17.53/5.43 YES 17.64/5.48 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 17.64/5.48 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 17.64/5.48 17.64/5.48 17.64/5.48 Termination w.r.t. Q of the given QTRS could be proven: 17.64/5.48 17.64/5.48 (0) QTRS 17.64/5.48 (1) FlatCCProof [EQUIVALENT, 0 ms] 17.64/5.48 (2) QTRS 17.64/5.48 (3) RootLabelingProof [EQUIVALENT, 0 ms] 17.64/5.48 (4) QTRS 17.64/5.48 (5) DependencyPairsProof [EQUIVALENT, 43 ms] 17.64/5.48 (6) QDP 17.64/5.48 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 17.64/5.48 (8) QDP 17.64/5.48 (9) QDPOrderProof [EQUIVALENT, 46 ms] 17.64/5.48 (10) QDP 17.64/5.48 (11) PisEmptyProof [EQUIVALENT, 0 ms] 17.64/5.48 (12) YES 17.64/5.48 17.64/5.48 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (0) 17.64/5.48 Obligation: 17.64/5.48 Q restricted rewrite system: 17.64/5.48 The TRS R consists of the following rules: 17.64/5.48 17.64/5.48 b(a(b(b(x1)))) -> b(b(b(b(x1)))) 17.64/5.48 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 17.64/5.48 b(a(b(b(x1)))) -> b(a(a(a(x1)))) 17.64/5.48 17.64/5.48 Q is empty. 17.64/5.48 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (1) FlatCCProof (EQUIVALENT) 17.64/5.48 We used flat context closure [ROOTLAB] 17.64/5.48 As Q is empty the flat context closure was sound AND complete. 17.64/5.48 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (2) 17.64/5.48 Obligation: 17.64/5.48 Q restricted rewrite system: 17.64/5.48 The TRS R consists of the following rules: 17.64/5.48 17.64/5.48 b(a(b(b(x1)))) -> b(b(b(b(x1)))) 17.64/5.48 b(a(b(b(x1)))) -> b(a(a(a(x1)))) 17.64/5.48 b(b(b(a(a(x1))))) -> b(a(a(b(b(x1))))) 17.64/5.48 a(b(b(a(a(x1))))) -> a(a(a(b(b(x1))))) 17.64/5.48 17.64/5.48 Q is empty. 17.64/5.48 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (3) RootLabelingProof (EQUIVALENT) 17.64/5.48 We used plain root labeling [ROOTLAB] with the following heuristic: 17.64/5.48 LabelAll: All function symbols get labeled 17.64/5.48 17.64/5.48 As Q is empty the root labeling was sound AND complete. 17.64/5.48 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (4) 17.64/5.48 Obligation: 17.64/5.48 Q restricted rewrite system: 17.64/5.48 The TRS R consists of the following rules: 17.64/5.48 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 17.64/5.48 Q is empty. 17.64/5.48 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (5) DependencyPairsProof (EQUIVALENT) 17.64/5.48 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (6) 17.64/5.48 Obligation: 17.64/5.48 Q DP problem: 17.64/5.48 The TRS P consists of the following rules: 17.64/5.48 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 17.64/5.48 17.64/5.48 The TRS R consists of the following rules: 17.64/5.48 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 17.64/5.48 Q is empty. 17.64/5.48 We have to consider all minimal (P,Q,R)-chains. 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (7) DependencyGraphProof (EQUIVALENT) 17.64/5.48 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (8) 17.64/5.48 Obligation: 17.64/5.48 Q DP problem: 17.64/5.48 The TRS P consists of the following rules: 17.64/5.48 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 17.64/5.48 17.64/5.48 The TRS R consists of the following rules: 17.64/5.48 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 17.64/5.48 Q is empty. 17.64/5.48 We have to consider all minimal (P,Q,R)-chains. 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (9) QDPOrderProof (EQUIVALENT) 17.64/5.48 We use the reduction pair processor [LPAR04,JAR06]. 17.64/5.48 17.64/5.48 17.64/5.48 The following pairs can be oriented strictly and are deleted. 17.64/5.48 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 17.64/5.48 B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 17.64/5.48 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 17.64/5.48 A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 17.64/5.48 The remaining pairs can at least be oriented weakly. 17.64/5.48 Used ordering: Polynomial interpretation [POLO]: 17.64/5.48 17.64/5.48 POL(A_{B_1}(x_1)) = x_1 17.64/5.48 POL(B_{A_1}(x_1)) = 1 + x_1 17.64/5.48 POL(B_{B_1}(x_1)) = x_1 17.64/5.48 POL(a_{a_1}(x_1)) = 1 + x_1 17.64/5.48 POL(a_{b_1}(x_1)) = 1 + x_1 17.64/5.48 POL(b_{a_1}(x_1)) = 1 + x_1 17.64/5.48 POL(b_{b_1}(x_1)) = 1 + x_1 17.64/5.48 17.64/5.48 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 17.64/5.48 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 17.64/5.48 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (10) 17.64/5.48 Obligation: 17.64/5.48 Q DP problem: 17.64/5.48 P is empty. 17.64/5.48 The TRS R consists of the following rules: 17.64/5.48 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 17.64/5.48 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 17.64/5.48 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 17.64/5.48 17.64/5.48 Q is empty. 17.64/5.48 We have to consider all minimal (P,Q,R)-chains. 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (11) PisEmptyProof (EQUIVALENT) 17.64/5.48 The TRS P is empty. Hence, there is no (P,Q,R) chain. 17.64/5.48 ---------------------------------------- 17.64/5.48 17.64/5.48 (12) 17.64/5.48 YES 18.00/5.59 EOF