154.28/40.33 YES 154.62/40.36 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 154.62/40.36 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 154.62/40.36 154.62/40.36 154.62/40.36 Termination w.r.t. Q of the given QTRS could be proven: 154.62/40.36 154.62/40.36 (0) QTRS 154.62/40.36 (1) FlatCCProof [EQUIVALENT, 0 ms] 154.62/40.36 (2) QTRS 154.62/40.36 (3) RootLabelingProof [EQUIVALENT, 0 ms] 154.62/40.36 (4) QTRS 154.62/40.36 (5) DependencyPairsProof [EQUIVALENT, 11 ms] 154.62/40.36 (6) QDP 154.62/40.36 (7) DependencyGraphProof [EQUIVALENT, 2 ms] 154.62/40.36 (8) QDP 154.62/40.36 (9) QDPOrderProof [EQUIVALENT, 303 ms] 154.62/40.36 (10) QDP 154.62/40.36 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 154.62/40.36 (12) QDP 154.62/40.36 (13) QDPOrderProof [EQUIVALENT, 5576 ms] 154.62/40.36 (14) QDP 154.62/40.36 (15) QDPOrderProof [EQUIVALENT, 2741 ms] 154.62/40.36 (16) QDP 154.62/40.36 (17) PisEmptyProof [EQUIVALENT, 0 ms] 154.62/40.36 (18) YES 154.62/40.36 154.62/40.36 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (0) 154.62/40.36 Obligation: 154.62/40.36 Q restricted rewrite system: 154.62/40.36 The TRS R consists of the following rules: 154.62/40.36 154.62/40.36 a(a(b(a(x1)))) -> a(a(b(b(x1)))) 154.62/40.36 b(b(b(b(x1)))) -> b(a(b(a(x1)))) 154.62/40.36 a(b(a(a(x1)))) -> a(a(a(b(x1)))) 154.62/40.36 b(a(b(b(x1)))) -> a(b(a(a(x1)))) 154.62/40.36 154.62/40.36 Q is empty. 154.62/40.36 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (1) FlatCCProof (EQUIVALENT) 154.62/40.36 We used flat context closure [ROOTLAB] 154.62/40.36 As Q is empty the flat context closure was sound AND complete. 154.62/40.36 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (2) 154.62/40.36 Obligation: 154.62/40.36 Q restricted rewrite system: 154.62/40.36 The TRS R consists of the following rules: 154.62/40.36 154.62/40.36 a(a(b(a(x1)))) -> a(a(b(b(x1)))) 154.62/40.36 b(b(b(b(x1)))) -> b(a(b(a(x1)))) 154.62/40.36 a(b(a(a(x1)))) -> a(a(a(b(x1)))) 154.62/40.36 a(b(a(b(b(x1))))) -> a(a(b(a(a(x1))))) 154.62/40.36 b(b(a(b(b(x1))))) -> b(a(b(a(a(x1))))) 154.62/40.36 154.62/40.36 Q is empty. 154.62/40.36 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (3) RootLabelingProof (EQUIVALENT) 154.62/40.36 We used plain root labeling [ROOTLAB] with the following heuristic: 154.62/40.36 LabelAll: All function symbols get labeled 154.62/40.36 154.62/40.36 As Q is empty the root labeling was sound AND complete. 154.62/40.36 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (4) 154.62/40.36 Obligation: 154.62/40.36 Q restricted rewrite system: 154.62/40.36 The TRS R consists of the following rules: 154.62/40.36 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 154.62/40.36 Q is empty. 154.62/40.36 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (5) DependencyPairsProof (EQUIVALENT) 154.62/40.36 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (6) 154.62/40.36 Obligation: 154.62/40.36 Q DP problem: 154.62/40.36 The TRS P consists of the following rules: 154.62/40.36 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 154.62/40.36 154.62/40.36 The TRS R consists of the following rules: 154.62/40.36 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 154.62/40.36 Q is empty. 154.62/40.36 We have to consider all minimal (P,Q,R)-chains. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (7) DependencyGraphProof (EQUIVALENT) 154.62/40.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (8) 154.62/40.36 Obligation: 154.62/40.36 Q DP problem: 154.62/40.36 The TRS P consists of the following rules: 154.62/40.36 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 154.62/40.36 154.62/40.36 The TRS R consists of the following rules: 154.62/40.36 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 154.62/40.36 Q is empty. 154.62/40.36 We have to consider all minimal (P,Q,R)-chains. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (9) QDPOrderProof (EQUIVALENT) 154.62/40.36 We use the reduction pair processor [LPAR04,JAR06]. 154.62/40.36 154.62/40.36 154.62/40.36 The following pairs can be oriented strictly and are deleted. 154.62/40.36 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1)) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1)) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 154.62/40.36 B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 154.62/40.36 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 154.62/40.36 The remaining pairs can at least be oriented weakly. 154.62/40.36 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 154.62/40.36 154.62/40.36 POL( A_{B_1}_1(x_1) ) = 2x_1 + 2 154.62/40.36 POL( B_{B_1}_1(x_1) ) = 2x_1 + 2 154.62/40.36 POL( A_{A_1}_1(x_1) ) = 2x_1 + 2 154.62/40.36 POL( a_{b_1}_1(x_1) ) = 2x_1 + 2 154.62/40.36 POL( b_{b_1}_1(x_1) ) = 2x_1 + 2 154.62/40.36 POL( b_{a_1}_1(x_1) ) = 2x_1 + 2 154.62/40.36 POL( a_{a_1}_1(x_1) ) = 2x_1 + 2 154.62/40.36 154.62/40.36 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 154.62/40.36 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 154.62/40.36 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (10) 154.62/40.36 Obligation: 154.62/40.36 Q DP problem: 154.62/40.36 The TRS P consists of the following rules: 154.62/40.36 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 154.62/40.36 The TRS R consists of the following rules: 154.62/40.36 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 154.62/40.36 Q is empty. 154.62/40.36 We have to consider all minimal (P,Q,R)-chains. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (11) DependencyGraphProof (EQUIVALENT) 154.62/40.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (12) 154.62/40.36 Obligation: 154.62/40.36 Q DP problem: 154.62/40.36 The TRS P consists of the following rules: 154.62/40.36 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 154.62/40.36 The TRS R consists of the following rules: 154.62/40.36 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 154.62/40.36 Q is empty. 154.62/40.36 We have to consider all minimal (P,Q,R)-chains. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (13) QDPOrderProof (EQUIVALENT) 154.62/40.36 We use the reduction pair processor [LPAR04,JAR06]. 154.62/40.36 154.62/40.36 154.62/40.36 The following pairs can be oriented strictly and are deleted. 154.62/40.36 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 The remaining pairs can at least be oriented weakly. 154.62/40.36 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(A_{A_1}(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(a_{b_1}(x_1)) = [[-I], [0A], [1A]] + [[0A, -I, -I], [-I, -I, 0A], [0A, 0A, 0A]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(b_{a_1}(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [-I, -I, -I], [1A, 0A, 0A]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(b_{b_1}(x_1)) = [[1A], [0A], [0A]] + [[1A, 0A, 0A], [0A, -I, -I], [-I, 0A, -I]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(a_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 154.62/40.36 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 154.62/40.36 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 154.62/40.36 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (14) 154.62/40.36 Obligation: 154.62/40.36 Q DP problem: 154.62/40.36 The TRS P consists of the following rules: 154.62/40.36 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 154.62/40.36 The TRS R consists of the following rules: 154.62/40.36 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 154.62/40.36 Q is empty. 154.62/40.36 We have to consider all minimal (P,Q,R)-chains. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (15) QDPOrderProof (EQUIVALENT) 154.62/40.36 We use the reduction pair processor [LPAR04,JAR06]. 154.62/40.36 154.62/40.36 154.62/40.36 The following pairs can be oriented strictly and are deleted. 154.62/40.36 154.62/40.36 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 The remaining pairs can at least be oriented weakly. 154.62/40.36 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(A_{A_1}(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, -I, 0A], [-I, 0A, 0A]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(b_{a_1}(x_1)) = [[1A], [0A], [0A]] + [[1A, -I, 0A], [0A, 1A, 0A], [1A, 0A, 0A]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(a_{a_1}(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, -I], [1A, -I, 0A], [-I, -I, -I]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 <<< 154.62/40.36 POL(b_{b_1}(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, 1A], [0A, -I, -I], [0A, -I, -I]] * x_1 154.62/40.36 >>> 154.62/40.36 154.62/40.36 154.62/40.36 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 154.62/40.36 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 154.62/40.36 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (16) 154.62/40.36 Obligation: 154.62/40.36 Q DP problem: 154.62/40.36 P is empty. 154.62/40.36 The TRS R consists of the following rules: 154.62/40.36 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) 154.62/40.36 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 154.62/40.36 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 154.62/40.36 154.62/40.36 Q is empty. 154.62/40.36 We have to consider all minimal (P,Q,R)-chains. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (17) PisEmptyProof (EQUIVALENT) 154.62/40.36 The TRS P is empty. Hence, there is no (P,Q,R) chain. 154.62/40.36 ---------------------------------------- 154.62/40.36 154.62/40.36 (18) 154.62/40.36 YES 154.76/40.44 EOF