27.29/7.95 YES 27.68/7.96 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 27.68/7.96 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 27.68/7.96 27.68/7.96 27.68/7.96 Termination w.r.t. Q of the given QTRS could be proven: 27.68/7.96 27.68/7.96 (0) QTRS 27.68/7.96 (1) FlatCCProof [EQUIVALENT, 0 ms] 27.68/7.96 (2) QTRS 27.68/7.96 (3) RootLabelingProof [EQUIVALENT, 0 ms] 27.68/7.96 (4) QTRS 27.68/7.96 (5) QTRSRRRProof [EQUIVALENT, 51 ms] 27.68/7.96 (6) QTRS 27.68/7.96 (7) DependencyPairsProof [EQUIVALENT, 30 ms] 27.68/7.96 (8) QDP 27.68/7.96 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 27.68/7.96 (10) QDP 27.68/7.96 (11) QDPOrderProof [EQUIVALENT, 57 ms] 27.68/7.96 (12) QDP 27.68/7.96 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 27.68/7.96 (14) AND 27.68/7.96 (15) QDP 27.68/7.96 (16) UsableRulesProof [EQUIVALENT, 0 ms] 27.68/7.96 (17) QDP 27.68/7.96 (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.68/7.96 (19) YES 27.68/7.96 (20) QDP 27.68/7.96 (21) UsableRulesProof [EQUIVALENT, 0 ms] 27.68/7.96 (22) QDP 27.68/7.96 (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] 27.68/7.96 (24) YES 27.68/7.96 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (0) 27.68/7.96 Obligation: 27.68/7.96 Q restricted rewrite system: 27.68/7.96 The TRS R consists of the following rules: 27.68/7.96 27.68/7.96 a(x1) -> x1 27.68/7.96 a(b(x1)) -> c(b(a(x1))) 27.68/7.96 b(x1) -> a(x1) 27.68/7.96 c(c(x1)) -> b(x1) 27.68/7.96 27.68/7.96 Q is empty. 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (1) FlatCCProof (EQUIVALENT) 27.68/7.96 We used flat context closure [ROOTLAB] 27.68/7.96 As Q is empty the flat context closure was sound AND complete. 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (2) 27.68/7.96 Obligation: 27.68/7.96 Q restricted rewrite system: 27.68/7.96 The TRS R consists of the following rules: 27.68/7.96 27.68/7.96 a(a(x1)) -> a(x1) 27.68/7.96 b(a(x1)) -> b(x1) 27.68/7.96 c(a(x1)) -> c(x1) 27.68/7.96 a(a(b(x1))) -> a(c(b(a(x1)))) 27.68/7.96 b(a(b(x1))) -> b(c(b(a(x1)))) 27.68/7.96 c(a(b(x1))) -> c(c(b(a(x1)))) 27.68/7.96 a(b(x1)) -> a(a(x1)) 27.68/7.96 b(b(x1)) -> b(a(x1)) 27.68/7.96 c(b(x1)) -> c(a(x1)) 27.68/7.96 a(c(c(x1))) -> a(b(x1)) 27.68/7.96 b(c(c(x1))) -> b(b(x1)) 27.68/7.96 c(c(c(x1))) -> c(b(x1)) 27.68/7.96 27.68/7.96 Q is empty. 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (3) RootLabelingProof (EQUIVALENT) 27.68/7.96 We used plain root labeling [ROOTLAB] with the following heuristic: 27.68/7.96 LabelAll: All function symbols get labeled 27.68/7.96 27.68/7.96 As Q is empty the root labeling was sound AND complete. 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (4) 27.68/7.96 Obligation: 27.68/7.96 Q restricted rewrite system: 27.68/7.96 The TRS R consists of the following rules: 27.68/7.96 27.68/7.96 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 27.68/7.96 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 27.68/7.96 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 27.68/7.96 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 27.68/7.96 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 27.68/7.96 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 27.68/7.96 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 27.68/7.96 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(a_{a_1}(x1)) 27.68/7.96 a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{b_1}(x1)) 27.68/7.96 a_{b_1}(b_{c_1}(x1)) -> a_{a_1}(a_{c_1}(x1)) 27.68/7.96 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(x1)) 27.68/7.96 b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{b_1}(x1)) 27.68/7.96 b_{b_1}(b_{c_1}(x1)) -> b_{a_1}(a_{c_1}(x1)) 27.68/7.96 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(a_{a_1}(x1)) 27.68/7.96 c_{b_1}(b_{b_1}(x1)) -> c_{a_1}(a_{b_1}(x1)) 27.68/7.96 c_{b_1}(b_{c_1}(x1)) -> c_{a_1}(a_{c_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1)) 27.68/7.96 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1)) 27.68/7.96 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 27.68/7.96 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1)) 27.68/7.96 27.68/7.96 Q is empty. 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (5) QTRSRRRProof (EQUIVALENT) 27.68/7.96 Used ordering: 27.68/7.96 Polynomial interpretation [POLO]: 27.68/7.96 27.68/7.96 POL(a_{a_1}(x_1)) = x_1 27.68/7.96 POL(a_{b_1}(x_1)) = x_1 27.68/7.96 POL(a_{c_1}(x_1)) = x_1 27.68/7.96 POL(b_{a_1}(x_1)) = 1 + x_1 27.68/7.96 POL(b_{b_1}(x_1)) = 1 + x_1 27.68/7.96 POL(b_{c_1}(x_1)) = 1 + x_1 27.68/7.96 POL(c_{a_1}(x_1)) = 1 + x_1 27.68/7.96 POL(c_{b_1}(x_1)) = x_1 27.68/7.96 POL(c_{c_1}(x_1)) = 1 + x_1 27.68/7.96 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 27.68/7.96 27.68/7.96 c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1) 27.68/7.96 a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(a_{a_1}(x1)) 27.68/7.96 a_{b_1}(b_{b_1}(x1)) -> a_{a_1}(a_{b_1}(x1)) 27.68/7.96 a_{b_1}(b_{c_1}(x1)) -> a_{a_1}(a_{c_1}(x1)) 27.68/7.96 b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(a_{a_1}(x1)) 27.68/7.96 b_{b_1}(b_{b_1}(x1)) -> b_{a_1}(a_{b_1}(x1)) 27.68/7.96 b_{b_1}(b_{c_1}(x1)) -> b_{a_1}(a_{c_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{c_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{a_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{c_1}(x1)) 27.68/7.96 c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{a_1}(x1)) 27.68/7.96 c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{b_1}(x1)) 27.68/7.96 c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{c_1}(x1)) 27.68/7.96 27.68/7.96 27.68/7.96 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (6) 27.68/7.96 Obligation: 27.68/7.96 Q restricted rewrite system: 27.68/7.96 The TRS R consists of the following rules: 27.68/7.96 27.68/7.96 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 27.68/7.96 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 27.68/7.96 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 27.68/7.96 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 27.68/7.96 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 27.68/7.96 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 27.68/7.96 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(a_{a_1}(x1)) 27.68/7.96 c_{b_1}(b_{b_1}(x1)) -> c_{a_1}(a_{b_1}(x1)) 27.68/7.96 c_{b_1}(b_{c_1}(x1)) -> c_{a_1}(a_{c_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 27.68/7.96 27.68/7.96 Q is empty. 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (7) DependencyPairsProof (EQUIVALENT) 27.68/7.96 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (8) 27.68/7.96 Obligation: 27.68/7.96 Q DP problem: 27.68/7.96 The TRS P consists of the following rules: 27.68/7.96 27.68/7.96 B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 27.68/7.96 B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1) 27.68/7.96 C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 27.68/7.96 A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 27.68/7.96 A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 27.68/7.96 A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{C_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1))) 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 27.68/7.96 B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{C_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1))) 27.68/7.96 B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{C_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1))) 27.68/7.96 B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 27.68/7.96 C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 27.68/7.96 C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 27.68/7.96 C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 27.68/7.96 C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 27.68/7.96 C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> A_{C_1}(x1) 27.68/7.96 C_{B_1}(b_{a_1}(x1)) -> C_{A_1}(a_{a_1}(x1)) 27.68/7.96 C_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1) 27.68/7.96 C_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 27.68/7.96 C_{B_1}(b_{c_1}(x1)) -> C_{A_1}(a_{c_1}(x1)) 27.68/7.96 C_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1) 27.68/7.96 27.68/7.96 The TRS R consists of the following rules: 27.68/7.96 27.68/7.96 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 27.68/7.96 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 27.68/7.96 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 27.68/7.96 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 27.68/7.96 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 27.68/7.96 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 27.68/7.96 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(a_{a_1}(x1)) 27.68/7.96 c_{b_1}(b_{b_1}(x1)) -> c_{a_1}(a_{b_1}(x1)) 27.68/7.96 c_{b_1}(b_{c_1}(x1)) -> c_{a_1}(a_{c_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 27.68/7.96 27.68/7.96 Q is empty. 27.68/7.96 We have to consider all minimal (P,Q,R)-chains. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (9) DependencyGraphProof (EQUIVALENT) 27.68/7.96 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (10) 27.68/7.96 Obligation: 27.68/7.96 Q DP problem: 27.68/7.96 The TRS P consists of the following rules: 27.68/7.96 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1))) 27.68/7.96 C_{B_1}(b_{a_1}(x1)) -> C_{A_1}(a_{a_1}(x1)) 27.68/7.96 C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) 27.68/7.96 C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1))) 27.68/7.96 C_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1))) 27.68/7.96 C_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 27.68/7.96 C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 27.68/7.96 B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1))) 27.68/7.96 C_{B_1}(b_{c_1}(x1)) -> C_{A_1}(a_{c_1}(x1)) 27.68/7.96 C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 27.68/7.96 A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1))) 27.68/7.96 B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 27.68/7.96 A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 27.68/7.96 C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 27.68/7.96 C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 27.68/7.96 27.68/7.96 The TRS R consists of the following rules: 27.68/7.96 27.68/7.96 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 27.68/7.96 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 27.68/7.96 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 27.68/7.96 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 27.68/7.96 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 27.68/7.96 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 27.68/7.96 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(a_{a_1}(x1)) 27.68/7.96 c_{b_1}(b_{b_1}(x1)) -> c_{a_1}(a_{b_1}(x1)) 27.68/7.96 c_{b_1}(b_{c_1}(x1)) -> c_{a_1}(a_{c_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 27.68/7.96 27.68/7.96 Q is empty. 27.68/7.96 We have to consider all minimal (P,Q,R)-chains. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (11) QDPOrderProof (EQUIVALENT) 27.68/7.96 We use the reduction pair processor [LPAR04,JAR06]. 27.68/7.96 27.68/7.96 27.68/7.96 The following pairs can be oriented strictly and are deleted. 27.68/7.96 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> C_{B_1}(b_{a_1}(a_{a_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 27.68/7.96 A_{A_1}(a_{b_1}(b_{a_1}(x1))) -> A_{A_1}(x1) 27.68/7.96 A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 27.68/7.96 B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1))) 27.68/7.96 B_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 27.68/7.96 A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1))) 27.68/7.96 A_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 27.68/7.96 C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> C_{B_1}(b_{a_1}(a_{b_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) 27.68/7.96 C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> C_{B_1}(b_{a_1}(a_{c_1}(x1))) 27.68/7.96 C_{A_1}(a_{b_1}(b_{c_1}(x1))) -> B_{A_1}(a_{c_1}(x1)) 27.68/7.96 The remaining pairs can at least be oriented weakly. 27.68/7.96 Used ordering: Polynomial interpretation [POLO]: 27.68/7.96 27.68/7.96 POL(A_{A_1}(x_1)) = x_1 27.68/7.96 POL(B_{A_1}(x_1)) = x_1 27.68/7.96 POL(C_{A_1}(x_1)) = x_1 27.68/7.96 POL(C_{B_1}(x_1)) = x_1 27.68/7.96 POL(a_{a_1}(x_1)) = x_1 27.68/7.96 POL(a_{b_1}(x_1)) = 1 + x_1 27.68/7.96 POL(a_{c_1}(x_1)) = x_1 27.68/7.96 POL(b_{a_1}(x_1)) = x_1 27.68/7.96 POL(b_{b_1}(x_1)) = 1 + x_1 27.68/7.96 POL(b_{c_1}(x_1)) = x_1 27.68/7.96 POL(c_{a_1}(x_1)) = 1 + x_1 27.68/7.96 POL(c_{b_1}(x_1)) = 1 + x_1 27.68/7.96 POL(c_{c_1}(x_1)) = 1 + x_1 27.68/7.96 27.68/7.96 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 27.68/7.96 27.68/7.96 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 27.68/7.96 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 27.68/7.96 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 27.68/7.96 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 27.68/7.96 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 27.68/7.96 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(a_{a_1}(x1)) 27.68/7.96 c_{b_1}(b_{b_1}(x1)) -> c_{a_1}(a_{b_1}(x1)) 27.68/7.96 c_{b_1}(b_{c_1}(x1)) -> c_{a_1}(a_{c_1}(x1)) 27.68/7.96 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 27.68/7.96 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 27.68/7.96 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 27.68/7.96 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (12) 27.68/7.96 Obligation: 27.68/7.96 Q DP problem: 27.68/7.96 The TRS P consists of the following rules: 27.68/7.96 27.68/7.96 C_{B_1}(b_{a_1}(x1)) -> C_{A_1}(a_{a_1}(x1)) 27.68/7.96 C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) 27.68/7.96 C_{B_1}(b_{a_1}(x1)) -> A_{A_1}(x1) 27.68/7.96 C_{B_1}(b_{b_1}(x1)) -> C_{A_1}(a_{b_1}(x1)) 27.68/7.96 B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 27.68/7.96 C_{B_1}(b_{c_1}(x1)) -> C_{A_1}(a_{c_1}(x1)) 27.68/7.96 27.68/7.96 The TRS R consists of the following rules: 27.68/7.96 27.68/7.96 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 27.68/7.96 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 27.68/7.96 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 27.68/7.96 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 27.68/7.96 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 27.68/7.96 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 27.68/7.96 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(a_{a_1}(x1)) 27.68/7.96 c_{b_1}(b_{b_1}(x1)) -> c_{a_1}(a_{b_1}(x1)) 27.68/7.96 c_{b_1}(b_{c_1}(x1)) -> c_{a_1}(a_{c_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 27.68/7.96 27.68/7.96 Q is empty. 27.68/7.96 We have to consider all minimal (P,Q,R)-chains. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (13) DependencyGraphProof (EQUIVALENT) 27.68/7.96 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (14) 27.68/7.96 Complex Obligation (AND) 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (15) 27.68/7.96 Obligation: 27.68/7.96 Q DP problem: 27.68/7.96 The TRS P consists of the following rules: 27.68/7.96 27.68/7.96 B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 27.68/7.96 27.68/7.96 The TRS R consists of the following rules: 27.68/7.96 27.68/7.96 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 27.68/7.96 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 27.68/7.96 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 27.68/7.96 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 27.68/7.96 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 27.68/7.96 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 27.68/7.96 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(a_{a_1}(x1)) 27.68/7.96 c_{b_1}(b_{b_1}(x1)) -> c_{a_1}(a_{b_1}(x1)) 27.68/7.96 c_{b_1}(b_{c_1}(x1)) -> c_{a_1}(a_{c_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 27.68/7.96 27.68/7.96 Q is empty. 27.68/7.96 We have to consider all minimal (P,Q,R)-chains. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (16) UsableRulesProof (EQUIVALENT) 27.68/7.96 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. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (17) 27.68/7.96 Obligation: 27.68/7.96 Q DP problem: 27.68/7.96 The TRS P consists of the following rules: 27.68/7.96 27.68/7.96 B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 27.68/7.96 27.68/7.96 R is empty. 27.68/7.96 Q is empty. 27.68/7.96 We have to consider all minimal (P,Q,R)-chains. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (18) QDPSizeChangeProof (EQUIVALENT) 27.68/7.96 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. 27.68/7.96 27.68/7.96 From the DPs we obtained the following set of size-change graphs: 27.68/7.96 *B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) 27.68/7.96 The graph contains the following edges 1 > 1 27.68/7.96 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (19) 27.68/7.96 YES 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (20) 27.68/7.96 Obligation: 27.68/7.96 Q DP problem: 27.68/7.96 The TRS P consists of the following rules: 27.68/7.96 27.68/7.96 C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) 27.68/7.96 27.68/7.96 The TRS R consists of the following rules: 27.68/7.96 27.68/7.96 a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1) 27.68/7.96 a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1) 27.68/7.96 b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1) 27.68/7.96 b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1) 27.68/7.96 b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1) 27.68/7.96 c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1) 27.68/7.96 c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1) 27.68/7.96 a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 27.68/7.96 c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 27.68/7.96 c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(a_{a_1}(x1)) 27.68/7.96 c_{b_1}(b_{b_1}(x1)) -> c_{a_1}(a_{b_1}(x1)) 27.68/7.96 c_{b_1}(b_{c_1}(x1)) -> c_{a_1}(a_{c_1}(x1)) 27.68/7.96 a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{b_1}(x1)) 27.68/7.96 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(x1)) 27.68/7.96 27.68/7.96 Q is empty. 27.68/7.96 We have to consider all minimal (P,Q,R)-chains. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (21) UsableRulesProof (EQUIVALENT) 27.68/7.96 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. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (22) 27.68/7.96 Obligation: 27.68/7.96 Q DP problem: 27.68/7.96 The TRS P consists of the following rules: 27.68/7.96 27.68/7.96 C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) 27.68/7.96 27.68/7.96 R is empty. 27.68/7.96 Q is empty. 27.68/7.96 We have to consider all minimal (P,Q,R)-chains. 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (23) QDPSizeChangeProof (EQUIVALENT) 27.68/7.96 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. 27.68/7.96 27.68/7.96 From the DPs we obtained the following set of size-change graphs: 27.68/7.96 *C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1) 27.68/7.96 The graph contains the following edges 1 > 1 27.68/7.96 27.68/7.96 27.68/7.96 ---------------------------------------- 27.68/7.96 27.68/7.96 (24) 27.68/7.96 YES 27.77/8.03 EOF