25.65/7.61 YES 25.65/7.64 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 25.65/7.64 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.65/7.64 25.65/7.64 25.65/7.64 Termination w.r.t. Q of the given QTRS could be proven: 25.65/7.64 25.65/7.64 (0) QTRS 25.65/7.64 (1) QTRS Reverse [EQUIVALENT, 0 ms] 25.65/7.64 (2) QTRS 25.65/7.64 (3) FlatCCProof [EQUIVALENT, 0 ms] 25.65/7.64 (4) QTRS 25.65/7.64 (5) RootLabelingProof [EQUIVALENT, 0 ms] 25.65/7.64 (6) QTRS 25.65/7.64 (7) DependencyPairsProof [EQUIVALENT, 12 ms] 25.65/7.64 (8) QDP 25.65/7.64 (9) DependencyGraphProof [EQUIVALENT, 2 ms] 25.65/7.64 (10) QDP 25.65/7.64 (11) QDPOrderProof [EQUIVALENT, 122 ms] 25.65/7.64 (12) QDP 25.65/7.64 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 25.65/7.64 (14) TRUE 25.65/7.64 25.65/7.64 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (0) 25.65/7.64 Obligation: 25.65/7.64 Q restricted rewrite system: 25.65/7.64 The TRS R consists of the following rules: 25.65/7.64 25.65/7.64 b(b(a(b(x1)))) -> b(a(a(b(x1)))) 25.65/7.64 a(a(a(a(x1)))) -> b(b(a(b(x1)))) 25.65/7.64 a(b(a(a(x1)))) -> a(a(a(b(x1)))) 25.65/7.64 25.65/7.64 Q is empty. 25.65/7.64 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (1) QTRS Reverse (EQUIVALENT) 25.65/7.64 We applied the QTRS Reverse Processor [REVERSE]. 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (2) 25.65/7.64 Obligation: 25.65/7.64 Q restricted rewrite system: 25.65/7.64 The TRS R consists of the following rules: 25.65/7.64 25.65/7.64 b(a(b(b(x1)))) -> b(a(a(b(x1)))) 25.65/7.64 a(a(a(a(x1)))) -> b(a(b(b(x1)))) 25.65/7.64 a(a(b(a(x1)))) -> b(a(a(a(x1)))) 25.65/7.64 25.65/7.64 Q is empty. 25.65/7.64 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (3) FlatCCProof (EQUIVALENT) 25.65/7.64 We used flat context closure [ROOTLAB] 25.65/7.64 As Q is empty the flat context closure was sound AND complete. 25.65/7.64 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (4) 25.65/7.64 Obligation: 25.65/7.64 Q restricted rewrite system: 25.65/7.64 The TRS R consists of the following rules: 25.65/7.64 25.65/7.64 b(a(b(b(x1)))) -> b(a(a(b(x1)))) 25.65/7.64 b(a(a(a(a(x1))))) -> b(b(a(b(b(x1))))) 25.65/7.64 a(a(a(a(a(x1))))) -> a(b(a(b(b(x1))))) 25.65/7.64 b(a(a(b(a(x1))))) -> b(b(a(a(a(x1))))) 25.65/7.64 a(a(a(b(a(x1))))) -> a(b(a(a(a(x1))))) 25.65/7.64 25.65/7.64 Q is empty. 25.65/7.64 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (5) RootLabelingProof (EQUIVALENT) 25.65/7.64 We used plain root labeling [ROOTLAB] with the following heuristic: 25.65/7.64 LabelAll: All function symbols get labeled 25.65/7.64 25.65/7.64 As Q is empty the root labeling was sound AND complete. 25.65/7.64 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (6) 25.65/7.64 Obligation: 25.65/7.64 Q restricted rewrite system: 25.65/7.64 The TRS R consists of the following rules: 25.65/7.64 25.65/7.64 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}(x1)))) 25.65/7.64 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 25.65/7.64 Q is empty. 25.65/7.64 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (7) DependencyPairsProof (EQUIVALENT) 25.65/7.64 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (8) 25.65/7.64 Obligation: 25.65/7.64 Q DP problem: 25.65/7.64 The TRS P consists of the following rules: 25.65/7.64 25.65/7.64 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}(x1)))) 25.65/7.64 B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 25.65/7.64 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.65/7.64 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 25.65/7.64 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 25.65/7.64 25.65/7.64 The TRS R consists of the following rules: 25.65/7.64 25.65/7.64 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}(x1)))) 25.65/7.64 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 25.65/7.64 Q is empty. 25.65/7.64 We have to consider all minimal (P,Q,R)-chains. 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (9) DependencyGraphProof (EQUIVALENT) 25.65/7.64 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes. 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (10) 25.65/7.64 Obligation: 25.65/7.64 Q DP problem: 25.65/7.64 The TRS P consists of the following rules: 25.65/7.64 25.65/7.64 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 25.65/7.64 25.65/7.64 The TRS R consists of the following rules: 25.65/7.64 25.65/7.64 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}(x1)))) 25.65/7.64 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 25.65/7.64 Q is empty. 25.65/7.64 We have to consider all minimal (P,Q,R)-chains. 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (11) QDPOrderProof (EQUIVALENT) 25.65/7.64 We use the reduction pair processor [LPAR04,JAR06]. 25.65/7.64 25.65/7.64 25.65/7.64 The following pairs can be oriented strictly and are deleted. 25.65/7.64 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 25.65/7.64 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 25.65/7.64 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 25.65/7.64 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 25.65/7.64 The remaining pairs can at least be oriented weakly. 25.65/7.64 Used ordering: Polynomial interpretation [POLO]: 25.65/7.64 25.65/7.64 POL(A_{A_1}(x_1)) = x_1 25.65/7.64 POL(B_{A_1}(x_1)) = x_1 25.65/7.64 POL(a_{a_1}(x_1)) = 1 + x_1 25.65/7.64 POL(a_{b_1}(x_1)) = 1 + x_1 25.65/7.64 POL(b_{a_1}(x_1)) = 1 + x_1 25.65/7.64 POL(b_{b_1}(x_1)) = 1 + x_1 25.65/7.64 25.65/7.64 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.65/7.64 25.65/7.64 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}(x1)))) 25.65/7.64 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 25.65/7.64 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (12) 25.65/7.64 Obligation: 25.65/7.64 Q DP problem: 25.65/7.64 The TRS P consists of the following rules: 25.65/7.64 25.65/7.64 B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.65/7.64 25.65/7.64 The TRS R consists of the following rules: 25.65/7.64 25.65/7.64 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}(x1)))) 25.65/7.64 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 25.65/7.64 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 25.65/7.64 25.65/7.64 Q is empty. 25.65/7.64 We have to consider all minimal (P,Q,R)-chains. 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (13) DependencyGraphProof (EQUIVALENT) 25.65/7.64 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 25.65/7.64 ---------------------------------------- 25.65/7.64 25.65/7.64 (14) 25.65/7.64 TRUE 26.46/7.71 EOF