27.75/8.06 YES 28.99/8.59 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 28.99/8.59 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.99/8.59 28.99/8.59 28.99/8.59 Termination w.r.t. Q of the given QTRS could be proven: 28.99/8.59 28.99/8.59 (0) QTRS 28.99/8.59 (1) QTRS Reverse [EQUIVALENT, 0 ms] 28.99/8.59 (2) QTRS 28.99/8.59 (3) FlatCCProof [EQUIVALENT, 0 ms] 28.99/8.59 (4) QTRS 28.99/8.59 (5) RootLabelingProof [EQUIVALENT, 0 ms] 28.99/8.59 (6) QTRS 28.99/8.59 (7) QTRSRRRProof [EQUIVALENT, 68 ms] 28.99/8.59 (8) QTRS 28.99/8.59 (9) DependencyPairsProof [EQUIVALENT, 40 ms] 28.99/8.59 (10) QDP 28.99/8.59 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 28.99/8.59 (12) QDP 28.99/8.59 (13) QDPOrderProof [EQUIVALENT, 82 ms] 28.99/8.59 (14) QDP 28.99/8.59 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 28.99/8.59 (16) AND 28.99/8.59 (17) QDP 28.99/8.59 (18) UsableRulesProof [EQUIVALENT, 0 ms] 28.99/8.59 (19) QDP 28.99/8.59 (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.99/8.59 (21) YES 28.99/8.59 (22) QDP 28.99/8.59 (23) UsableRulesProof [EQUIVALENT, 0 ms] 28.99/8.59 (24) QDP 28.99/8.59 (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] 28.99/8.59 (26) YES 28.99/8.59 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (0) 28.99/8.59 Obligation: 28.99/8.59 Q restricted rewrite system: 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 a(x1) -> x1 28.99/8.59 a(x1) -> b(x1) 28.99/8.59 a(b(x1)) -> b(a(c(x1))) 28.99/8.59 b(x1) -> x1 28.99/8.59 c(c(x1)) -> a(x1) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (1) QTRS Reverse (EQUIVALENT) 28.99/8.59 We applied the QTRS Reverse Processor [REVERSE]. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (2) 28.99/8.59 Obligation: 28.99/8.59 Q restricted rewrite system: 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 a(x1) -> x1 28.99/8.59 a(x1) -> b(x1) 28.99/8.59 b(a(x1)) -> c(a(b(x1))) 28.99/8.59 b(x1) -> x1 28.99/8.59 c(c(x1)) -> a(x1) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (3) FlatCCProof (EQUIVALENT) 28.99/8.59 We used flat context closure [ROOTLAB] 28.99/8.59 As Q is empty the flat context closure was sound AND complete. 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (4) 28.99/8.59 Obligation: 28.99/8.59 Q restricted rewrite system: 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 a(a(x1)) -> a(x1) 28.99/8.59 b(a(x1)) -> b(x1) 28.99/8.59 c(a(x1)) -> c(x1) 28.99/8.59 a(a(x1)) -> a(b(x1)) 28.99/8.59 b(a(x1)) -> b(b(x1)) 28.99/8.59 c(a(x1)) -> c(b(x1)) 28.99/8.59 a(b(a(x1))) -> a(c(a(b(x1)))) 28.99/8.59 b(b(a(x1))) -> b(c(a(b(x1)))) 28.99/8.59 c(b(a(x1))) -> c(c(a(b(x1)))) 28.99/8.59 a(b(x1)) -> a(x1) 28.99/8.59 b(b(x1)) -> b(x1) 28.99/8.59 c(b(x1)) -> c(x1) 28.99/8.59 a(c(c(x1))) -> a(a(x1)) 28.99/8.59 b(c(c(x1))) -> b(a(x1)) 28.99/8.59 c(c(c(x1))) -> c(a(x1)) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (5) RootLabelingProof (EQUIVALENT) 28.99/8.59 We used plain root labeling [ROOTLAB] with the following heuristic: 28.99/8.59 LabelAll: All function symbols get labeled 28.99/8.59 28.99/8.59 As Q is empty the root labeling was sound AND complete. 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (6) 28.99/8.59 Obligation: 28.99/8.59 Q restricted rewrite system: 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 28.99/8.59 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 28.99/8.59 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1)) 28.99/8.59 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 28.99/8.59 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 28.99/8.59 b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1)) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 28.99/8.59 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) 28.99/8.59 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 28.99/8.59 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 28.99/8.59 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 28.99/8.59 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 28.99/8.59 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 28.99/8.59 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (7) QTRSRRRProof (EQUIVALENT) 28.99/8.59 Used ordering: 28.99/8.59 Polynomial interpretation [POLO]: 28.99/8.59 28.99/8.59 POL(a_{a_1}(x_1)) = 1 + x_1 28.99/8.59 POL(a_{b_1}(x_1)) = x_1 28.99/8.59 POL(a_{c_1}(x_1)) = 1 + x_1 28.99/8.59 POL(b_{a_1}(x_1)) = 1 + x_1 28.99/8.59 POL(b_{b_1}(x_1)) = x_1 28.99/8.59 POL(b_{c_1}(x_1)) = 1 + x_1 28.99/8.59 POL(c_{a_1}(x_1)) = x_1 28.99/8.59 POL(c_{b_1}(x_1)) = x_1 28.99/8.59 POL(c_{c_1}(x_1)) = 1 + x_1 28.99/8.59 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 28.99/8.59 28.99/8.59 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 28.99/8.59 a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) 28.99/8.59 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) 28.99/8.59 a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(x1)) 28.99/8.59 b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) 28.99/8.59 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) 28.99/8.59 b_{a_1}(a_{c_1}(x1)) -> b_{b_1}(b_{c_1}(x1)) 28.99/8.59 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1) 28.99/8.59 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) 28.99/8.59 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) 28.99/8.59 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) 28.99/8.59 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) 28.99/8.59 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) 28.99/8.59 28.99/8.59 28.99/8.59 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (8) 28.99/8.59 Obligation: 28.99/8.59 Q restricted rewrite system: 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 28.99/8.59 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (9) DependencyPairsProof (EQUIVALENT) 28.99/8.59 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (10) 28.99/8.59 Obligation: 28.99/8.59 Q DP problem: 28.99/8.59 The TRS P consists of the following rules: 28.99/8.59 28.99/8.59 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(x1) 28.99/8.59 C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) 28.99/8.59 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1)) 28.99/8.59 C_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1) 28.99/8.59 C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) 28.99/8.59 C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1) 28.99/8.59 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 28.99/8.59 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.99/8.59 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 28.99/8.59 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 28.99/8.59 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.99/8.59 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 28.99/8.59 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 28.99/8.59 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 28.99/8.59 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 28.99/8.59 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 28.99/8.59 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 28.99/8.59 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.99/8.59 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 28.99/8.59 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 28.99/8.59 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1) 28.99/8.59 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.99/8.59 A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1) 28.99/8.59 C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 28.99/8.59 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 28.99/8.59 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 We have to consider all minimal (P,Q,R)-chains. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (11) DependencyGraphProof (EQUIVALENT) 28.99/8.59 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (12) 28.99/8.59 Obligation: 28.99/8.59 Q DP problem: 28.99/8.59 The TRS P consists of the following rules: 28.99/8.59 28.99/8.59 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 28.99/8.59 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(x1) 28.99/8.59 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 28.99/8.59 C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) 28.99/8.59 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 28.99/8.59 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1)) 28.99/8.59 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 28.99/8.59 C_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1) 28.99/8.59 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 28.99/8.59 C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) 28.99/8.59 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.99/8.59 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 28.99/8.59 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 28.99/8.59 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.99/8.59 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 28.99/8.59 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 28.99/8.59 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 28.99/8.59 C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 28.99/8.59 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 28.99/8.59 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 We have to consider all minimal (P,Q,R)-chains. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (13) QDPOrderProof (EQUIVALENT) 28.99/8.59 We use the reduction pair processor [LPAR04,JAR06]. 28.99/8.59 28.99/8.59 28.99/8.59 The following pairs can be oriented strictly and are deleted. 28.99/8.59 28.99/8.59 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(x1) 28.99/8.59 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 28.99/8.59 C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) 28.99/8.59 C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1)) 28.99/8.59 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 28.99/8.59 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 28.99/8.59 C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) 28.99/8.59 B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) 28.99/8.59 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 28.99/8.59 B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.99/8.59 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 28.99/8.59 B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 28.99/8.59 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{c_1}(x1)) 28.99/8.59 The remaining pairs can at least be oriented weakly. 28.99/8.59 Used ordering: Polynomial interpretation [POLO]: 28.99/8.59 28.99/8.59 POL(A_{B_1}(x_1)) = x_1 28.99/8.59 POL(B_{B_1}(x_1)) = 1 + x_1 28.99/8.59 POL(C_{A_1}(x_1)) = x_1 28.99/8.59 POL(C_{B_1}(x_1)) = x_1 28.99/8.59 POL(a_{a_1}(x_1)) = 1 + x_1 28.99/8.59 POL(a_{b_1}(x_1)) = 1 + x_1 28.99/8.59 POL(a_{c_1}(x_1)) = 1 + x_1 28.99/8.59 POL(b_{a_1}(x_1)) = x_1 28.99/8.59 POL(b_{b_1}(x_1)) = x_1 28.99/8.59 POL(b_{c_1}(x_1)) = x_1 28.99/8.59 POL(c_{a_1}(x_1)) = x_1 28.99/8.59 POL(c_{b_1}(x_1)) = 1 + x_1 28.99/8.59 POL(c_{c_1}(x_1)) = 1 + x_1 28.99/8.59 28.99/8.59 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.99/8.59 28.99/8.59 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 28.99/8.59 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 28.99/8.59 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (14) 28.99/8.59 Obligation: 28.99/8.59 Q DP problem: 28.99/8.59 The TRS P consists of the following rules: 28.99/8.59 28.99/8.59 C_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 28.99/8.59 A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{a_1}(x1))) 28.99/8.59 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(x1))) 28.99/8.59 C_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1) 28.99/8.59 C_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.99/8.59 A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) 28.99/8.59 A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 28.99/8.59 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.99/8.59 C_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{c_1}(x1))) 28.99/8.59 C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 28.99/8.59 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 28.99/8.59 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 We have to consider all minimal (P,Q,R)-chains. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (15) DependencyGraphProof (EQUIVALENT) 28.99/8.59 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 9 less nodes. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (16) 28.99/8.59 Complex Obligation (AND) 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (17) 28.99/8.59 Obligation: 28.99/8.59 Q DP problem: 28.99/8.59 The TRS P consists of the following rules: 28.99/8.59 28.99/8.59 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.99/8.59 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 28.99/8.59 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 We have to consider all minimal (P,Q,R)-chains. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (18) UsableRulesProof (EQUIVALENT) 28.99/8.59 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. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (19) 28.99/8.59 Obligation: 28.99/8.59 Q DP problem: 28.99/8.59 The TRS P consists of the following rules: 28.99/8.59 28.99/8.59 A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.99/8.59 28.99/8.59 R is empty. 28.99/8.59 Q is empty. 28.99/8.59 We have to consider all minimal (P,Q,R)-chains. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (20) QDPSizeChangeProof (EQUIVALENT) 28.99/8.59 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. 28.99/8.59 28.99/8.59 From the DPs we obtained the following set of size-change graphs: 28.99/8.59 *A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1) 28.99/8.59 The graph contains the following edges 1 > 1 28.99/8.59 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (21) 28.99/8.59 YES 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (22) 28.99/8.59 Obligation: 28.99/8.59 Q DP problem: 28.99/8.59 The TRS P consists of the following rules: 28.99/8.59 28.99/8.59 C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 28.99/8.59 28.99/8.59 The TRS R consists of the following rules: 28.99/8.59 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 c_{a_1}(a_{a_1}(x1)) -> c_{b_1}(b_{a_1}(x1)) 28.99/8.59 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(b_{b_1}(x1)) 28.99/8.59 c_{a_1}(a_{c_1}(x1)) -> c_{b_1}(b_{c_1}(x1)) 28.99/8.59 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 28.99/8.59 c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 28.99/8.59 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1) 28.99/8.59 a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1) 28.99/8.59 a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1) 28.99/8.59 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1) 28.99/8.59 b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1) 28.99/8.59 b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1) 28.99/8.59 c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1) 28.99/8.59 c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1) 28.99/8.59 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) 28.99/8.59 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) 28.99/8.59 28.99/8.59 Q is empty. 28.99/8.59 We have to consider all minimal (P,Q,R)-chains. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (23) UsableRulesProof (EQUIVALENT) 28.99/8.59 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. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (24) 28.99/8.59 Obligation: 28.99/8.59 Q DP problem: 28.99/8.59 The TRS P consists of the following rules: 28.99/8.59 28.99/8.59 C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 28.99/8.59 28.99/8.59 R is empty. 28.99/8.59 Q is empty. 28.99/8.59 We have to consider all minimal (P,Q,R)-chains. 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (25) QDPSizeChangeProof (EQUIVALENT) 28.99/8.59 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. 28.99/8.59 28.99/8.59 From the DPs we obtained the following set of size-change graphs: 28.99/8.59 *C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1) 28.99/8.59 The graph contains the following edges 1 > 1 28.99/8.59 28.99/8.59 28.99/8.59 ---------------------------------------- 28.99/8.59 28.99/8.59 (26) 28.99/8.59 YES 29.10/9.45 EOF