30.31/8.74 YES 30.31/8.75 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 30.31/8.75 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.31/8.75 30.31/8.75 30.31/8.75 Termination w.r.t. Q of the given QTRS could be proven: 30.31/8.75 30.31/8.75 (0) QTRS 30.31/8.75 (1) QTRS Reverse [EQUIVALENT, 0 ms] 30.31/8.75 (2) QTRS 30.31/8.75 (3) FlatCCProof [EQUIVALENT, 0 ms] 30.31/8.75 (4) QTRS 30.31/8.75 (5) RootLabelingProof [EQUIVALENT, 1 ms] 30.31/8.75 (6) QTRS 30.31/8.75 (7) QTRSRRRProof [EQUIVALENT, 73 ms] 30.31/8.75 (8) QTRS 30.31/8.75 (9) DependencyPairsProof [EQUIVALENT, 35 ms] 30.31/8.75 (10) QDP 30.31/8.75 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 30.31/8.75 (12) QDP 30.31/8.75 (13) QDPOrderProof [EQUIVALENT, 131 ms] 30.31/8.75 (14) QDP 30.31/8.75 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 30.31/8.75 (16) AND 30.31/8.75 (17) QDP 30.31/8.75 (18) UsableRulesProof [EQUIVALENT, 0 ms] 30.31/8.75 (19) QDP 30.31/8.75 (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.31/8.75 (21) YES 30.31/8.75 (22) QDP 30.31/8.75 (23) UsableRulesProof [EQUIVALENT, 0 ms] 30.31/8.75 (24) QDP 30.31/8.75 (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.31/8.75 (26) YES 30.31/8.75 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (0) 30.31/8.75 Obligation: 30.31/8.75 Q restricted rewrite system: 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 a(x1) -> b(x1) 30.31/8.75 a(b(x1)) -> b(a(c(x1))) 30.31/8.75 b(b(x1)) -> x1 30.31/8.75 c(c(x1)) -> a(x1) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (1) QTRS Reverse (EQUIVALENT) 30.31/8.75 We applied the QTRS Reverse Processor [REVERSE]. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (2) 30.31/8.75 Obligation: 30.31/8.75 Q restricted rewrite system: 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 a(x1) -> b(x1) 30.31/8.75 b(a(x1)) -> c(a(b(x1))) 30.31/8.75 b(b(x1)) -> x1 30.31/8.75 c(c(x1)) -> a(x1) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (3) FlatCCProof (EQUIVALENT) 30.31/8.75 We used flat context closure [ROOTLAB] 30.31/8.75 As Q is empty the flat context closure was sound AND complete. 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (4) 30.31/8.75 Obligation: 30.31/8.75 Q restricted rewrite system: 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 a(a(x1)) -> a(b(x1)) 30.31/8.75 b(a(x1)) -> b(b(x1)) 30.31/8.75 c(a(x1)) -> c(b(x1)) 30.31/8.75 a(b(a(x1))) -> a(c(a(b(x1)))) 30.31/8.75 b(b(a(x1))) -> b(c(a(b(x1)))) 30.31/8.75 c(b(a(x1))) -> c(c(a(b(x1)))) 30.31/8.75 a(b(b(x1))) -> a(x1) 30.31/8.75 b(b(b(x1))) -> b(x1) 30.31/8.75 c(b(b(x1))) -> c(x1) 30.31/8.75 a(c(c(x1))) -> a(a(x1)) 30.31/8.75 b(c(c(x1))) -> b(a(x1)) 30.31/8.75 c(c(c(x1))) -> c(a(x1)) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (5) RootLabelingProof (EQUIVALENT) 30.31/8.75 We used plain root labeling [ROOTLAB] with the following heuristic: 30.31/8.75 LabelAll: All function symbols get labeled 30.31/8.75 30.31/8.75 As Q is empty the root labeling was sound AND complete. 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (6) 30.31/8.75 Obligation: 30.31/8.75 Q restricted rewrite system: 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 30.31/8.75 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 30.31/8.75 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1)) 30.31/8.75 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 30.31/8.75 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 30.31/8.75 b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1)) 30.31/8.75 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 30.31/8.75 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 30.31/8.75 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 30.31/8.75 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.31/8.75 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 30.31/8.75 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 30.31/8.75 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 30.31/8.75 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 30.31/8.75 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 30.31/8.75 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (7) QTRSRRRProof (EQUIVALENT) 30.31/8.75 Used ordering: 30.31/8.75 Polynomial interpretation [POLO]: 30.31/8.75 30.31/8.75 POL(a_{a_1}(x_1)) = 1 + x_1 30.31/8.75 POL(a_{b_1}(x_1)) = 1 + x_1 30.31/8.75 POL(a_{c_1}(x_1)) = 1 + x_1 30.31/8.75 POL(b_{a_1}(x_1)) = x_1 30.31/8.75 POL(b_{b_1}(x_1)) = x_1 30.31/8.75 POL(b_{c_1}(x_1)) = x_1 30.31/8.75 POL(c_{a_1}(x_1)) = x_1 30.31/8.75 POL(c_{b_1}(x_1)) = 1 + x_1 30.31/8.75 POL(c_{c_1}(x_1)) = 1 + x_1 30.31/8.75 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 30.31/8.75 30.31/8.75 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 30.31/8.75 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 30.31/8.75 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1)) 30.31/8.75 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 30.31/8.75 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 30.31/8.75 b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1)) 30.31/8.75 c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(x1) 30.31/8.75 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 30.31/8.75 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 30.31/8.75 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 30.31/8.75 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 30.31/8.75 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 30.31/8.75 30.31/8.75 30.31/8.75 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (8) 30.31/8.75 Obligation: 30.31/8.75 Q restricted rewrite system: 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 30.31/8.75 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 30.31/8.75 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 30.31/8.75 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.31/8.75 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (9) DependencyPairsProof (EQUIVALENT) 30.31/8.75 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (10) 30.31/8.75 Obligation: 30.31/8.75 Q DP problem: 30.31/8.75 The TRS P consists of the following rules: 30.31/8.75 30.31/8.75 C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) 30.31/8.75 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1)) 30.31/8.75 C_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1) 30.31/8.75 C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) 30.31/8.75 C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1) 30.31/8.75 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 30.31/8.75 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.31/8.75 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 30.31/8.75 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 30.31/8.75 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.31/8.75 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 30.31/8.75 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 30.31/8.75 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 30.31/8.75 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 30.31/8.75 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 30.31/8.75 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 30.31/8.75 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 30.31/8.75 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 30.31/8.75 A_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1) 30.31/8.75 A_{B_1}(b_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 30.31/8.75 C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{B_1}(x1) 30.31/8.75 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 30.31/8.75 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 30.31/8.75 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 30.31/8.75 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.31/8.75 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 We have to consider all minimal (P,Q,R)-chains. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (11) DependencyGraphProof (EQUIVALENT) 30.31/8.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (12) 30.31/8.75 Obligation: 30.31/8.75 Q DP problem: 30.31/8.75 The TRS P consists of the following rules: 30.31/8.75 30.31/8.75 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 30.31/8.75 C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) 30.31/8.75 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 30.31/8.75 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1)) 30.31/8.75 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 30.31/8.75 C_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1) 30.31/8.75 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 30.31/8.75 C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) 30.31/8.75 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 30.31/8.75 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 30.31/8.75 A_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 30.31/8.75 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 30.31/8.75 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 30.31/8.75 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 30.31/8.75 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 30.31/8.75 C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{B_1}(x1) 30.31/8.75 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 30.31/8.75 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 30.31/8.75 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 30.31/8.75 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.31/8.75 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 We have to consider all minimal (P,Q,R)-chains. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (13) QDPOrderProof (EQUIVALENT) 30.31/8.75 We use the reduction pair processor [LPAR04,JAR06]. 30.31/8.75 30.31/8.75 30.31/8.75 The following pairs can be oriented strictly and are deleted. 30.31/8.75 30.31/8.75 C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 30.31/8.75 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1)) 30.31/8.75 C_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1) 30.31/8.75 C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 30.31/8.75 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 30.31/8.75 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 30.31/8.75 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 30.31/8.75 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 30.31/8.75 The remaining pairs can at least be oriented weakly. 30.31/8.75 Used ordering: Polynomial interpretation [POLO]: 30.31/8.75 30.31/8.75 POL(A_{B_1}(x_1)) = 1 + x_1 30.31/8.75 POL(B_{B_1}(x_1)) = x_1 30.31/8.75 POL(C_{A_1}(x_1)) = x_1 30.31/8.75 POL(C_{B_1}(x_1)) = x_1 30.31/8.75 POL(a_{a_1}(x_1)) = 1 + x_1 30.31/8.75 POL(a_{b_1}(x_1)) = 1 + x_1 30.31/8.75 POL(a_{c_1}(x_1)) = 1 + x_1 30.31/8.75 POL(b_{a_1}(x_1)) = x_1 30.31/8.75 POL(b_{b_1}(x_1)) = x_1 30.31/8.75 POL(b_{c_1}(x_1)) = x_1 30.31/8.75 POL(c_{a_1}(x_1)) = x_1 30.31/8.75 POL(c_{b_1}(x_1)) = 1 + x_1 30.31/8.75 POL(c_{c_1}(x_1)) = 1 + x_1 30.31/8.75 30.31/8.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 30.31/8.75 30.31/8.75 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.31/8.75 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 30.31/8.75 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 30.31/8.75 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 30.31/8.75 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 30.31/8.75 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.31/8.75 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 30.31/8.75 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.31/8.75 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (14) 30.31/8.75 Obligation: 30.31/8.75 Q DP problem: 30.31/8.75 The TRS P consists of the following rules: 30.31/8.75 30.31/8.75 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 30.31/8.75 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 30.31/8.75 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 30.31/8.75 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 30.31/8.75 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 30.31/8.75 A_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 30.31/8.75 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 30.31/8.75 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 30.31/8.75 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 30.31/8.75 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 30.31/8.75 C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{B_1}(x1) 30.31/8.75 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 30.31/8.75 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 30.31/8.75 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 30.31/8.75 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.31/8.75 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 We have to consider all minimal (P,Q,R)-chains. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (15) DependencyGraphProof (EQUIVALENT) 30.31/8.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 12 less nodes. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (16) 30.31/8.75 Complex Obligation (AND) 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (17) 30.31/8.75 Obligation: 30.31/8.75 Q DP problem: 30.31/8.75 The TRS P consists of the following rules: 30.31/8.75 30.31/8.75 A_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1) 30.31/8.75 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 30.31/8.75 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 30.31/8.75 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 30.31/8.75 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.31/8.75 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 We have to consider all minimal (P,Q,R)-chains. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (18) UsableRulesProof (EQUIVALENT) 30.31/8.75 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. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (19) 30.31/8.75 Obligation: 30.31/8.75 Q DP problem: 30.31/8.75 The TRS P consists of the following rules: 30.31/8.75 30.31/8.75 A_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1) 30.31/8.75 30.31/8.75 R is empty. 30.31/8.75 Q is empty. 30.31/8.75 We have to consider all minimal (P,Q,R)-chains. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (20) QDPSizeChangeProof (EQUIVALENT) 30.31/8.75 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. 30.31/8.75 30.31/8.75 From the DPs we obtained the following set of size-change graphs: 30.31/8.75 *A_{B_1}(b_{b_1}(b_{b_1}(x1))) -> A_{B_1}(x1) 30.31/8.75 The graph contains the following edges 1 > 1 30.31/8.75 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (21) 30.31/8.75 YES 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (22) 30.31/8.75 Obligation: 30.31/8.75 Q DP problem: 30.31/8.75 The TRS P consists of the following rules: 30.31/8.75 30.31/8.75 C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{B_1}(x1) 30.31/8.75 30.31/8.75 The TRS R consists of the following rules: 30.31/8.75 30.31/8.75 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 30.31/8.75 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 30.31/8.75 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 30.31/8.75 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 30.31/8.75 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 30.31/8.75 a_{b_1}(b_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 30.31/8.75 a_{b_1}(b_{b_1}(b_{c_1}(x1))) -> a_{c_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 30.31/8.75 b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{b_1}(x1) 30.31/8.75 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(x1) 30.31/8.75 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 30.31/8.75 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 30.31/8.75 30.31/8.75 Q is empty. 30.31/8.75 We have to consider all minimal (P,Q,R)-chains. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (23) UsableRulesProof (EQUIVALENT) 30.31/8.75 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. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (24) 30.31/8.75 Obligation: 30.31/8.75 Q DP problem: 30.31/8.75 The TRS P consists of the following rules: 30.31/8.75 30.31/8.75 C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{B_1}(x1) 30.31/8.75 30.31/8.75 R is empty. 30.31/8.75 Q is empty. 30.31/8.75 We have to consider all minimal (P,Q,R)-chains. 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (25) QDPSizeChangeProof (EQUIVALENT) 30.31/8.75 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. 30.31/8.75 30.31/8.75 From the DPs we obtained the following set of size-change graphs: 30.31/8.75 *C_{B_1}(b_{b_1}(b_{b_1}(x1))) -> C_{B_1}(x1) 30.31/8.75 The graph contains the following edges 1 > 1 30.31/8.75 30.31/8.75 30.31/8.75 ---------------------------------------- 30.31/8.75 30.31/8.75 (26) 30.31/8.75 YES 30.52/8.82 EOF