16.29/5.02 YES 16.57/5.11 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 16.57/5.11 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 16.57/5.11 16.57/5.11 16.57/5.11 Termination w.r.t. Q of the given QTRS could be proven: 16.57/5.11 16.57/5.11 (0) QTRS 16.57/5.11 (1) FlatCCProof [EQUIVALENT, 0 ms] 16.57/5.11 (2) QTRS 16.57/5.11 (3) RootLabelingProof [EQUIVALENT, 0 ms] 16.57/5.11 (4) QTRS 16.57/5.11 (5) DependencyPairsProof [EQUIVALENT, 0 ms] 16.57/5.11 (6) QDP 16.57/5.11 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 16.57/5.11 (8) QDP 16.57/5.11 (9) QDPOrderProof [EQUIVALENT, 165 ms] 16.57/5.11 (10) QDP 16.57/5.11 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 16.57/5.11 (12) TRUE 16.57/5.11 16.57/5.11 16.57/5.11 ---------------------------------------- 16.57/5.11 16.57/5.11 (0) 16.57/5.11 Obligation: 16.57/5.11 Q restricted rewrite system: 16.57/5.11 The TRS R consists of the following rules: 16.57/5.11 16.57/5.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 16.57/5.11 b(b(a(b(x1)))) -> a(b(b(a(x1)))) 16.57/5.11 b(b(a(b(x1)))) -> a(a(a(a(x1)))) 16.57/5.11 16.57/5.11 Q is empty. 16.57/5.11 16.57/5.11 ---------------------------------------- 16.57/5.11 16.57/5.11 (1) FlatCCProof (EQUIVALENT) 16.57/5.11 We used flat context closure [ROOTLAB] 16.57/5.11 As Q is empty the flat context closure was sound AND complete. 16.57/5.11 16.57/5.11 ---------------------------------------- 16.57/5.11 16.57/5.11 (2) 16.57/5.11 Obligation: 16.57/5.11 Q restricted rewrite system: 16.57/5.11 The TRS R consists of the following rules: 16.57/5.11 16.57/5.11 a(a(b(a(x1)))) -> a(b(a(b(x1)))) 16.57/5.11 a(b(b(a(b(x1))))) -> a(a(b(b(a(x1))))) 16.57/5.11 b(b(b(a(b(x1))))) -> b(a(b(b(a(x1))))) 16.57/5.11 a(b(b(a(b(x1))))) -> a(a(a(a(a(x1))))) 16.57/5.11 b(b(b(a(b(x1))))) -> b(a(a(a(a(x1))))) 16.57/5.11 16.57/5.11 Q is empty. 16.57/5.11 16.57/5.11 ---------------------------------------- 16.57/5.11 16.57/5.11 (3) RootLabelingProof (EQUIVALENT) 16.57/5.11 We used plain root labeling [ROOTLAB] with the following heuristic: 16.57/5.11 LabelAll: All function symbols get labeled 16.57/5.11 16.57/5.11 As Q is empty the root labeling was sound AND complete. 16.57/5.11 16.57/5.11 ---------------------------------------- 16.57/5.11 16.57/5.11 (4) 16.57/5.11 Obligation: 16.57/5.11 Q restricted rewrite system: 16.57/5.11 The TRS R consists of the following rules: 16.57/5.11 16.57/5.11 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 16.57/5.11 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 16.57/5.11 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.11 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.11 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.11 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.11 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.11 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.11 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.11 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.11 16.57/5.11 Q is empty. 16.57/5.11 16.57/5.11 ---------------------------------------- 16.57/5.11 16.57/5.11 (5) DependencyPairsProof (EQUIVALENT) 16.57/5.11 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 16.57/5.11 ---------------------------------------- 16.57/5.11 16.57/5.11 (6) 16.57/5.11 Obligation: 16.57/5.11 Q DP problem: 16.57/5.11 The TRS P consists of the following rules: 16.57/5.11 16.57/5.11 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 16.57/5.11 A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) 16.57/5.11 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 16.57/5.11 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) 16.57/5.11 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 16.57/5.11 16.57/5.11 The TRS R consists of the following rules: 16.57/5.11 16.57/5.11 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 16.57/5.11 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 16.57/5.11 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.11 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.11 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.11 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.11 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.11 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.11 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.11 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.11 16.57/5.11 Q is empty. 16.57/5.11 We have to consider all minimal (P,Q,R)-chains. 16.57/5.11 ---------------------------------------- 16.57/5.11 16.57/5.11 (7) DependencyGraphProof (EQUIVALENT) 16.57/5.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes. 16.57/5.11 ---------------------------------------- 16.57/5.11 16.57/5.11 (8) 16.57/5.11 Obligation: 16.57/5.11 Q DP problem: 16.57/5.11 The TRS P consists of the following rules: 16.57/5.11 16.57/5.11 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.11 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 16.57/5.11 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 16.57/5.11 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 16.57/5.13 16.57/5.13 The TRS R consists of the following rules: 16.57/5.13 16.57/5.13 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 16.57/5.13 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.13 16.57/5.13 Q is empty. 16.57/5.13 We have to consider all minimal (P,Q,R)-chains. 16.57/5.13 ---------------------------------------- 16.57/5.13 16.57/5.13 (9) QDPOrderProof (EQUIVALENT) 16.57/5.13 We use the reduction pair processor [LPAR04,JAR06]. 16.57/5.13 16.57/5.13 16.57/5.13 The following pairs can be oriented strictly and are deleted. 16.57/5.13 16.57/5.13 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) 16.57/5.13 A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) 16.57/5.13 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 16.57/5.13 The remaining pairs can at least be oriented weakly. 16.57/5.13 Used ordering: Polynomial interpretation [POLO]: 16.57/5.13 16.57/5.13 POL(A_{A_1}(x_1)) = x_1 16.57/5.13 POL(A_{B_1}(x_1)) = x_1 16.57/5.13 POL(B_{B_1}(x_1)) = x_1 16.57/5.13 POL(a_{a_1}(x_1)) = 1 + x_1 16.57/5.13 POL(a_{b_1}(x_1)) = 1 + x_1 16.57/5.13 POL(b_{a_1}(x_1)) = 1 + x_1 16.57/5.13 POL(b_{b_1}(x_1)) = 1 + x_1 16.57/5.13 16.57/5.13 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 16.57/5.13 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.13 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 16.57/5.13 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.13 16.57/5.13 16.57/5.13 ---------------------------------------- 16.57/5.13 16.57/5.13 (10) 16.57/5.13 Obligation: 16.57/5.13 Q DP problem: 16.57/5.13 The TRS P consists of the following rules: 16.57/5.13 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.13 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.13 16.57/5.13 The TRS R consists of the following rules: 16.57/5.13 16.57/5.13 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 16.57/5.13 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.13 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) 16.57/5.13 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) 16.57/5.13 16.57/5.13 Q is empty. 16.57/5.13 We have to consider all minimal (P,Q,R)-chains. 16.57/5.13 ---------------------------------------- 16.57/5.13 16.57/5.13 (11) DependencyGraphProof (EQUIVALENT) 16.57/5.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. 16.57/5.13 ---------------------------------------- 16.57/5.13 16.57/5.13 (12) 16.57/5.13 TRUE 16.97/5.24 EOF