139.35/36.38 YES 139.67/36.46 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 139.67/36.46 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 139.67/36.46 139.67/36.46 139.67/36.46 Termination w.r.t. Q of the given QTRS could be proven: 139.67/36.46 139.67/36.46 (0) QTRS 139.67/36.46 (1) QTRS Reverse [EQUIVALENT, 0 ms] 139.67/36.46 (2) QTRS 139.67/36.46 (3) FlatCCProof [EQUIVALENT, 0 ms] 139.67/36.46 (4) QTRS 139.67/36.46 (5) RootLabelingProof [EQUIVALENT, 0 ms] 139.67/36.46 (6) QTRS 139.67/36.46 (7) DependencyPairsProof [EQUIVALENT, 109 ms] 139.67/36.46 (8) QDP 139.67/36.46 (9) QDPOrderProof [EQUIVALENT, 183 ms] 139.67/36.46 (10) QDP 139.67/36.46 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 139.67/36.46 (12) AND 139.67/36.46 (13) QDP 139.67/36.46 (14) QDPOrderProof [EQUIVALENT, 2174 ms] 139.67/36.46 (15) QDP 139.67/36.46 (16) QDPOrderProof [EQUIVALENT, 1069 ms] 139.67/36.46 (17) QDP 139.67/36.46 (18) QDPOrderProof [EQUIVALENT, 1333 ms] 139.67/36.46 (19) QDP 139.67/36.46 (20) DependencyGraphProof [EQUIVALENT, 0 ms] 139.67/36.46 (21) TRUE 139.67/36.46 (22) QDP 139.67/36.46 (23) QDPOrderProof [EQUIVALENT, 2140 ms] 139.67/36.46 (24) QDP 139.67/36.46 (25) QDPOrderProof [EQUIVALENT, 1040 ms] 139.67/36.46 (26) QDP 139.67/36.46 (27) QDPOrderProof [EQUIVALENT, 1308 ms] 139.67/36.46 (28) QDP 139.67/36.46 (29) DependencyGraphProof [EQUIVALENT, 0 ms] 139.67/36.46 (30) TRUE 139.67/36.46 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (0) 139.67/36.46 Obligation: 139.67/36.46 Q restricted rewrite system: 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a(b(a(a(x1)))) -> b(b(b(b(x1)))) 139.67/36.46 b(b(a(b(x1)))) -> a(b(a(a(x1)))) 139.67/36.46 b(a(b(b(x1)))) -> a(a(b(a(x1)))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (1) QTRS Reverse (EQUIVALENT) 139.67/36.46 We applied the QTRS Reverse Processor [REVERSE]. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (2) 139.67/36.46 Obligation: 139.67/36.46 Q restricted rewrite system: 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a(a(b(a(x1)))) -> b(b(b(b(x1)))) 139.67/36.46 b(a(b(b(x1)))) -> a(a(b(a(x1)))) 139.67/36.46 b(b(a(b(x1)))) -> a(b(a(a(x1)))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (3) FlatCCProof (EQUIVALENT) 139.67/36.46 We used flat context closure [ROOTLAB] 139.67/36.46 As Q is empty the flat context closure was sound AND complete. 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (4) 139.67/36.46 Obligation: 139.67/36.46 Q restricted rewrite system: 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a(a(a(b(a(x1))))) -> a(b(b(b(b(x1))))) 139.67/36.46 b(a(a(b(a(x1))))) -> b(b(b(b(b(x1))))) 139.67/36.46 a(b(a(b(b(x1))))) -> a(a(a(b(a(x1))))) 139.67/36.46 b(b(a(b(b(x1))))) -> b(a(a(b(a(x1))))) 139.67/36.46 a(b(b(a(b(x1))))) -> a(a(b(a(a(x1))))) 139.67/36.46 b(b(b(a(b(x1))))) -> b(a(b(a(a(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (5) RootLabelingProof (EQUIVALENT) 139.67/36.46 We used plain root labeling [ROOTLAB] with the following heuristic: 139.67/36.46 LabelAll: All function symbols get labeled 139.67/36.46 139.67/36.46 As Q is empty the root labeling was sound AND complete. 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (6) 139.67/36.46 Obligation: 139.67/36.46 Q restricted rewrite system: 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (7) DependencyPairsProof (EQUIVALENT) 139.67/36.46 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (8) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 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}(x1)))) 139.67/36.46 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}(x1))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1)) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 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_{b_1}(x1)))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1)) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 139.67/36.46 B_{B_1}(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}(a_{a_1}(x1))))) 139.67/36.46 B_{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}(x1)))) 139.67/36.46 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}(x1))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1)) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 B_{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_{b_1}(x1)))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1)) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 139.67/36.46 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}(x1))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (9) QDPOrderProof (EQUIVALENT) 139.67/36.46 We use the reduction pair processor [LPAR04,JAR06]. 139.67/36.46 139.67/36.46 139.67/36.46 The following pairs can be oriented strictly and are deleted. 139.67/36.46 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 139.67/36.46 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}(x1)))) 139.67/36.46 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}(x1))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1)) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 139.67/36.46 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_{b_1}(x1)))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1)) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 139.67/36.46 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}(x1))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1)) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1)) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 139.67/36.46 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}(x1))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) 139.67/36.46 The remaining pairs can at least be oriented weakly. 139.67/36.46 Used ordering: Polynomial interpretation [POLO]: 139.67/36.46 139.67/36.46 POL(A_{A_1}(x_1)) = 1 + x_1 139.67/36.46 POL(A_{B_1}(x_1)) = 1 + x_1 139.67/36.46 POL(B_{A_1}(x_1)) = x_1 139.67/36.46 POL(B_{B_1}(x_1)) = x_1 139.67/36.46 POL(a_{a_1}(x_1)) = 1 + x_1 139.67/36.46 POL(a_{b_1}(x_1)) = 1 + x_1 139.67/36.46 POL(b_{a_1}(x_1)) = 1 + x_1 139.67/36.46 POL(b_{b_1}(x_1)) = 1 + x_1 139.67/36.46 139.67/36.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 139.67/36.46 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (10) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 B_{B_1}(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}(a_{a_1}(x1))))) 139.67/36.46 B_{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}(x1)))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 B_{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_{b_1}(x1)))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (11) DependencyGraphProof (EQUIVALENT) 139.67/36.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (12) 139.67/36.46 Complex Obligation (AND) 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (13) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (14) QDPOrderProof (EQUIVALENT) 139.67/36.46 We use the reduction pair processor [LPAR04,JAR06]. 139.67/36.46 139.67/36.46 139.67/36.46 The following pairs can be oriented strictly and are deleted. 139.67/36.46 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 The remaining pairs can at least be oriented weakly. 139.67/36.46 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(A_{B_1}(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 1A, -I], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{b_1}(x_1)) = [[0A], [0A], [1A]] + [[0A, -I, -I], [-I, -I, 0A], [-I, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 1A], [0A, 0A, 1A], [-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(A_{A_1}(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, 0A, 0A], [0A, -I, -I]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 139.67/36.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 139.67/36.46 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (15) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (16) QDPOrderProof (EQUIVALENT) 139.67/36.46 We use the reduction pair processor [LPAR04,JAR06]. 139.67/36.46 139.67/36.46 139.67/36.46 The following pairs can be oriented strictly and are deleted. 139.67/36.46 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 The remaining pairs can at least be oriented weakly. 139.67/36.46 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(A_{B_1}(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[1A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(A_{A_1}(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, 0A], [1A, 0A, 0A], [-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 139.67/36.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 139.67/36.46 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (17) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (18) QDPOrderProof (EQUIVALENT) 139.67/36.46 We use the reduction pair processor [LPAR04,JAR06]. 139.67/36.46 139.67/36.46 139.67/36.46 The following pairs can be oriented strictly and are deleted. 139.67/36.46 139.67/36.46 A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 The remaining pairs can at least be oriented weakly. 139.67/36.46 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(A_{B_1}(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, -I], [0A, -I, 0A], [0A, -I, -I]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{b_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [1A, 0A, 0A], [1A, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(A_{A_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{a_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 139.67/36.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 139.67/36.46 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (19) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (20) DependencyGraphProof (EQUIVALENT) 139.67/36.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (21) 139.67/36.46 TRUE 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (22) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 B_{B_1}(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}(a_{a_1}(x1))))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (23) QDPOrderProof (EQUIVALENT) 139.67/36.46 We use the reduction pair processor [LPAR04,JAR06]. 139.67/36.46 139.67/36.46 139.67/36.46 The following pairs can be oriented strictly and are deleted. 139.67/36.46 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 The remaining pairs can at least be oriented weakly. 139.67/36.46 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(B_{B_1}(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 1A, -I], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{b_1}(x_1)) = [[0A], [0A], [1A]] + [[0A, -I, -I], [-I, -I, 0A], [-I, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 1A], [0A, 0A, 1A], [-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(B_{A_1}(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 0A], [-I, 0A, 0A], [0A, -I, -I]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 139.67/36.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 139.67/36.46 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (24) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 B_{B_1}(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}(a_{a_1}(x1))))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (25) QDPOrderProof (EQUIVALENT) 139.67/36.46 We use the reduction pair processor [LPAR04,JAR06]. 139.67/36.46 139.67/36.46 139.67/36.46 The following pairs can be oriented strictly and are deleted. 139.67/36.46 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 The remaining pairs can at least be oriented weakly. 139.67/36.46 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(B_{B_1}(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[1A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(B_{A_1}(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, 0A], [1A, 0A, 0A], [-I, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 139.67/36.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 139.67/36.46 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (26) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 B_{B_1}(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}(a_{a_1}(x1))))) 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (27) QDPOrderProof (EQUIVALENT) 139.67/36.46 We use the reduction pair processor [LPAR04,JAR06]. 139.67/36.46 139.67/36.46 139.67/36.46 The following pairs can be oriented strictly and are deleted. 139.67/36.46 139.67/36.46 B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 The remaining pairs can at least be oriented weakly. 139.67/36.46 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(B_{B_1}(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, -I], [0A, -I, 0A], [0A, -I, -I]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(b_{b_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [1A, 0A, 0A], [1A, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(B_{A_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 <<< 139.67/36.46 POL(a_{a_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 139.67/36.46 >>> 139.67/36.46 139.67/36.46 139.67/36.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 139.67/36.46 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (28) 139.67/36.46 Obligation: 139.67/36.46 Q DP problem: 139.67/36.46 The TRS P consists of the following rules: 139.67/36.46 139.67/36.46 B_{B_1}(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}(a_{a_1}(x1))))) 139.67/36.46 B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 The TRS R consists of the following rules: 139.67/36.46 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 139.67/36.46 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(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}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) 139.67/36.46 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) 139.67/36.46 139.67/36.46 Q is empty. 139.67/36.46 We have to consider all minimal (P,Q,R)-chains. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (29) DependencyGraphProof (EQUIVALENT) 139.67/36.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 139.67/36.46 ---------------------------------------- 139.67/36.46 139.67/36.46 (30) 139.67/36.46 TRUE 139.93/36.55 EOF