18.87/5.66 YES 18.87/5.67 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 18.87/5.67 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 18.87/5.67 18.87/5.67 18.87/5.67 Termination w.r.t. Q of the given QTRS could be proven: 18.87/5.67 18.87/5.67 (0) QTRS 18.87/5.67 (1) QTRS Reverse [EQUIVALENT, 0 ms] 18.87/5.67 (2) QTRS 18.87/5.67 (3) FlatCCProof [EQUIVALENT, 1 ms] 18.87/5.67 (4) QTRS 18.87/5.67 (5) RootLabelingProof [EQUIVALENT, 0 ms] 18.87/5.67 (6) QTRS 18.87/5.67 (7) QTRSRRRProof [EQUIVALENT, 52 ms] 18.87/5.67 (8) QTRS 18.87/5.67 (9) DependencyPairsProof [EQUIVALENT, 0 ms] 18.87/5.67 (10) QDP 18.87/5.67 (11) QDPOrderProof [EQUIVALENT, 47 ms] 18.87/5.67 (12) QDP 18.87/5.67 (13) QDPOrderProof [EQUIVALENT, 25 ms] 18.87/5.67 (14) QDP 18.87/5.67 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 18.87/5.67 (16) TRUE 18.87/5.67 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (0) 18.87/5.67 Obligation: 18.87/5.67 Q restricted rewrite system: 18.87/5.67 The TRS R consists of the following rules: 18.87/5.67 18.87/5.67 b(a(a(b(x1)))) -> b(a(a(a(x1)))) 18.87/5.67 a(b(b(a(x1)))) -> a(a(b(b(x1)))) 18.87/5.67 b(a(a(a(x1)))) -> b(b(b(a(x1)))) 18.87/5.67 18.87/5.67 Q is empty. 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (1) QTRS Reverse (EQUIVALENT) 18.87/5.67 We applied the QTRS Reverse Processor [REVERSE]. 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (2) 18.87/5.67 Obligation: 18.87/5.67 Q restricted rewrite system: 18.87/5.67 The TRS R consists of the following rules: 18.87/5.67 18.87/5.67 b(a(a(b(x1)))) -> a(a(a(b(x1)))) 18.87/5.67 a(b(b(a(x1)))) -> b(b(a(a(x1)))) 18.87/5.67 a(a(a(b(x1)))) -> a(b(b(b(x1)))) 18.87/5.67 18.87/5.67 Q is empty. 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (3) FlatCCProof (EQUIVALENT) 18.87/5.67 We used flat context closure [ROOTLAB] 18.87/5.67 As Q is empty the flat context closure was sound AND complete. 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (4) 18.87/5.67 Obligation: 18.87/5.67 Q restricted rewrite system: 18.87/5.67 The TRS R consists of the following rules: 18.87/5.67 18.87/5.67 a(a(a(b(x1)))) -> a(b(b(b(x1)))) 18.87/5.67 b(b(a(a(b(x1))))) -> b(a(a(a(b(x1))))) 18.87/5.67 a(b(a(a(b(x1))))) -> a(a(a(a(b(x1))))) 18.87/5.67 b(a(b(b(a(x1))))) -> b(b(b(a(a(x1))))) 18.87/5.67 a(a(b(b(a(x1))))) -> a(b(b(a(a(x1))))) 18.87/5.67 18.87/5.67 Q is empty. 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (5) RootLabelingProof (EQUIVALENT) 18.87/5.67 We used plain root labeling [ROOTLAB] with the following heuristic: 18.87/5.67 LabelAll: All function symbols get labeled 18.87/5.67 18.87/5.67 As Q is empty the root labeling was sound AND complete. 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (6) 18.87/5.67 Obligation: 18.87/5.67 Q restricted rewrite system: 18.87/5.67 The TRS R consists of the following rules: 18.87/5.67 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 18.87/5.67 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 18.87/5.67 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 18.87/5.67 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 18.87/5.67 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 18.87/5.67 18.87/5.67 Q is empty. 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (7) QTRSRRRProof (EQUIVALENT) 18.87/5.67 Used ordering: 18.87/5.67 Polynomial interpretation [POLO]: 18.87/5.67 18.87/5.67 POL(a_{a_1}(x_1)) = x_1 18.87/5.67 POL(a_{b_1}(x_1)) = x_1 18.87/5.67 POL(b_{a_1}(x_1)) = 1 + x_1 18.87/5.67 POL(b_{b_1}(x_1)) = x_1 18.87/5.67 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 18.87/5.67 18.87/5.67 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 18.87/5.67 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 18.87/5.67 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 18.87/5.67 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 18.87/5.67 18.87/5.67 18.87/5.67 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (8) 18.87/5.67 Obligation: 18.87/5.67 Q restricted rewrite system: 18.87/5.67 The TRS R consists of the following rules: 18.87/5.67 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 18.87/5.67 18.87/5.67 Q is empty. 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (9) DependencyPairsProof (EQUIVALENT) 18.87/5.67 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (10) 18.87/5.67 Obligation: 18.87/5.67 Q DP problem: 18.87/5.67 The TRS P consists of the following rules: 18.87/5.67 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 18.87/5.67 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 18.87/5.67 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 18.87/5.67 18.87/5.67 The TRS R consists of the following rules: 18.87/5.67 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 18.87/5.67 18.87/5.67 Q is empty. 18.87/5.67 We have to consider all minimal (P,Q,R)-chains. 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (11) QDPOrderProof (EQUIVALENT) 18.87/5.67 We use the reduction pair processor [LPAR04,JAR06]. 18.87/5.67 18.87/5.67 18.87/5.67 The following pairs can be oriented strictly and are deleted. 18.87/5.67 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 18.87/5.67 The remaining pairs can at least be oriented weakly. 18.87/5.67 Used ordering: Polynomial interpretation [POLO]: 18.87/5.67 18.87/5.67 POL(A_{A_1}(x_1)) = x_1 18.87/5.67 POL(B_{B_1}(x_1)) = x_1 18.87/5.67 POL(a_{a_1}(x_1)) = 1 + x_1 18.87/5.67 POL(a_{b_1}(x_1)) = x_1 18.87/5.67 POL(b_{a_1}(x_1)) = x_1 18.87/5.67 POL(b_{b_1}(x_1)) = 1 + x_1 18.87/5.67 18.87/5.67 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 18.87/5.67 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 18.87/5.67 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (12) 18.87/5.67 Obligation: 18.87/5.67 Q DP problem: 18.87/5.67 The TRS P consists of the following rules: 18.87/5.67 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 18.87/5.67 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 18.87/5.67 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 18.87/5.67 18.87/5.67 The TRS R consists of the following rules: 18.87/5.67 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 18.87/5.67 18.87/5.67 Q is empty. 18.87/5.67 We have to consider all minimal (P,Q,R)-chains. 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (13) QDPOrderProof (EQUIVALENT) 18.87/5.67 We use the reduction pair processor [LPAR04,JAR06]. 18.87/5.67 18.87/5.67 18.87/5.67 The following pairs can be oriented strictly and are deleted. 18.87/5.67 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 18.87/5.67 A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 18.87/5.67 A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 18.87/5.67 The remaining pairs can at least be oriented weakly. 18.87/5.67 Used ordering: Polynomial interpretation [POLO]: 18.87/5.67 18.87/5.67 POL(A_{A_1}(x_1)) = x_1 18.87/5.67 POL(B_{B_1}(x_1)) = x_1 18.87/5.67 POL(a_{a_1}(x_1)) = 1 + x_1 18.87/5.67 POL(a_{b_1}(x_1)) = 1 + x_1 18.87/5.67 POL(b_{a_1}(x_1)) = x_1 18.87/5.67 POL(b_{b_1}(x_1)) = 1 + x_1 18.87/5.67 18.87/5.67 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 18.87/5.67 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 18.87/5.67 18.87/5.67 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (14) 18.87/5.67 Obligation: 18.87/5.67 Q DP problem: 18.87/5.67 The TRS P consists of the following rules: 18.87/5.67 18.87/5.67 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 18.87/5.67 B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 18.87/5.67 18.87/5.67 The TRS R consists of the following rules: 18.87/5.67 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 18.87/5.67 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) 18.87/5.67 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 18.87/5.67 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 18.87/5.67 18.87/5.67 Q is empty. 18.87/5.67 We have to consider all minimal (P,Q,R)-chains. 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (15) DependencyGraphProof (EQUIVALENT) 18.87/5.67 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 18.87/5.67 ---------------------------------------- 18.87/5.67 18.87/5.67 (16) 18.87/5.67 TRUE 19.03/5.72 EOF