14.31/4.59 YES 14.47/4.65 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 14.47/4.65 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 14.47/4.65 14.47/4.65 14.47/4.65 Termination w.r.t. Q of the given QTRS could be proven: 14.47/4.65 14.47/4.65 (0) QTRS 14.47/4.65 (1) QTRS Reverse [EQUIVALENT, 0 ms] 14.47/4.65 (2) QTRS 14.47/4.65 (3) FlatCCProof [EQUIVALENT, 0 ms] 14.47/4.65 (4) QTRS 14.47/4.65 (5) RootLabelingProof [EQUIVALENT, 4 ms] 14.47/4.65 (6) QTRS 14.47/4.65 (7) DependencyPairsProof [EQUIVALENT, 26 ms] 14.47/4.65 (8) QDP 14.47/4.65 (9) DependencyGraphProof [EQUIVALENT, 3 ms] 14.47/4.65 (10) QDP 14.47/4.65 (11) QDPOrderProof [EQUIVALENT, 51 ms] 14.47/4.65 (12) QDP 14.47/4.65 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 14.47/4.65 (14) TRUE 14.47/4.65 14.47/4.65 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (0) 14.47/4.65 Obligation: 14.47/4.65 Q restricted rewrite system: 14.47/4.65 The TRS R consists of the following rules: 14.47/4.65 14.47/4.65 a(b(b(a(x1)))) -> b(b(a(a(x1)))) 14.47/4.65 b(b(b(a(x1)))) -> a(a(a(a(x1)))) 14.47/4.65 a(b(a(a(x1)))) -> b(a(b(b(x1)))) 14.47/4.65 14.47/4.65 Q is empty. 14.47/4.65 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (1) QTRS Reverse (EQUIVALENT) 14.47/4.65 We applied the QTRS Reverse Processor [REVERSE]. 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (2) 14.47/4.65 Obligation: 14.47/4.65 Q restricted rewrite system: 14.47/4.65 The TRS R consists of the following rules: 14.47/4.65 14.47/4.65 a(b(b(a(x1)))) -> a(a(b(b(x1)))) 14.47/4.65 a(b(b(b(x1)))) -> a(a(a(a(x1)))) 14.47/4.65 a(a(b(a(x1)))) -> b(b(a(b(x1)))) 14.47/4.65 14.47/4.65 Q is empty. 14.47/4.65 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (3) FlatCCProof (EQUIVALENT) 14.47/4.65 We used flat context closure [ROOTLAB] 14.47/4.65 As Q is empty the flat context closure was sound AND complete. 14.47/4.65 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (4) 14.47/4.65 Obligation: 14.47/4.65 Q restricted rewrite system: 14.47/4.65 The TRS R consists of the following rules: 14.47/4.65 14.47/4.65 a(b(b(a(x1)))) -> a(a(b(b(x1)))) 14.47/4.65 a(b(b(b(x1)))) -> a(a(a(a(x1)))) 14.47/4.65 a(a(a(b(a(x1))))) -> a(b(b(a(b(x1))))) 14.47/4.65 b(a(a(b(a(x1))))) -> b(b(b(a(b(x1))))) 14.47/4.65 14.47/4.65 Q is empty. 14.47/4.65 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (5) RootLabelingProof (EQUIVALENT) 14.47/4.65 We used plain root labeling [ROOTLAB] with the following heuristic: 14.47/4.65 LabelAll: All function symbols get labeled 14.47/4.65 14.47/4.65 As Q is empty the root labeling was sound AND complete. 14.47/4.65 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (6) 14.47/4.65 Obligation: 14.47/4.65 Q restricted rewrite system: 14.47/4.65 The TRS R consists of the following rules: 14.47/4.65 14.47/4.65 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)))) 14.47/4.65 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)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 14.47/4.65 Q is empty. 14.47/4.65 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (7) DependencyPairsProof (EQUIVALENT) 14.47/4.65 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (8) 14.47/4.65 Obligation: 14.47/4.65 Q DP problem: 14.47/4.65 The TRS P consists of the following rules: 14.47/4.65 14.47/4.65 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)))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1) 14.47/4.65 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)))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1)) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1)) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1))) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1))) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1))) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1))) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 14.47/4.65 14.47/4.65 The TRS R consists of the following rules: 14.47/4.65 14.47/4.65 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)))) 14.47/4.65 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)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 14.47/4.65 Q is empty. 14.47/4.65 We have to consider all minimal (P,Q,R)-chains. 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (9) DependencyGraphProof (EQUIVALENT) 14.47/4.65 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes. 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (10) 14.47/4.65 Obligation: 14.47/4.65 Q DP problem: 14.47/4.65 The TRS P consists of the following rules: 14.47/4.65 14.47/4.65 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1)) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 14.47/4.65 14.47/4.65 The TRS R consists of the following rules: 14.47/4.65 14.47/4.65 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)))) 14.47/4.65 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)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 14.47/4.65 Q is empty. 14.47/4.65 We have to consider all minimal (P,Q,R)-chains. 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (11) QDPOrderProof (EQUIVALENT) 14.47/4.65 We use the reduction pair processor [LPAR04,JAR06]. 14.47/4.65 14.47/4.65 14.47/4.65 The following pairs can be oriented strictly and are deleted. 14.47/4.65 14.47/4.65 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1)) 14.47/4.65 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 14.47/4.65 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 14.47/4.65 A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 14.47/4.65 The remaining pairs can at least be oriented weakly. 14.47/4.65 Used ordering: Polynomial interpretation [POLO]: 14.47/4.65 14.47/4.65 POL(A_{A_1}(x_1)) = 1 + x_1 14.47/4.65 POL(A_{B_1}(x_1)) = 1 + x_1 14.47/4.65 POL(B_{A_1}(x_1)) = x_1 14.47/4.65 POL(a_{a_1}(x_1)) = 1 + x_1 14.47/4.65 POL(a_{b_1}(x_1)) = 1 + x_1 14.47/4.65 POL(b_{a_1}(x_1)) = 1 + x_1 14.47/4.65 POL(b_{b_1}(x_1)) = 1 + x_1 14.47/4.65 14.47/4.65 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 14.47/4.65 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 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)))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 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)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 14.47/4.65 14.47/4.65 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (12) 14.47/4.65 Obligation: 14.47/4.65 Q DP problem: 14.47/4.65 The TRS P consists of the following rules: 14.47/4.65 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 14.47/4.65 The TRS R consists of the following rules: 14.47/4.65 14.47/4.65 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)))) 14.47/4.65 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)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 14.47/4.65 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) 14.47/4.65 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 14.47/4.65 14.47/4.65 Q is empty. 14.47/4.65 We have to consider all minimal (P,Q,R)-chains. 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (13) DependencyGraphProof (EQUIVALENT) 14.47/4.65 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 14.47/4.65 ---------------------------------------- 14.47/4.65 14.47/4.65 (14) 14.47/4.65 TRUE 14.70/4.74 EOF