15.11/4.70 YES 15.11/4.75 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 15.11/4.75 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 15.11/4.75 15.11/4.75 15.11/4.75 Termination w.r.t. Q of the given QTRS could be proven: 15.11/4.75 15.11/4.75 (0) QTRS 15.11/4.75 (1) FlatCCProof [EQUIVALENT, 0 ms] 15.11/4.75 (2) QTRS 15.11/4.75 (3) RootLabelingProof [EQUIVALENT, 0 ms] 15.11/4.75 (4) QTRS 15.11/4.75 (5) DependencyPairsProof [EQUIVALENT, 46 ms] 15.11/4.75 (6) QDP 15.11/4.75 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 15.11/4.75 (8) QDP 15.11/4.75 (9) QDPOrderProof [EQUIVALENT, 108 ms] 15.11/4.75 (10) QDP 15.11/4.75 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 15.11/4.75 (12) TRUE 15.11/4.75 15.11/4.75 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (0) 15.11/4.75 Obligation: 15.11/4.75 Q restricted rewrite system: 15.11/4.75 The TRS R consists of the following rules: 15.11/4.75 15.11/4.75 a(b(a(a(x1)))) -> a(a(b(a(x1)))) 15.11/4.75 b(b(a(a(x1)))) -> a(a(a(b(x1)))) 15.11/4.75 b(a(a(b(x1)))) -> b(b(b(a(x1)))) 15.11/4.75 15.11/4.75 Q is empty. 15.11/4.75 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (1) FlatCCProof (EQUIVALENT) 15.11/4.75 We used flat context closure [ROOTLAB] 15.11/4.75 As Q is empty the flat context closure was sound AND complete. 15.11/4.75 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (2) 15.11/4.75 Obligation: 15.11/4.75 Q restricted rewrite system: 15.11/4.75 The TRS R consists of the following rules: 15.11/4.75 15.11/4.75 a(b(a(a(x1)))) -> a(a(b(a(x1)))) 15.11/4.75 b(a(a(b(x1)))) -> b(b(b(a(x1)))) 15.11/4.75 a(b(b(a(a(x1))))) -> a(a(a(a(b(x1))))) 15.11/4.75 b(b(b(a(a(x1))))) -> b(a(a(a(b(x1))))) 15.11/4.75 15.11/4.75 Q is empty. 15.11/4.75 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (3) RootLabelingProof (EQUIVALENT) 15.11/4.75 We used plain root labeling [ROOTLAB] with the following heuristic: 15.11/4.75 LabelAll: All function symbols get labeled 15.11/4.75 15.11/4.75 As Q is empty the root labeling was sound AND complete. 15.11/4.75 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (4) 15.11/4.75 Obligation: 15.11/4.75 Q restricted rewrite system: 15.11/4.75 The TRS R consists of the following rules: 15.11/4.75 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 15.11/4.75 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)))) 15.11/4.75 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)))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 15.11/4.75 Q is empty. 15.11/4.75 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (5) DependencyPairsProof (EQUIVALENT) 15.11/4.75 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (6) 15.11/4.75 Obligation: 15.11/4.75 Q DP problem: 15.11/4.75 The TRS P consists of the following rules: 15.11/4.75 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 15.11/4.75 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)))) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 15.11/4.75 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)))) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 15.11/4.75 15.11/4.75 The TRS R consists of the following rules: 15.11/4.75 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 15.11/4.75 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)))) 15.11/4.75 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)))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 15.11/4.75 Q is empty. 15.11/4.75 We have to consider all minimal (P,Q,R)-chains. 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (7) DependencyGraphProof (EQUIVALENT) 15.11/4.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (8) 15.11/4.75 Obligation: 15.11/4.75 Q DP problem: 15.11/4.75 The TRS P consists of the following rules: 15.11/4.75 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 15.11/4.75 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)))) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 15.11/4.75 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)))) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 15.11/4.75 15.11/4.75 The TRS R consists of the following rules: 15.11/4.75 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 15.11/4.75 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)))) 15.11/4.75 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)))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 15.11/4.75 Q is empty. 15.11/4.75 We have to consider all minimal (P,Q,R)-chains. 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (9) QDPOrderProof (EQUIVALENT) 15.11/4.75 We use the reduction pair processor [LPAR04,JAR06]. 15.11/4.75 15.11/4.75 15.11/4.75 The following pairs can be oriented strictly and are deleted. 15.11/4.75 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 15.11/4.75 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 15.11/4.75 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) 15.11/4.75 B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 15.11/4.75 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 15.11/4.75 The remaining pairs can at least be oriented weakly. 15.11/4.75 Used ordering: Polynomial interpretation [POLO]: 15.11/4.75 15.11/4.75 POL(A_{B_1}(x_1)) = 1 + x_1 15.11/4.75 POL(B_{A_1}(x_1)) = 1 + x_1 15.11/4.75 POL(B_{B_1}(x_1)) = 1 + x_1 15.11/4.75 POL(a_{a_1}(x_1)) = 1 + x_1 15.11/4.75 POL(a_{b_1}(x_1)) = 1 + x_1 15.11/4.75 POL(b_{a_1}(x_1)) = 1 + x_1 15.11/4.75 POL(b_{b_1}(x_1)) = 1 + x_1 15.11/4.75 15.11/4.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 15.11/4.75 15.11/4.75 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)))) 15.11/4.75 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)))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 15.11/4.75 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (10) 15.11/4.75 Obligation: 15.11/4.75 Q DP problem: 15.11/4.75 The TRS P consists of the following rules: 15.11/4.75 15.11/4.75 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)))) 15.11/4.75 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)))) 15.11/4.75 15.11/4.75 The TRS R consists of the following rules: 15.11/4.75 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 15.11/4.75 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 15.11/4.75 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)))) 15.11/4.75 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)))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 15.11/4.75 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 15.11/4.75 15.11/4.75 Q is empty. 15.11/4.75 We have to consider all minimal (P,Q,R)-chains. 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (11) DependencyGraphProof (EQUIVALENT) 15.11/4.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 15.11/4.75 ---------------------------------------- 15.11/4.75 15.11/4.75 (12) 15.11/4.75 TRUE 15.53/4.87 EOF