 25.88/7.50 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 25.88/7.50 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.88/7.50 25.88/7.50 25.88/7.50 Termination w.r.t. Q of the given QTRS could be proven: 25.88/7.50 25.88/7.50 (0) QTRS 25.88/7.50 (1) QTRS Reverse [EQUIVALENT, 0 ms] 25.88/7.50 (2) QTRS 25.88/7.50 (3) FlatCCProof [EQUIVALENT, 0 ms] 25.88/7.50 (4) QTRS 25.88/7.50 (5) RootLabelingProof [EQUIVALENT, 2 ms] 25.88/7.50 (6) QTRS 25.88/7.50 (7) QTRSRRRProof [EQUIVALENT, 9 ms] 25.88/7.50 (8) QTRS 25.88/7.50 (9) DependencyPairsProof [EQUIVALENT, 35 ms] 25.88/7.50 (10) QDP 25.88/7.50 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 25.88/7.50 (12) QDP 25.88/7.50 (13) QDPOrderProof [EQUIVALENT, 144 ms] 25.88/7.50 (14) QDP 25.88/7.50 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 25.88/7.50 (16) AND 25.88/7.50 (17) QDP 25.88/7.50 (18) UsableRulesProof [EQUIVALENT, 0 ms] 25.88/7.50 (19) QDP 25.88/7.50 (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] 25.88/7.50 (21) YES 25.88/7.50 (22) QDP 25.88/7.50 (23) UsableRulesProof [EQUIVALENT, 0 ms] 25.88/7.50 (24) QDP 25.88/7.50 (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] 25.88/7.50 (26) YES 25.88/7.50 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (0) 25.88/7.50 Obligation: 25.88/7.50 Q restricted rewrite system: 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 a(x1) -> x1 25.88/7.50 a(a(x1)) -> b(x1) 25.88/7.50 b(x1) -> c(x1) 25.88/7.50 b(c(x1)) -> c(b(a(x1))) 25.88/7.50 c(x1) -> x1 25.88/7.50 25.88/7.50 Q is empty. 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (1) QTRS Reverse (EQUIVALENT) 25.88/7.50 We applied the QTRS Reverse Processor [REVERSE]. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (2) 25.88/7.50 Obligation: 25.88/7.50 Q restricted rewrite system: 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 a(x1) -> x1 25.88/7.50 a(a(x1)) -> b(x1) 25.88/7.50 b(x1) -> c(x1) 25.88/7.50 c(b(x1)) -> a(b(c(x1))) 25.88/7.50 c(x1) -> x1 25.88/7.50 25.88/7.50 Q is empty. 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (3) FlatCCProof (EQUIVALENT) 25.88/7.50 We used flat context closure [ROOTLAB] 25.88/7.50 As Q is empty the flat context closure was sound AND complete. 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (4) 25.88/7.50 Obligation: 25.88/7.50 Q restricted rewrite system: 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 a(a(x1)) -> a(x1) 25.88/7.50 b(a(x1)) -> b(x1) 25.88/7.50 c(a(x1)) -> c(x1) 25.88/7.50 a(a(a(x1))) -> a(b(x1)) 25.88/7.50 b(a(a(x1))) -> b(b(x1)) 25.88/7.50 c(a(a(x1))) -> c(b(x1)) 25.88/7.50 a(b(x1)) -> a(c(x1)) 25.88/7.50 b(b(x1)) -> b(c(x1)) 25.88/7.50 c(b(x1)) -> c(c(x1)) 25.88/7.50 a(c(b(x1))) -> a(a(b(c(x1)))) 25.88/7.50 b(c(b(x1))) -> b(a(b(c(x1)))) 25.88/7.50 c(c(b(x1))) -> c(a(b(c(x1)))) 25.88/7.50 a(c(x1)) -> a(x1) 25.88/7.50 b(c(x1)) -> b(x1) 25.88/7.50 c(c(x1)) -> c(x1) 25.88/7.50 25.88/7.50 Q is empty. 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (5) RootLabelingProof (EQUIVALENT) 25.88/7.50 We used plain root labeling [ROOTLAB] with the following heuristic: 25.88/7.50 LabelAll: All function symbols get labeled 25.88/7.50 25.88/7.50 As Q is empty the root labeling was sound AND complete. 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (6) 25.88/7.50 Obligation: 25.88/7.50 Q restricted rewrite system: 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.50 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 25.88/7.50 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.50 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.50 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.50 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.50 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.50 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1)) 25.88/7.50 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 25.88/7.50 a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1)) 25.88/7.50 b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1)) 25.88/7.50 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 25.88/7.50 b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1)) 25.88/7.50 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(x1)) 25.88/7.50 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{c_1}(x1)) 25.88/7.50 b_{b_1}(b_{a_1}(x1)) -> b_{c_1}(c_{a_1}(x1)) 25.88/7.50 b_{b_1}(b_{b_1}(x1)) -> b_{c_1}(c_{b_1}(x1)) 25.88/7.50 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(c_{c_1}(x1)) 25.88/7.50 c_{b_1}(b_{a_1}(x1)) -> c_{c_1}(c_{a_1}(x1)) 25.88/7.50 c_{b_1}(b_{b_1}(x1)) -> c_{c_1}(c_{b_1}(x1)) 25.88/7.50 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(c_{c_1}(x1)) 25.88/7.50 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.50 a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) 25.88/7.50 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.50 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.50 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.50 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.50 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.50 25.88/7.50 Q is empty. 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (7) QTRSRRRProof (EQUIVALENT) 25.88/7.50 Used ordering: 25.88/7.50 Polynomial interpretation [POLO]: 25.88/7.50 25.88/7.50 POL(a_{a_1}(x_1)) = 1 + x_1 25.88/7.50 POL(a_{b_1}(x_1)) = x_1 25.88/7.50 POL(a_{c_1}(x_1)) = 1 + x_1 25.88/7.50 POL(b_{a_1}(x_1)) = 1 + x_1 25.88/7.50 POL(b_{b_1}(x_1)) = 1 + x_1 25.88/7.50 POL(b_{c_1}(x_1)) = 1 + x_1 25.88/7.50 POL(c_{a_1}(x_1)) = x_1 25.88/7.50 POL(c_{b_1}(x_1)) = x_1 25.88/7.50 POL(c_{c_1}(x_1)) = x_1 25.88/7.50 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 25.88/7.50 25.88/7.50 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.50 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 25.88/7.50 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.50 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.50 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.50 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.50 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.50 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1)) 25.88/7.50 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 25.88/7.50 a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1)) 25.88/7.50 b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1)) 25.88/7.50 b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1)) 25.88/7.50 b_{b_1}(b_{a_1}(x1)) -> b_{c_1}(c_{a_1}(x1)) 25.88/7.50 b_{b_1}(b_{b_1}(x1)) -> b_{c_1}(c_{b_1}(x1)) 25.88/7.50 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(c_{c_1}(x1)) 25.88/7.50 c_{b_1}(b_{a_1}(x1)) -> c_{c_1}(c_{a_1}(x1)) 25.88/7.50 c_{b_1}(b_{b_1}(x1)) -> c_{c_1}(c_{b_1}(x1)) 25.88/7.50 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(c_{c_1}(x1)) 25.88/7.50 a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) 25.88/7.50 25.88/7.50 25.88/7.50 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (8) 25.88/7.50 Obligation: 25.88/7.50 Q restricted rewrite system: 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(x1)) 25.88/7.50 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{c_1}(x1)) 25.88/7.50 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.50 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.50 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.50 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.50 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.50 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.50 25.88/7.50 Q is empty. 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (9) DependencyPairsProof (EQUIVALENT) 25.88/7.50 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (10) 25.88/7.50 Obligation: 25.88/7.50 Q DP problem: 25.88/7.50 The TRS P consists of the following rules: 25.88/7.50 25.88/7.50 A_{B_1}(b_{a_1}(x1)) -> A_{C_1}(c_{a_1}(x1)) 25.88/7.50 A_{B_1}(b_{a_1}(x1)) -> C_{A_1}(x1) 25.88/7.50 A_{B_1}(b_{b_1}(x1)) -> A_{C_1}(c_{b_1}(x1)) 25.88/7.50 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(c_{c_1}(x1)) 25.88/7.50 A_{B_1}(b_{c_1}(x1)) -> C_{C_1}(x1) 25.88/7.50 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.88/7.50 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.88/7.50 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1) 25.88/7.50 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.88/7.50 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.88/7.50 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.88/7.50 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.88/7.50 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 25.88/7.50 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.88/7.50 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1) 25.88/7.50 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.88/7.50 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 25.88/7.50 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.88/7.50 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1) 25.88/7.50 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 25.88/7.50 A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 25.88/7.50 B_{C_1}(c_{a_1}(x1)) -> B_{A_1}(x1) 25.88/7.50 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 25.88/7.50 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(x1)) 25.88/7.50 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{c_1}(x1)) 25.88/7.50 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.50 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.50 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.50 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.50 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.50 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.50 25.88/7.50 Q is empty. 25.88/7.50 We have to consider all minimal (P,Q,R)-chains. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (11) DependencyGraphProof (EQUIVALENT) 25.88/7.50 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (12) 25.88/7.50 Obligation: 25.88/7.50 Q DP problem: 25.88/7.50 The TRS P consists of the following rules: 25.88/7.50 25.88/7.50 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.88/7.50 A_{B_1}(b_{a_1}(x1)) -> A_{C_1}(c_{a_1}(x1)) 25.88/7.50 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.88/7.50 A_{B_1}(b_{b_1}(x1)) -> A_{C_1}(c_{b_1}(x1)) 25.88/7.50 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.88/7.50 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(c_{c_1}(x1)) 25.88/7.50 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.88/7.50 A_{B_1}(b_{c_1}(x1)) -> C_{C_1}(x1) 25.88/7.50 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 25.88/7.50 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.88/7.50 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 25.88/7.50 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.88/7.50 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.88/7.50 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 25.88/7.50 A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 25.88/7.50 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(x1)) 25.88/7.50 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{c_1}(x1)) 25.88/7.50 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.50 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.50 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.50 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.50 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.50 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.50 25.88/7.50 Q is empty. 25.88/7.50 We have to consider all minimal (P,Q,R)-chains. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (13) QDPOrderProof (EQUIVALENT) 25.88/7.50 We use the reduction pair processor [LPAR04,JAR06]. 25.88/7.50 25.88/7.50 25.88/7.50 The following pairs can be oriented strictly and are deleted. 25.88/7.50 25.88/7.50 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.88/7.50 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.88/7.50 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.88/7.50 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.88/7.50 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 25.88/7.50 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.88/7.50 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 25.88/7.50 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 25.88/7.50 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 25.88/7.50 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 25.88/7.50 The remaining pairs can at least be oriented weakly. 25.88/7.50 Used ordering: Polynomial interpretation [POLO]: 25.88/7.50 25.88/7.50 POL(A_{B_1}(x_1)) = x_1 25.88/7.50 POL(A_{C_1}(x_1)) = x_1 25.88/7.50 POL(B_{C_1}(x_1)) = x_1 25.88/7.50 POL(C_{C_1}(x_1)) = x_1 25.88/7.50 POL(a_{a_1}(x_1)) = 1 + x_1 25.88/7.50 POL(a_{b_1}(x_1)) = 1 + x_1 25.88/7.50 POL(a_{c_1}(x_1)) = 1 + x_1 25.88/7.50 POL(b_{a_1}(x_1)) = x_1 25.88/7.50 POL(b_{b_1}(x_1)) = 1 + x_1 25.88/7.50 POL(b_{c_1}(x_1)) = x_1 25.88/7.50 POL(c_{a_1}(x_1)) = x_1 25.88/7.50 POL(c_{b_1}(x_1)) = 1 + x_1 25.88/7.50 POL(c_{c_1}(x_1)) = x_1 25.88/7.50 25.88/7.50 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.88/7.50 25.88/7.50 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 25.88/7.50 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.50 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.50 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.50 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.50 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(x1)) 25.88/7.50 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{c_1}(x1)) 25.88/7.50 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.50 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 25.88/7.50 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.50 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (14) 25.88/7.50 Obligation: 25.88/7.50 Q DP problem: 25.88/7.50 The TRS P consists of the following rules: 25.88/7.50 25.88/7.50 A_{B_1}(b_{a_1}(x1)) -> A_{C_1}(c_{a_1}(x1)) 25.88/7.50 A_{B_1}(b_{b_1}(x1)) -> A_{C_1}(c_{b_1}(x1)) 25.88/7.50 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(c_{c_1}(x1)) 25.88/7.50 A_{B_1}(b_{c_1}(x1)) -> C_{C_1}(x1) 25.88/7.50 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 25.88/7.50 A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 25.88/7.50 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(x1)) 25.88/7.50 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{c_1}(x1)) 25.88/7.50 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.50 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.50 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.50 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.50 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.50 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.50 25.88/7.50 Q is empty. 25.88/7.50 We have to consider all minimal (P,Q,R)-chains. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (15) DependencyGraphProof (EQUIVALENT) 25.88/7.50 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (16) 25.88/7.50 Complex Obligation (AND) 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (17) 25.88/7.50 Obligation: 25.88/7.50 Q DP problem: 25.88/7.50 The TRS P consists of the following rules: 25.88/7.50 25.88/7.50 A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 25.88/7.50 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(x1)) 25.88/7.50 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{c_1}(x1)) 25.88/7.50 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.50 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.50 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.50 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.50 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.50 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.50 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.50 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.50 25.88/7.50 Q is empty. 25.88/7.50 We have to consider all minimal (P,Q,R)-chains. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (18) UsableRulesProof (EQUIVALENT) 25.88/7.50 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. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (19) 25.88/7.50 Obligation: 25.88/7.50 Q DP problem: 25.88/7.50 The TRS P consists of the following rules: 25.88/7.50 25.88/7.50 A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 25.88/7.50 25.88/7.50 R is empty. 25.88/7.50 Q is empty. 25.88/7.50 We have to consider all minimal (P,Q,R)-chains. 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (20) QDPSizeChangeProof (EQUIVALENT) 25.88/7.50 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. 25.88/7.50 25.88/7.50 From the DPs we obtained the following set of size-change graphs: 25.88/7.50 *A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 25.88/7.50 The graph contains the following edges 1 > 1 25.88/7.50 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (21) 25.88/7.50 YES 25.88/7.50 25.88/7.50 ---------------------------------------- 25.88/7.50 25.88/7.50 (22) 25.88/7.50 Obligation: 25.88/7.50 Q DP problem: 25.88/7.50 The TRS P consists of the following rules: 25.88/7.50 25.88/7.50 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 25.88/7.50 25.88/7.50 The TRS R consists of the following rules: 25.88/7.50 25.88/7.50 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.50 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.50 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 25.88/7.50 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 25.88/7.50 a_{b_1}(b_{a_1}(x1)) -> a_{c_1}(c_{a_1}(x1)) 25.88/7.50 a_{b_1}(b_{b_1}(x1)) -> a_{c_1}(c_{b_1}(x1)) 25.88/7.51 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(c_{c_1}(x1)) 25.88/7.51 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.51 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.51 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.51 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.51 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.51 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.51 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 25.88/7.51 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 25.88/7.51 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 25.88/7.51 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 25.88/7.51 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 25.88/7.51 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 25.88/7.51 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 25.88/7.51 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 25.88/7.51 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 25.88/7.51 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 25.88/7.51 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 25.88/7.51 25.88/7.51 Q is empty. 25.88/7.51 We have to consider all minimal (P,Q,R)-chains. 25.88/7.51 ---------------------------------------- 25.88/7.51 25.88/7.51 (23) UsableRulesProof (EQUIVALENT) 25.88/7.51 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. 25.88/7.51 ---------------------------------------- 25.88/7.51 25.88/7.51 (24) 25.88/7.51 Obligation: 25.88/7.51 Q DP problem: 25.88/7.51 The TRS P consists of the following rules: 25.88/7.51 25.88/7.51 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 25.88/7.51 25.88/7.51 R is empty. 25.88/7.51 Q is empty. 25.88/7.51 We have to consider all minimal (P,Q,R)-chains. 25.88/7.51 ---------------------------------------- 25.88/7.51 25.88/7.51 (25) QDPSizeChangeProof (EQUIVALENT) 25.88/7.51 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. 25.88/7.51 25.88/7.51 From the DPs we obtained the following set of size-change graphs: 25.88/7.51 *B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 25.88/7.51 The graph contains the following edges 1 > 1 25.88/7.51 25.88/7.51 25.88/7.51 ---------------------------------------- 25.88/7.51 25.88/7.51 (26) 25.88/7.51 YES 26.16/7.58 EOF