26.25/7.64 YES 26.83/9.31 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 26.83/9.31 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 26.83/9.31 26.83/9.31 26.83/9.31 Termination w.r.t. Q of the given QTRS could be proven: 26.83/9.31 26.83/9.31 (0) QTRS 26.83/9.31 (1) QTRS Reverse [EQUIVALENT, 0 ms] 26.83/9.31 (2) QTRS 26.83/9.31 (3) FlatCCProof [EQUIVALENT, 0 ms] 26.83/9.31 (4) QTRS 26.83/9.31 (5) RootLabelingProof [EQUIVALENT, 0 ms] 26.83/9.31 (6) QTRS 26.83/9.31 (7) DependencyPairsProof [EQUIVALENT, 28 ms] 26.83/9.31 (8) QDP 26.83/9.31 (9) DependencyGraphProof [EQUIVALENT, 6 ms] 26.83/9.31 (10) QDP 26.83/9.31 (11) QDPOrderProof [EQUIVALENT, 59 ms] 26.83/9.31 (12) QDP 26.83/9.31 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 26.83/9.31 (14) TRUE 26.83/9.31 26.83/9.31 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (0) 26.83/9.31 Obligation: 26.83/9.31 Q restricted rewrite system: 26.83/9.31 The TRS R consists of the following rules: 26.83/9.31 26.83/9.31 a(b(a(b(x1)))) -> b(b(b(a(x1)))) 26.83/9.31 a(b(b(b(x1)))) -> a(b(a(b(x1)))) 26.83/9.31 b(a(b(a(x1)))) -> b(a(a(a(x1)))) 26.83/9.31 a(b(b(a(x1)))) -> b(b(a(a(x1)))) 26.83/9.31 26.83/9.31 Q is empty. 26.83/9.31 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (1) QTRS Reverse (EQUIVALENT) 26.83/9.31 We applied the QTRS Reverse Processor [REVERSE]. 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (2) 26.83/9.31 Obligation: 26.83/9.31 Q restricted rewrite system: 26.83/9.31 The TRS R consists of the following rules: 26.83/9.31 26.83/9.31 b(a(b(a(x1)))) -> a(b(b(b(x1)))) 26.83/9.31 b(b(b(a(x1)))) -> b(a(b(a(x1)))) 26.83/9.31 a(b(a(b(x1)))) -> a(a(a(b(x1)))) 26.83/9.31 a(b(b(a(x1)))) -> a(a(b(b(x1)))) 26.83/9.31 26.83/9.31 Q is empty. 26.83/9.31 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (3) FlatCCProof (EQUIVALENT) 26.83/9.31 We used flat context closure [ROOTLAB] 26.83/9.31 As Q is empty the flat context closure was sound AND complete. 26.83/9.31 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (4) 26.83/9.31 Obligation: 26.83/9.31 Q restricted rewrite system: 26.83/9.31 The TRS R consists of the following rules: 26.83/9.31 26.83/9.31 b(b(b(a(x1)))) -> b(a(b(a(x1)))) 26.83/9.31 a(b(a(b(x1)))) -> a(a(a(b(x1)))) 26.83/9.31 a(b(b(a(x1)))) -> a(a(b(b(x1)))) 26.83/9.31 b(b(a(b(a(x1))))) -> b(a(b(b(b(x1))))) 26.83/9.31 a(b(a(b(a(x1))))) -> a(a(b(b(b(x1))))) 26.83/9.31 26.83/9.31 Q is empty. 26.83/9.31 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (5) RootLabelingProof (EQUIVALENT) 26.83/9.31 We used plain root labeling [ROOTLAB] with the following heuristic: 26.83/9.31 LabelAll: All function symbols get labeled 26.83/9.31 26.83/9.31 As Q is empty the root labeling was sound AND complete. 26.83/9.31 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (6) 26.83/9.31 Obligation: 26.83/9.31 Q restricted rewrite system: 26.83/9.31 The TRS R consists of the following rules: 26.83/9.31 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 26.83/9.31 Q is empty. 26.83/9.31 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (7) DependencyPairsProof (EQUIVALENT) 26.83/9.31 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (8) 26.83/9.31 Obligation: 26.83/9.31 Q DP problem: 26.83/9.31 The TRS P consists of the following rules: 26.83/9.31 26.83/9.31 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 26.83/9.31 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 26.83/9.31 26.83/9.31 The TRS R consists of the following rules: 26.83/9.31 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 26.83/9.31 Q is empty. 26.83/9.31 We have to consider all minimal (P,Q,R)-chains. 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (9) DependencyGraphProof (EQUIVALENT) 26.83/9.31 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (10) 26.83/9.31 Obligation: 26.83/9.31 Q DP problem: 26.83/9.31 The TRS P consists of the following rules: 26.83/9.31 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 26.83/9.31 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 26.83/9.31 26.83/9.31 The TRS R consists of the following rules: 26.83/9.31 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 26.83/9.31 Q is empty. 26.83/9.31 We have to consider all minimal (P,Q,R)-chains. 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (11) QDPOrderProof (EQUIVALENT) 26.83/9.31 We use the reduction pair processor [LPAR04,JAR06]. 26.83/9.31 26.83/9.31 26.83/9.31 The following pairs can be oriented strictly and are deleted. 26.83/9.31 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 26.83/9.31 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 26.83/9.31 A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 26.83/9.31 The remaining pairs can at least be oriented weakly. 26.83/9.31 Used ordering: Polynomial interpretation [POLO]: 26.83/9.31 26.83/9.31 POL(A_{B_1}(x_1)) = 1 + x_1 26.83/9.31 POL(B_{B_1}(x_1)) = x_1 26.83/9.31 POL(a_{a_1}(x_1)) = 1 + x_1 26.83/9.31 POL(a_{b_1}(x_1)) = 1 + x_1 26.83/9.31 POL(b_{a_1}(x_1)) = 1 + x_1 26.83/9.31 POL(b_{b_1}(x_1)) = 1 + x_1 26.83/9.31 26.83/9.31 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 26.83/9.31 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 26.83/9.31 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (12) 26.83/9.31 Obligation: 26.83/9.31 Q DP problem: 26.83/9.31 The TRS P consists of the following rules: 26.83/9.31 26.83/9.31 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 26.83/9.31 The TRS R consists of the following rules: 26.83/9.31 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 26.83/9.31 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 26.83/9.31 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 26.83/9.31 a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 26.83/9.31 26.83/9.31 Q is empty. 26.83/9.31 We have to consider all minimal (P,Q,R)-chains. 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (13) DependencyGraphProof (EQUIVALENT) 26.83/9.31 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. 26.83/9.31 ---------------------------------------- 26.83/9.31 26.83/9.31 (14) 26.83/9.31 TRUE 26.91/9.63 EOF