175.12/45.52 YES 175.45/45.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 175.45/45.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 175.45/45.57 175.45/45.57 175.45/45.57 Termination w.r.t. Q of the given QTRS could be proven: 175.45/45.57 175.45/45.57 (0) QTRS 175.45/45.57 (1) FlatCCProof [EQUIVALENT, 0 ms] 175.45/45.57 (2) QTRS 175.45/45.57 (3) RootLabelingProof [EQUIVALENT, 0 ms] 175.45/45.57 (4) QTRS 175.45/45.57 (5) DependencyPairsProof [EQUIVALENT, 56 ms] 175.45/45.57 (6) QDP 175.45/45.57 (7) QDPOrderProof [EQUIVALENT, 249 ms] 175.45/45.57 (8) QDP 175.45/45.57 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 175.45/45.57 (10) AND 175.45/45.57 (11) QDP 175.45/45.57 (12) QDPOrderProof [EQUIVALENT, 4803 ms] 175.45/45.57 (13) QDP 175.45/45.57 (14) QDPOrderProof [EQUIVALENT, 1517 ms] 175.45/45.57 (15) QDP 175.45/45.57 (16) QDPOrderProof [EQUIVALENT, 2164 ms] 175.45/45.57 (17) QDP 175.45/45.57 (18) DependencyGraphProof [EQUIVALENT, 0 ms] 175.45/45.57 (19) TRUE 175.45/45.57 (20) QDP 175.45/45.57 (21) QDPOrderProof [EQUIVALENT, 4855 ms] 175.45/45.57 (22) QDP 175.45/45.57 (23) QDPOrderProof [EQUIVALENT, 1721 ms] 175.45/45.57 (24) QDP 175.45/45.57 (25) QDPOrderProof [EQUIVALENT, 1847 ms] 175.45/45.57 (26) QDP 175.45/45.57 (27) DependencyGraphProof [EQUIVALENT, 0 ms] 175.45/45.57 (28) TRUE 175.45/45.57 175.45/45.57 175.45/45.57 ---------------------------------------- 175.45/45.57 175.45/45.57 (0) 175.45/45.57 Obligation: 175.45/45.57 Q restricted rewrite system: 175.45/45.57 The TRS R consists of the following rules: 175.45/45.57 175.45/45.57 b(b(a(b(x1)))) -> a(b(a(b(x1)))) 175.45/45.57 a(a(a(a(x1)))) -> b(b(b(b(x1)))) 175.45/45.57 b(b(a(b(x1)))) -> a(a(a(b(x1)))) 175.45/45.57 175.45/45.57 Q is empty. 175.45/45.57 175.45/45.57 ---------------------------------------- 175.45/45.57 175.45/45.57 (1) FlatCCProof (EQUIVALENT) 175.45/45.57 We used flat context closure [ROOTLAB] 175.45/45.57 As Q is empty the flat context closure was sound AND complete. 175.45/45.57 175.45/45.57 ---------------------------------------- 175.45/45.57 175.45/45.57 (2) 175.45/45.57 Obligation: 175.45/45.57 Q restricted rewrite system: 175.45/45.57 The TRS R consists of the following rules: 175.45/45.57 175.45/45.57 b(b(b(a(b(x1))))) -> b(a(b(a(b(x1))))) 175.45/45.57 a(b(b(a(b(x1))))) -> a(a(b(a(b(x1))))) 175.45/45.57 b(a(a(a(a(x1))))) -> b(b(b(b(b(x1))))) 175.45/45.57 a(a(a(a(a(x1))))) -> a(b(b(b(b(x1))))) 175.45/45.57 b(b(b(a(b(x1))))) -> b(a(a(a(b(x1))))) 175.45/45.57 a(b(b(a(b(x1))))) -> a(a(a(a(b(x1))))) 175.45/45.57 175.45/45.57 Q is empty. 175.45/45.57 175.45/45.57 ---------------------------------------- 175.45/45.57 175.45/45.57 (3) RootLabelingProof (EQUIVALENT) 175.45/45.57 We used plain root labeling [ROOTLAB] with the following heuristic: 175.45/45.57 LabelAll: All function symbols get labeled 175.45/45.57 175.45/45.57 As Q is empty the root labeling was sound AND complete. 175.45/45.57 175.45/45.57 ---------------------------------------- 175.45/45.57 175.45/45.57 (4) 175.45/45.57 Obligation: 175.45/45.57 Q restricted rewrite system: 175.45/45.57 The TRS R consists of the following rules: 175.45/45.57 175.45/45.57 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_{b_1}(b_{b_1}(x1))))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 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_{b_1}(b_{b_1}(x1))))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.45/45.57 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.45/45.57 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.45/45.57 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.45/45.57 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_{b_1}(b_{b_1}(x1))))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 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}(b_{b_1}(x1))))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 175.45/45.57 Q is empty. 175.45/45.57 175.45/45.57 ---------------------------------------- 175.45/45.57 175.45/45.57 (5) DependencyPairsProof (EQUIVALENT) 175.45/45.57 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 175.45/45.57 ---------------------------------------- 175.45/45.57 175.45/45.57 (6) 175.45/45.57 Obligation: 175.45/45.57 Q DP problem: 175.45/45.57 The TRS P consists of the following rules: 175.45/45.57 175.45/45.57 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_{b_1}(b_{b_1}(x1))))) 175.45/45.57 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 175.45/45.57 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_{b_1}(b_{b_1}(x1))))) 175.45/45.57 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 175.45/45.57 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 175.45/45.57 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 175.45/45.57 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_{b_1}(b_{b_1}(x1))))) 175.45/45.57 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}(b_{b_1}(x1)))) 175.45/45.57 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 175.45/45.57 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 175.45/45.57 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}(b_{b_1}(x1))))) 175.45/45.57 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}(b_{b_1}(x1)))) 175.45/45.57 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}(x1))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 175.45/45.57 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}(x1))) 175.45/45.57 175.45/45.57 The TRS R consists of the following rules: 175.45/45.57 175.45/45.57 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_{b_1}(b_{b_1}(x1))))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 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_{b_1}(b_{b_1}(x1))))) 175.45/45.57 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_{b_1}(b_{a_1}(x1))))) 175.45/45.57 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.45/45.57 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.45/45.57 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.45/45.57 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (7) QDPOrderProof (EQUIVALENT) 175.55/45.58 We use the reduction pair processor [LPAR04,JAR06]. 175.55/45.58 175.55/45.58 175.55/45.58 The following pairs can be oriented strictly and are deleted. 175.55/45.58 175.55/45.58 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 175.55/45.58 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 175.55/45.58 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) 175.55/45.58 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1)) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1)) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) 175.55/45.58 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}(b_{b_1}(x1)))) 175.55/45.58 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) 175.55/45.58 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 175.55/45.58 B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) 175.55/45.58 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}(b_{b_1}(x1)))) 175.55/45.58 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}(x1))) 175.55/45.58 A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 175.55/45.58 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}(x1))) 175.55/45.58 The remaining pairs can at least be oriented weakly. 175.55/45.58 Used ordering: Polynomial interpretation [POLO]: 175.55/45.58 175.55/45.58 POL(A_{A_1}(x_1)) = x_1 175.55/45.58 POL(A_{B_1}(x_1)) = x_1 175.55/45.58 POL(B_{A_1}(x_1)) = x_1 175.55/45.58 POL(B_{B_1}(x_1)) = x_1 175.55/45.58 POL(a_{a_1}(x_1)) = 1 + x_1 175.55/45.58 POL(a_{b_1}(x_1)) = 1 + x_1 175.55/45.58 POL(b_{a_1}(x_1)) = 1 + x_1 175.55/45.58 POL(b_{b_1}(x_1)) = 1 + x_1 175.55/45.58 175.55/45.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 175.55/45.58 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (8) 175.55/45.58 Obligation: 175.55/45.58 Q DP problem: 175.55/45.58 The TRS P consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 The TRS R consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (9) DependencyGraphProof (EQUIVALENT) 175.55/45.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (10) 175.55/45.58 Complex Obligation (AND) 175.55/45.58 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (11) 175.55/45.58 Obligation: 175.55/45.58 Q DP problem: 175.55/45.58 The TRS P consists of the following rules: 175.55/45.58 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 The TRS R consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (12) QDPOrderProof (EQUIVALENT) 175.55/45.58 We use the reduction pair processor [LPAR04,JAR06]. 175.55/45.58 175.55/45.58 175.55/45.58 The following pairs can be oriented strictly and are deleted. 175.55/45.58 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 The remaining pairs can at least be oriented weakly. 175.55/45.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(A_{A_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [1A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, -I], [0A, 0A, -I], [1A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(A_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{b_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{a_1}(x_1)) = [[0A], [0A], [1A]] + [[-I, 0A, -I], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 175.55/45.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 175.55/45.58 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (13) 175.55/45.58 Obligation: 175.55/45.58 Q DP problem: 175.55/45.58 The TRS P consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 The TRS R consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (14) QDPOrderProof (EQUIVALENT) 175.55/45.58 We use the reduction pair processor [LPAR04,JAR06]. 175.55/45.58 175.55/45.58 175.55/45.58 The following pairs can be oriented strictly and are deleted. 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 The remaining pairs can at least be oriented weakly. 175.55/45.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(A_{B_1}(x_1)) = [[0A]] + [[1A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [0A, -I, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{a_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{b_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 1A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(A_{A_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 175.55/45.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 175.55/45.58 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (15) 175.55/45.58 Obligation: 175.55/45.58 Q DP problem: 175.55/45.58 The TRS P consists of the following rules: 175.55/45.58 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 The TRS R consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (16) QDPOrderProof (EQUIVALENT) 175.55/45.58 We use the reduction pair processor [LPAR04,JAR06]. 175.55/45.58 175.55/45.58 175.55/45.58 The following pairs can be oriented strictly and are deleted. 175.55/45.58 175.55/45.58 A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 The remaining pairs can at least be oriented weakly. 175.55/45.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(A_{A_1}(x_1)) = [[1A]] + [[-I, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{a_1}(x_1)) = [[0A], [-I], [-I]] + [[-I, 1A, 0A], [-I, -I, 0A], [0A, 0A, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(A_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, 0A, -I], [-I, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [-I, 1A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 175.55/45.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 175.55/45.58 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (17) 175.55/45.58 Obligation: 175.55/45.58 Q DP problem: 175.55/45.58 The TRS P consists of the following rules: 175.55/45.58 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 The TRS R consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (18) DependencyGraphProof (EQUIVALENT) 175.55/45.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (19) 175.55/45.58 TRUE 175.55/45.58 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (20) 175.55/45.58 Obligation: 175.55/45.58 Q DP problem: 175.55/45.58 The TRS P consists of the following rules: 175.55/45.58 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 The TRS R consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (21) QDPOrderProof (EQUIVALENT) 175.55/45.58 We use the reduction pair processor [LPAR04,JAR06]. 175.55/45.58 175.55/45.58 175.55/45.58 The following pairs can be oriented strictly and are deleted. 175.55/45.58 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 The remaining pairs can at least be oriented weakly. 175.55/45.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(B_{A_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [1A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, -I], [0A, 0A, -I], [1A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(B_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{b_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{a_1}(x_1)) = [[0A], [0A], [1A]] + [[-I, 0A, -I], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 175.55/45.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 175.55/45.58 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (22) 175.55/45.58 Obligation: 175.55/45.58 Q DP problem: 175.55/45.58 The TRS P consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 The TRS R consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (23) QDPOrderProof (EQUIVALENT) 175.55/45.58 We use the reduction pair processor [LPAR04,JAR06]. 175.55/45.58 175.55/45.58 175.55/45.58 The following pairs can be oriented strictly and are deleted. 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 The remaining pairs can at least be oriented weakly. 175.55/45.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(B_{B_1}(x_1)) = [[0A]] + [[1A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [0A, -I, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{a_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{b_1}(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 1A], [-I, -I, 0A], [0A, 0A, 1A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(B_{A_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 175.55/45.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 175.55/45.58 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (24) 175.55/45.58 Obligation: 175.55/45.58 Q DP problem: 175.55/45.58 The TRS P consists of the following rules: 175.55/45.58 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 The TRS R consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (25) QDPOrderProof (EQUIVALENT) 175.55/45.58 We use the reduction pair processor [LPAR04,JAR06]. 175.55/45.58 175.55/45.58 175.55/45.58 The following pairs can be oriented strictly and are deleted. 175.55/45.58 175.55/45.58 B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 The remaining pairs can at least be oriented weakly. 175.55/45.58 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(B_{A_1}(x_1)) = [[1A]] + [[-I, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{a_1}(x_1)) = [[0A], [-I], [-I]] + [[-I, 1A, 0A], [-I, -I, 0A], [0A, 0A, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(B_{B_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, 0A, -I], [-I, 0A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 <<< 175.55/45.58 POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [-I, 1A, 0A]] * x_1 175.55/45.58 >>> 175.55/45.58 175.55/45.58 175.55/45.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 175.55/45.58 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (26) 175.55/45.58 Obligation: 175.55/45.58 Q DP problem: 175.55/45.58 The TRS P consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 The TRS R consists of the following rules: 175.55/45.58 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) 175.55/45.58 a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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_{b_1}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 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}(b_{b_1}(x1))))) 175.55/45.58 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_{b_1}(b_{a_1}(x1))))) 175.55/45.58 175.55/45.58 Q is empty. 175.55/45.58 We have to consider all minimal (P,Q,R)-chains. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (27) DependencyGraphProof (EQUIVALENT) 175.55/45.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 175.55/45.58 ---------------------------------------- 175.55/45.58 175.55/45.58 (28) 175.55/45.58 TRUE 175.73/45.65 EOF