13.55/4.44 YES 13.92/4.48 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 13.92/4.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 13.92/4.48 13.92/4.48 13.92/4.48 Termination w.r.t. Q of the given QTRS could be proven: 13.92/4.48 13.92/4.48 (0) QTRS 13.92/4.48 (1) QTRS Reverse [EQUIVALENT, 0 ms] 13.92/4.48 (2) QTRS 13.92/4.48 (3) FlatCCProof [EQUIVALENT, 0 ms] 13.92/4.48 (4) QTRS 13.92/4.48 (5) RootLabelingProof [EQUIVALENT, 0 ms] 13.92/4.48 (6) QTRS 13.92/4.48 (7) DependencyPairsProof [EQUIVALENT, 25 ms] 13.92/4.48 (8) QDP 13.92/4.48 (9) DependencyGraphProof [EQUIVALENT, 6 ms] 13.92/4.48 (10) QDP 13.92/4.48 (11) QDPOrderProof [EQUIVALENT, 71 ms] 13.92/4.48 (12) QDP 13.92/4.48 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 13.92/4.48 (14) TRUE 13.92/4.48 13.92/4.48 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (0) 13.92/4.48 Obligation: 13.92/4.48 Q restricted rewrite system: 13.92/4.48 The TRS R consists of the following rules: 13.92/4.48 13.92/4.48 a(b(a(b(x1)))) -> a(b(b(b(x1)))) 13.92/4.48 b(b(a(b(x1)))) -> b(b(a(a(x1)))) 13.92/4.48 a(a(a(b(x1)))) -> b(a(a(a(x1)))) 13.92/4.48 13.92/4.48 Q is empty. 13.92/4.48 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (1) QTRS Reverse (EQUIVALENT) 13.92/4.48 We applied the QTRS Reverse Processor [REVERSE]. 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (2) 13.92/4.48 Obligation: 13.92/4.48 Q restricted rewrite system: 13.92/4.48 The TRS R consists of the following rules: 13.92/4.48 13.92/4.48 b(a(b(a(x1)))) -> b(b(b(a(x1)))) 13.92/4.48 b(a(b(b(x1)))) -> a(a(b(b(x1)))) 13.92/4.48 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 13.92/4.48 13.92/4.48 Q is empty. 13.92/4.48 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (3) FlatCCProof (EQUIVALENT) 13.92/4.48 We used flat context closure [ROOTLAB] 13.92/4.48 As Q is empty the flat context closure was sound AND complete. 13.92/4.48 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (4) 13.92/4.48 Obligation: 13.92/4.48 Q restricted rewrite system: 13.92/4.48 The TRS R consists of the following rules: 13.92/4.48 13.92/4.48 b(a(b(a(x1)))) -> b(b(b(a(x1)))) 13.92/4.48 b(b(a(b(b(x1))))) -> b(a(a(b(b(x1))))) 13.92/4.48 a(b(a(b(b(x1))))) -> a(a(a(b(b(x1))))) 13.92/4.48 b(b(a(a(a(x1))))) -> b(a(a(a(b(x1))))) 13.92/4.48 a(b(a(a(a(x1))))) -> a(a(a(a(b(x1))))) 13.92/4.48 13.92/4.48 Q is empty. 13.92/4.48 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (5) RootLabelingProof (EQUIVALENT) 13.92/4.48 We used plain root labeling [ROOTLAB] with the following heuristic: 13.92/4.48 LabelAll: All function symbols get labeled 13.92/4.48 13.92/4.48 As Q is empty the root labeling was sound AND complete. 13.92/4.48 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (6) 13.92/4.48 Obligation: 13.92/4.48 Q restricted rewrite system: 13.92/4.48 The TRS R consists of the following rules: 13.92/4.48 13.92/4.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.92/4.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.92/4.48 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.48 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.48 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.48 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.48 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.48 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.48 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.48 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.48 13.92/4.48 Q is empty. 13.92/4.48 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (7) DependencyPairsProof (EQUIVALENT) 13.92/4.48 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (8) 13.92/4.48 Obligation: 13.92/4.48 Q DP problem: 13.92/4.48 The TRS P consists of the following rules: 13.92/4.48 13.92/4.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.92/4.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 13.92/4.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.92/4.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 13.92/4.48 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.48 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.48 B_{B_1}(b_{a_1}(a_{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))))) 13.92/4.48 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 13.92/4.48 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 13.92/4.48 B_{B_1}(b_{a_1}(a_{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))))) 13.92/4.48 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 13.92/4.48 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 13.92/4.48 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 13.92/4.48 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 13.92/4.48 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 13.92/4.48 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 13.92/4.48 13.92/4.48 The TRS R consists of the following rules: 13.92/4.48 13.92/4.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.92/4.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.92/4.48 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.48 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.48 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.48 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.48 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.48 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.48 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.48 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.48 13.92/4.48 Q is empty. 13.92/4.48 We have to consider all minimal (P,Q,R)-chains. 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (9) DependencyGraphProof (EQUIVALENT) 13.92/4.48 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 13.92/4.48 ---------------------------------------- 13.92/4.48 13.92/4.48 (10) 13.92/4.48 Obligation: 13.92/4.48 Q DP problem: 13.92/4.48 The TRS P consists of the following rules: 13.92/4.48 13.92/4.48 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 13.92/4.48 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 13.92/4.48 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 13.92/4.48 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 13.92/4.48 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 13.92/4.48 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 13.92/4.48 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 13.92/4.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.92/4.48 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 13.92/4.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 13.92/4.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.92/4.48 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 13.92/4.48 13.92/4.48 The TRS R consists of the following rules: 13.92/4.48 13.92/4.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.92/4.48 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.92/4.48 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 13.92/4.49 Q is empty. 13.92/4.49 We have to consider all minimal (P,Q,R)-chains. 13.92/4.49 ---------------------------------------- 13.92/4.49 13.92/4.49 (11) QDPOrderProof (EQUIVALENT) 13.92/4.49 We use the reduction pair processor [LPAR04,JAR06]. 13.92/4.49 13.92/4.49 13.92/4.49 The following pairs can be oriented strictly and are deleted. 13.92/4.49 13.92/4.49 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 13.92/4.49 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) 13.92/4.49 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 13.92/4.49 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 13.92/4.49 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 13.92/4.49 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) 13.92/4.49 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 13.92/4.49 B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 13.92/4.49 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 13.92/4.49 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 13.92/4.49 The remaining pairs can at least be oriented weakly. 13.92/4.49 Used ordering: Polynomial interpretation [POLO]: 13.92/4.49 13.92/4.49 POL(A_{B_1}(x_1)) = x_1 13.92/4.49 POL(B_{A_1}(x_1)) = x_1 13.92/4.49 POL(B_{B_1}(x_1)) = x_1 13.92/4.49 POL(a_{a_1}(x_1)) = 1 + x_1 13.92/4.49 POL(a_{b_1}(x_1)) = 1 + x_1 13.92/4.49 POL(b_{a_1}(x_1)) = 1 + x_1 13.92/4.49 POL(b_{b_1}(x_1)) = 1 + x_1 13.92/4.49 13.92/4.49 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 13.92/4.49 13.92/4.49 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.92/4.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 13.92/4.49 13.92/4.49 ---------------------------------------- 13.92/4.49 13.92/4.49 (12) 13.92/4.49 Obligation: 13.92/4.49 Q DP problem: 13.92/4.49 The TRS P consists of the following rules: 13.92/4.49 13.92/4.49 B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.92/4.49 B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.92/4.49 13.92/4.49 The TRS R consists of the following rules: 13.92/4.49 13.92/4.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 13.92/4.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 b_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 a_{b_1}(b_{a_1}(a_{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))))) 13.92/4.49 13.92/4.49 Q is empty. 13.92/4.49 We have to consider all minimal (P,Q,R)-chains. 13.92/4.49 ---------------------------------------- 13.92/4.49 13.92/4.49 (13) DependencyGraphProof (EQUIVALENT) 13.92/4.49 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 13.92/4.49 ---------------------------------------- 13.92/4.49 13.92/4.49 (14) 13.92/4.49 TRUE 14.17/4.57 EOF