26.00/7.64 YES 26.35/7.66 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 26.35/7.66 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 26.35/7.66 26.35/7.66 26.35/7.66 Termination w.r.t. Q of the given QTRS could be proven: 26.35/7.66 26.35/7.66 (0) QTRS 26.35/7.66 (1) FlatCCProof [EQUIVALENT, 0 ms] 26.35/7.66 (2) QTRS 26.35/7.66 (3) RootLabelingProof [EQUIVALENT, 0 ms] 26.35/7.66 (4) QTRS 26.35/7.66 (5) DependencyPairsProof [EQUIVALENT, 53 ms] 26.35/7.66 (6) QDP 26.35/7.66 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 26.35/7.66 (8) QDP 26.35/7.66 (9) QDPOrderProof [EQUIVALENT, 68 ms] 26.35/7.66 (10) QDP 26.35/7.66 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 26.35/7.66 (12) TRUE 26.35/7.66 26.35/7.66 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (0) 26.35/7.66 Obligation: 26.35/7.66 Q restricted rewrite system: 26.35/7.66 The TRS R consists of the following rules: 26.35/7.66 26.35/7.66 a(a(b(a(x1)))) -> a(a(a(b(x1)))) 26.35/7.66 b(b(b(a(x1)))) -> a(a(a(b(x1)))) 26.35/7.66 b(a(a(b(x1)))) -> b(b(b(a(x1)))) 26.35/7.66 26.35/7.66 Q is empty. 26.35/7.66 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (1) FlatCCProof (EQUIVALENT) 26.35/7.66 We used flat context closure [ROOTLAB] 26.35/7.66 As Q is empty the flat context closure was sound AND complete. 26.35/7.66 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (2) 26.35/7.66 Obligation: 26.35/7.66 Q restricted rewrite system: 26.35/7.66 The TRS R consists of the following rules: 26.35/7.66 26.35/7.66 a(a(b(a(x1)))) -> a(a(a(b(x1)))) 26.35/7.66 b(a(a(b(x1)))) -> b(b(b(a(x1)))) 26.35/7.66 a(b(b(b(a(x1))))) -> a(a(a(a(b(x1))))) 26.35/7.66 b(b(b(b(a(x1))))) -> b(a(a(a(b(x1))))) 26.35/7.66 26.35/7.66 Q is empty. 26.35/7.66 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (3) RootLabelingProof (EQUIVALENT) 26.35/7.66 We used plain root labeling [ROOTLAB] with the following heuristic: 26.35/7.66 LabelAll: All function symbols get labeled 26.35/7.66 26.35/7.66 As Q is empty the root labeling was sound AND complete. 26.35/7.66 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (4) 26.35/7.66 Obligation: 26.35/7.66 Q restricted rewrite system: 26.35/7.66 The TRS R consists of the following rules: 26.35/7.66 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 26.35/7.66 Q is empty. 26.35/7.66 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (5) DependencyPairsProof (EQUIVALENT) 26.35/7.66 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (6) 26.35/7.66 Obligation: 26.35/7.66 Q DP problem: 26.35/7.66 The TRS P consists of the following rules: 26.35/7.66 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.35/7.66 26.35/7.66 The TRS R consists of the following rules: 26.35/7.66 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 26.35/7.66 Q is empty. 26.35/7.66 We have to consider all minimal (P,Q,R)-chains. 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (7) DependencyGraphProof (EQUIVALENT) 26.35/7.66 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 11 less nodes. 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (8) 26.35/7.66 Obligation: 26.35/7.66 Q DP problem: 26.35/7.66 The TRS P consists of the following rules: 26.35/7.66 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 26.35/7.66 26.35/7.66 The TRS R consists of the following rules: 26.35/7.66 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 26.35/7.66 Q is empty. 26.35/7.66 We have to consider all minimal (P,Q,R)-chains. 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (9) QDPOrderProof (EQUIVALENT) 26.35/7.66 We use the reduction pair processor [LPAR04,JAR06]. 26.35/7.66 26.35/7.66 26.35/7.66 The following pairs can be oriented strictly and are deleted. 26.35/7.66 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 26.35/7.66 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) 26.35/7.66 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 26.35/7.66 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 26.35/7.66 The remaining pairs can at least be oriented weakly. 26.35/7.66 Used ordering: Polynomial interpretation [POLO]: 26.35/7.66 26.35/7.66 POL(A_{A_1}(x_1)) = x_1 26.35/7.66 POL(A_{B_1}(x_1)) = x_1 26.35/7.66 POL(B_{A_1}(x_1)) = 1 + x_1 26.35/7.66 POL(B_{B_1}(x_1)) = 1 + x_1 26.35/7.66 POL(a_{a_1}(x_1)) = 1 + x_1 26.35/7.66 POL(a_{b_1}(x_1)) = 1 + x_1 26.35/7.66 POL(b_{a_1}(x_1)) = 1 + x_1 26.35/7.66 POL(b_{b_1}(x_1)) = 1 + x_1 26.35/7.66 26.35/7.66 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.35/7.66 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.35/7.66 26.35/7.66 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (10) 26.35/7.66 Obligation: 26.35/7.66 Q DP problem: 26.35/7.66 The TRS P consists of the following rules: 26.35/7.66 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.35/7.66 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.35/7.66 26.35/7.66 The TRS R consists of the following rules: 26.35/7.66 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.35/7.66 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.35/7.66 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 26.35/7.66 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 26.35/7.66 26.35/7.66 Q is empty. 26.35/7.66 We have to consider all minimal (P,Q,R)-chains. 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (11) DependencyGraphProof (EQUIVALENT) 26.35/7.66 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 26.35/7.66 ---------------------------------------- 26.35/7.66 26.35/7.66 (12) 26.35/7.66 TRUE 26.59/7.80 EOF