26.65/7.72 YES 29.40/9.22 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 29.40/9.22 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.40/9.22 29.40/9.22 29.40/9.22 Termination w.r.t. Q of the given QTRS could be proven: 29.40/9.22 29.40/9.22 (0) QTRS 29.40/9.22 (1) FlatCCProof [EQUIVALENT, 0 ms] 29.40/9.22 (2) QTRS 29.40/9.22 (3) RootLabelingProof [EQUIVALENT, 0 ms] 29.40/9.22 (4) QTRS 29.40/9.22 (5) QTRSRRRProof [EQUIVALENT, 28 ms] 29.40/9.22 (6) QTRS 29.40/9.22 (7) DependencyPairsProof [EQUIVALENT, 13 ms] 29.40/9.22 (8) QDP 29.40/9.22 (9) QDPOrderProof [EQUIVALENT, 176 ms] 29.40/9.22 (10) QDP 29.40/9.22 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 29.40/9.22 (12) AND 29.40/9.22 (13) QDP 29.40/9.22 (14) UsableRulesProof [EQUIVALENT, 0 ms] 29.40/9.22 (15) QDP 29.40/9.22 (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] 29.40/9.22 (17) YES 29.40/9.22 (18) QDP 29.40/9.22 (19) UsableRulesProof [EQUIVALENT, 0 ms] 29.40/9.22 (20) QDP 29.40/9.22 (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] 29.40/9.22 (22) YES 29.40/9.22 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (0) 29.40/9.22 Obligation: 29.40/9.22 Q restricted rewrite system: 29.40/9.22 The TRS R consists of the following rules: 29.40/9.22 29.40/9.22 a(x1) -> x1 29.40/9.22 a(a(x1)) -> b(c(x1)) 29.40/9.22 b(x1) -> x1 29.40/9.22 c(x1) -> x1 29.40/9.22 c(b(x1)) -> a(b(c(x1))) 29.40/9.22 29.40/9.22 Q is empty. 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (1) FlatCCProof (EQUIVALENT) 29.40/9.22 We used flat context closure [ROOTLAB] 29.40/9.22 As Q is empty the flat context closure was sound AND complete. 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (2) 29.40/9.22 Obligation: 29.40/9.22 Q restricted rewrite system: 29.40/9.22 The TRS R consists of the following rules: 29.40/9.22 29.40/9.22 a(a(x1)) -> a(x1) 29.40/9.22 b(a(x1)) -> b(x1) 29.40/9.22 c(a(x1)) -> c(x1) 29.40/9.22 a(a(a(x1))) -> a(b(c(x1))) 29.40/9.22 b(a(a(x1))) -> b(b(c(x1))) 29.40/9.22 c(a(a(x1))) -> c(b(c(x1))) 29.40/9.22 a(b(x1)) -> a(x1) 29.40/9.22 b(b(x1)) -> b(x1) 29.40/9.22 c(b(x1)) -> c(x1) 29.40/9.22 a(c(x1)) -> a(x1) 29.40/9.22 b(c(x1)) -> b(x1) 29.40/9.22 c(c(x1)) -> c(x1) 29.40/9.22 a(c(b(x1))) -> a(a(b(c(x1)))) 29.40/9.22 b(c(b(x1))) -> b(a(b(c(x1)))) 29.40/9.22 c(c(b(x1))) -> c(a(b(c(x1)))) 29.40/9.22 29.40/9.22 Q is empty. 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (3) RootLabelingProof (EQUIVALENT) 29.40/9.22 We used plain root labeling [ROOTLAB] with the following heuristic: 29.40/9.22 LabelAll: All function symbols get labeled 29.40/9.22 29.40/9.22 As Q is empty the root labeling was sound AND complete. 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (4) 29.40/9.22 Obligation: 29.40/9.22 Q restricted rewrite system: 29.40/9.22 The TRS R consists of the following rules: 29.40/9.22 29.40/9.22 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 29.40/9.22 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 29.40/9.22 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) 29.40/9.22 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 29.40/9.22 Q is empty. 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (5) QTRSRRRProof (EQUIVALENT) 29.40/9.22 Used ordering: 29.40/9.22 Polynomial interpretation [POLO]: 29.40/9.22 29.40/9.22 POL(a_{a_1}(x_1)) = 1 + x_1 29.40/9.22 POL(a_{b_1}(x_1)) = x_1 29.40/9.22 POL(a_{c_1}(x_1)) = 1 + x_1 29.40/9.22 POL(b_{a_1}(x_1)) = 1 + x_1 29.40/9.22 POL(b_{b_1}(x_1)) = 1 + x_1 29.40/9.22 POL(b_{c_1}(x_1)) = 1 + x_1 29.40/9.22 POL(c_{a_1}(x_1)) = x_1 29.40/9.22 POL(c_{b_1}(x_1)) = x_1 29.40/9.22 POL(c_{c_1}(x_1)) = x_1 29.40/9.22 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 29.40/9.22 29.40/9.22 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 29.40/9.22 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 b_{a_1}(a_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 29.40/9.22 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1) 29.40/9.22 29.40/9.22 29.40/9.22 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (6) 29.40/9.22 Obligation: 29.40/9.22 Q restricted rewrite system: 29.40/9.22 The TRS R consists of the following rules: 29.40/9.22 29.40/9.22 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 29.40/9.22 Q is empty. 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (7) DependencyPairsProof (EQUIVALENT) 29.40/9.22 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (8) 29.40/9.22 Obligation: 29.40/9.22 Q DP problem: 29.40/9.22 The TRS P consists of the following rules: 29.40/9.22 29.40/9.22 B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 C_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1) 29.40/9.22 A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 29.40/9.22 B_{C_1}(c_{a_1}(x1)) -> B_{A_1}(x1) 29.40/9.22 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 29.40/9.22 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 29.40/9.22 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1) 29.40/9.22 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 29.40/9.22 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 29.40/9.22 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 29.40/9.22 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1) 29.40/9.22 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 29.40/9.22 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 29.40/9.22 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 29.40/9.22 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1) 29.40/9.22 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 29.40/9.22 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 29.40/9.22 29.40/9.22 The TRS R consists of the following rules: 29.40/9.22 29.40/9.22 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 29.40/9.22 Q is empty. 29.40/9.22 We have to consider all minimal (P,Q,R)-chains. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (9) QDPOrderProof (EQUIVALENT) 29.40/9.22 We use the reduction pair processor [LPAR04,JAR06]. 29.40/9.22 29.40/9.22 29.40/9.22 The following pairs can be oriented strictly and are deleted. 29.40/9.22 29.40/9.22 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1) 29.40/9.22 B_{C_1}(c_{a_1}(x1)) -> B_{A_1}(x1) 29.40/9.22 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1) 29.40/9.22 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 29.40/9.22 B_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1) 29.40/9.22 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 29.40/9.22 B_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 29.40/9.22 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 29.40/9.22 C_{C_1}(c_{b_1}(b_{a_1}(x1))) -> C_{A_1}(x1) 29.40/9.22 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 29.40/9.22 C_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 29.40/9.22 The remaining pairs can at least be oriented weakly. 29.40/9.22 Used ordering: Polynomial interpretation [POLO]: 29.40/9.22 29.40/9.22 POL(A_{B_1}(x_1)) = 1 + x_1 29.40/9.22 POL(A_{C_1}(x_1)) = x_1 29.40/9.22 POL(B_{A_1}(x_1)) = 1 + x_1 29.40/9.22 POL(B_{C_1}(x_1)) = 1 + x_1 29.40/9.22 POL(C_{A_1}(x_1)) = 1 + x_1 29.40/9.22 POL(C_{C_1}(x_1)) = 1 + x_1 29.40/9.22 POL(a_{a_1}(x_1)) = 1 + x_1 29.40/9.22 POL(a_{b_1}(x_1)) = x_1 29.40/9.22 POL(a_{c_1}(x_1)) = x_1 29.40/9.22 POL(b_{a_1}(x_1)) = 1 + x_1 29.40/9.22 POL(b_{b_1}(x_1)) = 1 + x_1 29.40/9.22 POL(b_{c_1}(x_1)) = x_1 29.40/9.22 POL(c_{a_1}(x_1)) = 1 + x_1 29.40/9.22 POL(c_{b_1}(x_1)) = 1 + x_1 29.40/9.22 POL(c_{c_1}(x_1)) = x_1 29.40/9.22 29.40/9.22 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.40/9.22 29.40/9.22 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (10) 29.40/9.22 Obligation: 29.40/9.22 Q DP problem: 29.40/9.22 The TRS P consists of the following rules: 29.40/9.22 29.40/9.22 B_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 C_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 29.40/9.22 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 29.40/9.22 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> A_{B_1}(b_{c_1}(c_{a_1}(x1))) 29.40/9.22 A_{C_1}(c_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{a_1}(x1)) 29.40/9.22 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> A_{B_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 A_{C_1}(c_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(x1)) 29.40/9.22 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> A_{B_1}(b_{c_1}(c_{c_1}(x1))) 29.40/9.22 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{c_1}(x1)) 29.40/9.22 A_{C_1}(c_{b_1}(b_{c_1}(x1))) -> C_{C_1}(x1) 29.40/9.22 29.40/9.22 The TRS R consists of the following rules: 29.40/9.22 29.40/9.22 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 29.40/9.22 Q is empty. 29.40/9.22 We have to consider all minimal (P,Q,R)-chains. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (11) DependencyGraphProof (EQUIVALENT) 29.40/9.22 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 9 less nodes. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (12) 29.40/9.22 Complex Obligation (AND) 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (13) 29.40/9.22 Obligation: 29.40/9.22 Q DP problem: 29.40/9.22 The TRS P consists of the following rules: 29.40/9.22 29.40/9.22 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 29.40/9.22 29.40/9.22 The TRS R consists of the following rules: 29.40/9.22 29.40/9.22 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 29.40/9.22 Q is empty. 29.40/9.22 We have to consider all minimal (P,Q,R)-chains. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (14) UsableRulesProof (EQUIVALENT) 29.40/9.22 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. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (15) 29.40/9.22 Obligation: 29.40/9.22 Q DP problem: 29.40/9.22 The TRS P consists of the following rules: 29.40/9.22 29.40/9.22 B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 29.40/9.22 29.40/9.22 R is empty. 29.40/9.22 Q is empty. 29.40/9.22 We have to consider all minimal (P,Q,R)-chains. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (16) QDPSizeChangeProof (EQUIVALENT) 29.40/9.22 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. 29.40/9.22 29.40/9.22 From the DPs we obtained the following set of size-change graphs: 29.40/9.22 *B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1) 29.40/9.22 The graph contains the following edges 1 > 1 29.40/9.22 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (17) 29.40/9.22 YES 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (18) 29.40/9.22 Obligation: 29.40/9.22 Q DP problem: 29.40/9.22 The TRS P consists of the following rules: 29.40/9.22 29.40/9.22 A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 29.40/9.22 29.40/9.22 The TRS R consists of the following rules: 29.40/9.22 29.40/9.22 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{c_1}(c_{b_1}(x1))) 29.40/9.22 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1) 29.40/9.22 a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1) 29.40/9.22 b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1) 29.40/9.22 b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1) 29.40/9.22 b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1) 29.40/9.22 c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1) 29.40/9.22 c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1) 29.40/9.22 c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1) 29.40/9.22 a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 29.40/9.22 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 29.40/9.22 29.40/9.22 Q is empty. 29.40/9.22 We have to consider all minimal (P,Q,R)-chains. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (19) UsableRulesProof (EQUIVALENT) 29.40/9.22 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. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (20) 29.40/9.22 Obligation: 29.40/9.22 Q DP problem: 29.40/9.22 The TRS P consists of the following rules: 29.40/9.22 29.40/9.22 A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 29.40/9.22 29.40/9.22 R is empty. 29.40/9.22 Q is empty. 29.40/9.22 We have to consider all minimal (P,Q,R)-chains. 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (21) QDPSizeChangeProof (EQUIVALENT) 29.40/9.22 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. 29.40/9.22 29.40/9.22 From the DPs we obtained the following set of size-change graphs: 29.40/9.22 *A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1) 29.40/9.22 The graph contains the following edges 1 > 1 29.40/9.22 29.40/9.22 29.40/9.22 ---------------------------------------- 29.40/9.22 29.40/9.22 (22) 29.40/9.22 YES 29.56/9.35 EOF