19.12/5.80 YES 19.12/5.84 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 19.12/5.84 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 19.12/5.84 19.12/5.84 19.12/5.84 Termination w.r.t. Q of the given QTRS could be proven: 19.12/5.84 19.12/5.84 (0) QTRS 19.12/5.84 (1) FlatCCProof [EQUIVALENT, 0 ms] 19.12/5.84 (2) QTRS 19.12/5.84 (3) RootLabelingProof [EQUIVALENT, 0 ms] 19.12/5.84 (4) QTRS 19.12/5.84 (5) QTRSRRRProof [EQUIVALENT, 94 ms] 19.12/5.84 (6) QTRS 19.12/5.84 (7) QTRSRRRProof [EQUIVALENT, 20 ms] 19.12/5.84 (8) QTRS 19.12/5.84 (9) DependencyPairsProof [EQUIVALENT, 42 ms] 19.12/5.84 (10) QDP 19.12/5.84 (11) QDPOrderProof [EQUIVALENT, 266 ms] 19.12/5.84 (12) QDP 19.12/5.84 (13) DependencyGraphProof [EQUIVALENT, 0 ms] 19.12/5.84 (14) TRUE 19.12/5.84 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (0) 19.12/5.84 Obligation: 19.12/5.84 Q restricted rewrite system: 19.12/5.84 The TRS R consists of the following rules: 19.12/5.84 19.12/5.84 A(b(x1)) -> b(a(B(A(x1)))) 19.12/5.84 B(a(x1)) -> a(b(A(B(x1)))) 19.12/5.84 A(a(x1)) -> x1 19.12/5.84 B(b(x1)) -> x1 19.12/5.84 19.12/5.84 Q is empty. 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (1) FlatCCProof (EQUIVALENT) 19.12/5.84 We used flat context closure [ROOTLAB] 19.12/5.84 As Q is empty the flat context closure was sound AND complete. 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (2) 19.12/5.84 Obligation: 19.12/5.84 Q restricted rewrite system: 19.12/5.84 The TRS R consists of the following rules: 19.12/5.84 19.12/5.84 A(A(b(x1))) -> A(b(a(B(A(x1))))) 19.12/5.84 b(A(b(x1))) -> b(b(a(B(A(x1))))) 19.12/5.84 a(A(b(x1))) -> a(b(a(B(A(x1))))) 19.12/5.84 B(A(b(x1))) -> B(b(a(B(A(x1))))) 19.12/5.84 A(B(a(x1))) -> A(a(b(A(B(x1))))) 19.12/5.84 b(B(a(x1))) -> b(a(b(A(B(x1))))) 19.12/5.84 a(B(a(x1))) -> a(a(b(A(B(x1))))) 19.12/5.84 B(B(a(x1))) -> B(a(b(A(B(x1))))) 19.12/5.84 A(A(a(x1))) -> A(x1) 19.12/5.84 b(A(a(x1))) -> b(x1) 19.12/5.84 a(A(a(x1))) -> a(x1) 19.12/5.84 B(A(a(x1))) -> B(x1) 19.12/5.84 A(B(b(x1))) -> A(x1) 19.12/5.84 b(B(b(x1))) -> b(x1) 19.12/5.84 a(B(b(x1))) -> a(x1) 19.12/5.84 B(B(b(x1))) -> B(x1) 19.12/5.84 19.12/5.84 Q is empty. 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (3) RootLabelingProof (EQUIVALENT) 19.12/5.84 We used plain root labeling [ROOTLAB] with the following heuristic: 19.12/5.84 LabelAll: All function symbols get labeled 19.12/5.84 19.12/5.84 As Q is empty the root labeling was sound AND complete. 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (4) 19.12/5.84 Obligation: 19.12/5.84 Q restricted rewrite system: 19.12/5.84 The TRS R consists of the following rules: 19.12/5.84 19.12/5.84 A_{A_1}(A_{b_1}(b_{A_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{b_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{a_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{B_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{A_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{B_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 a_{A_1}(A_{b_1}(b_{A_1}(x1))) -> a_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 a_{A_1}(A_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 a_{A_1}(A_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 a_{A_1}(A_{b_1}(b_{B_1}(x1))) -> a_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{A_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{b_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{a_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{B_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{A_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{A_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{b_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{a_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{B_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{A_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{A_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{B_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{A_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{A_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{B_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{A_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{A_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{b_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{a_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{B_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 A_{A_1}(A_{a_1}(a_{A_1}(x1))) -> A_{A_1}(x1) 19.12/5.84 A_{A_1}(A_{a_1}(a_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 A_{A_1}(A_{a_1}(a_{a_1}(x1))) -> A_{a_1}(x1) 19.12/5.84 A_{A_1}(A_{a_1}(a_{B_1}(x1))) -> A_{B_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{A_1}(x1))) -> b_{A_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{B_1}(x1))) -> b_{B_1}(x1) 19.12/5.84 a_{A_1}(A_{a_1}(a_{A_1}(x1))) -> a_{A_1}(x1) 19.12/5.84 a_{A_1}(A_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1) 19.12/5.84 a_{A_1}(A_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1) 19.12/5.84 a_{A_1}(A_{a_1}(a_{B_1}(x1))) -> a_{B_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{A_1}(x1))) -> B_{A_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{b_1}(x1))) -> B_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{B_1}(x1))) -> B_{B_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{A_1}(x1))) -> A_{A_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{a_1}(x1))) -> A_{a_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{B_1}(x1))) -> A_{B_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{A_1}(x1))) -> b_{A_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{B_1}(x1))) -> b_{B_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{A_1}(x1))) -> a_{A_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{B_1}(x1))) -> a_{B_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{A_1}(x1))) -> B_{A_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{b_1}(x1))) -> B_{b_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{B_1}(x1))) -> B_{B_1}(x1) 19.12/5.84 19.12/5.84 Q is empty. 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (5) QTRSRRRProof (EQUIVALENT) 19.12/5.84 Used ordering: 19.12/5.84 Polynomial interpretation [POLO]: 19.12/5.84 19.12/5.84 POL(A_{A_1}(x_1)) = 1 + x_1 19.12/5.84 POL(A_{B_1}(x_1)) = x_1 19.12/5.84 POL(A_{a_1}(x_1)) = x_1 19.12/5.84 POL(A_{b_1}(x_1)) = 1 + x_1 19.12/5.84 POL(B_{A_1}(x_1)) = x_1 19.12/5.84 POL(B_{B_1}(x_1)) = 1 + x_1 19.12/5.84 POL(B_{a_1}(x_1)) = 1 + x_1 19.12/5.84 POL(B_{b_1}(x_1)) = x_1 19.12/5.84 POL(a_{A_1}(x_1)) = 1 + x_1 19.12/5.84 POL(a_{B_1}(x_1)) = 1 + x_1 19.12/5.84 POL(a_{a_1}(x_1)) = 1 + x_1 19.12/5.84 POL(a_{b_1}(x_1)) = x_1 19.12/5.84 POL(b_{A_1}(x_1)) = 1 + x_1 19.12/5.84 POL(b_{B_1}(x_1)) = x_1 19.12/5.84 POL(b_{a_1}(x_1)) = x_1 19.12/5.84 POL(b_{b_1}(x_1)) = 1 + x_1 19.12/5.84 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 19.12/5.84 19.12/5.84 a_{A_1}(A_{b_1}(b_{A_1}(x1))) -> a_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 a_{A_1}(A_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 a_{A_1}(A_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 a_{A_1}(A_{b_1}(b_{B_1}(x1))) -> a_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{A_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{A_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{A_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{A_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{A_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{A_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{A_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{A_1}(x1))))) 19.12/5.84 A_{A_1}(A_{a_1}(a_{A_1}(x1))) -> A_{A_1}(x1) 19.12/5.84 A_{A_1}(A_{a_1}(a_{a_1}(x1))) -> A_{a_1}(x1) 19.12/5.84 A_{A_1}(A_{a_1}(a_{B_1}(x1))) -> A_{B_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{A_1}(x1))) -> b_{A_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{B_1}(x1))) -> b_{B_1}(x1) 19.12/5.84 a_{A_1}(A_{a_1}(a_{A_1}(x1))) -> a_{A_1}(x1) 19.12/5.84 a_{A_1}(A_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1) 19.12/5.84 a_{A_1}(A_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1) 19.12/5.84 a_{A_1}(A_{a_1}(a_{B_1}(x1))) -> a_{B_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{A_1}(x1))) -> B_{A_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{A_1}(x1))) -> a_{A_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{b_1}(x1))) -> a_{b_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{A_1}(x1))) -> B_{A_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{b_1}(x1))) -> B_{b_1}(x1) 19.12/5.84 19.12/5.84 19.12/5.84 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (6) 19.12/5.84 Obligation: 19.12/5.84 Q restricted rewrite system: 19.12/5.84 The TRS R consists of the following rules: 19.12/5.84 19.12/5.84 A_{A_1}(A_{b_1}(b_{A_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{b_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{a_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{B_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{A_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{B_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{A_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{b_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{a_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{B_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{b_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{a_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{B_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{B_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{B_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{b_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{a_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{B_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 A_{A_1}(A_{a_1}(a_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{b_1}(x1))) -> B_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{B_1}(x1))) -> B_{B_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{A_1}(x1))) -> A_{A_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{a_1}(x1))) -> A_{a_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{B_1}(x1))) -> A_{B_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{A_1}(x1))) -> b_{A_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{B_1}(x1))) -> b_{B_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{B_1}(x1))) -> a_{B_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{B_1}(x1))) -> B_{B_1}(x1) 19.12/5.84 19.12/5.84 Q is empty. 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (7) QTRSRRRProof (EQUIVALENT) 19.12/5.84 Used ordering: 19.12/5.84 Polynomial interpretation [POLO]: 19.12/5.84 19.12/5.84 POL(A_{A_1}(x_1)) = x_1 19.12/5.84 POL(A_{B_1}(x_1)) = x_1 19.12/5.84 POL(A_{a_1}(x_1)) = x_1 19.12/5.84 POL(A_{b_1}(x_1)) = x_1 19.12/5.84 POL(B_{A_1}(x_1)) = x_1 19.12/5.84 POL(B_{B_1}(x_1)) = x_1 19.12/5.84 POL(B_{a_1}(x_1)) = x_1 19.12/5.84 POL(B_{b_1}(x_1)) = x_1 19.12/5.84 POL(a_{B_1}(x_1)) = x_1 19.12/5.84 POL(a_{a_1}(x_1)) = x_1 19.12/5.84 POL(a_{b_1}(x_1)) = x_1 19.12/5.84 POL(b_{A_1}(x_1)) = x_1 19.12/5.84 POL(b_{B_1}(x_1)) = 1 + x_1 19.12/5.84 POL(b_{a_1}(x_1)) = x_1 19.12/5.84 POL(b_{b_1}(x_1)) = x_1 19.12/5.84 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 19.12/5.84 19.12/5.84 A_{A_1}(A_{b_1}(b_{B_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{B_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{B_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{B_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 b_{B_1}(B_{a_1}(a_{B_1}(x1))) -> b_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 A_{B_1}(B_{b_1}(b_{B_1}(x1))) -> A_{B_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{A_1}(x1))) -> b_{A_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{b_1}(x1))) -> b_{b_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{a_1}(x1))) -> b_{a_1}(x1) 19.12/5.84 b_{B_1}(B_{b_1}(b_{B_1}(x1))) -> b_{B_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{B_1}(x1))) -> a_{B_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{B_1}(x1))) -> B_{B_1}(x1) 19.12/5.84 19.12/5.84 19.12/5.84 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (8) 19.12/5.84 Obligation: 19.12/5.84 Q restricted rewrite system: 19.12/5.84 The TRS R consists of the following rules: 19.12/5.84 19.12/5.84 A_{A_1}(A_{b_1}(b_{A_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{b_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{a_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{A_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{A_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{b_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{a_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{b_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{a_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{B_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{B_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{b_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{a_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{B_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 A_{A_1}(A_{a_1}(a_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{b_1}(x1))) -> B_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{B_1}(x1))) -> B_{B_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{A_1}(x1))) -> A_{A_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{a_1}(x1))) -> A_{a_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 19.12/5.84 Q is empty. 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (9) DependencyPairsProof (EQUIVALENT) 19.12/5.84 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (10) 19.12/5.84 Obligation: 19.12/5.84 Q DP problem: 19.12/5.84 The TRS P consists of the following rules: 19.12/5.84 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{A_1}(x1))) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> B_{A_1}^1(A_{A_1}(x1)) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> A_{A_1}^1(x1) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{b_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{b_1}(x1))) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{b_1}(x1))) -> B_{A_1}^1(A_{b_1}(x1)) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{a_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{a_1}(x1))) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{a_1}(x1))) -> B_{A_1}^1(A_{a_1}(x1)) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{A_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{A_1}(x1))) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{A_1}(x1))) -> B_{A_1}^1(A_{A_1}(x1)) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{A_1}(x1))) -> A_{A_1}^1(x1) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{b_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{b_1}(x1))) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{b_1}(x1))) -> B_{A_1}^1(A_{b_1}(x1)) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{a_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{a_1}(x1))) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{a_1}(x1))) -> B_{A_1}^1(A_{a_1}(x1)) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{A_1}(x1))) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> B_{A_1}^1(A_{A_1}(x1)) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> A_{A_1}^1(x1) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{b_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{b_1}(x1))) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{b_1}(x1))) -> B_{A_1}^1(A_{b_1}(x1)) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{a_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{a_1}(x1))) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{a_1}(x1))) -> B_{A_1}^1(A_{a_1}(x1)) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{b_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{b_1}(x1))) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{b_1}(x1))) -> A_{B_1}^2(B_{b_1}(x1)) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{a_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{a_1}(x1))) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{a_1}(x1))) -> A_{B_1}^2(B_{a_1}(x1)) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{B_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{B_1}(x1))) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{B_1}(x1))) -> A_{B_1}^2(B_{B_1}(x1)) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{B_1}(x1))) -> B_{B_1}^1(x1) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{b_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{b_1}(x1))) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{b_1}(x1))) -> A_{B_1}^2(B_{b_1}(x1)) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{a_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{a_1}(x1))) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{a_1}(x1))) -> A_{B_1}^2(B_{a_1}(x1)) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{B_1}(x1))) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> A_{B_1}^2(B_{B_1}(x1)) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> B_{B_1}^1(x1) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{b_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{b_1}(x1))) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{b_1}(x1))) -> A_{B_1}^2(B_{b_1}(x1)) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{a_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{a_1}(x1))) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{a_1}(x1))) -> A_{B_1}^2(B_{a_1}(x1)) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{B_1}(x1))) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> A_{B_1}^2(B_{B_1}(x1)) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> B_{B_1}^1(x1) 19.12/5.84 B_{A_1}^1(A_{a_1}(a_{B_1}(x1))) -> B_{B_1}^1(x1) 19.12/5.84 A_{B_1}^2(B_{b_1}(b_{A_1}(x1))) -> A_{A_1}^1(x1) 19.12/5.84 19.12/5.84 The TRS R consists of the following rules: 19.12/5.84 19.12/5.84 A_{A_1}(A_{b_1}(b_{A_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{b_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{a_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{A_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{A_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{b_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{a_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{b_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{a_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{B_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{B_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{b_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{a_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{B_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 A_{A_1}(A_{a_1}(a_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{b_1}(x1))) -> B_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{B_1}(x1))) -> B_{B_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{A_1}(x1))) -> A_{A_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{a_1}(x1))) -> A_{a_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 19.12/5.84 Q is empty. 19.12/5.84 We have to consider all minimal (P,Q,R)-chains. 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (11) QDPOrderProof (EQUIVALENT) 19.12/5.84 We use the reduction pair processor [LPAR04,JAR06]. 19.12/5.84 19.12/5.84 19.12/5.84 The following pairs can be oriented strictly and are deleted. 19.12/5.84 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> B_{A_1}^1(A_{A_1}(x1)) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> A_{A_1}^1(x1) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{b_1}(x1))) -> B_{A_1}^1(A_{b_1}(x1)) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{a_1}(x1))) -> B_{A_1}^1(A_{a_1}(x1)) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{A_1}(x1))) -> B_{A_1}^1(A_{A_1}(x1)) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{A_1}(x1))) -> A_{A_1}^1(x1) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{b_1}(x1))) -> B_{A_1}^1(A_{b_1}(x1)) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{a_1}(x1))) -> B_{A_1}^1(A_{a_1}(x1)) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> B_{A_1}^1(A_{A_1}(x1)) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> A_{A_1}^1(x1) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{b_1}(x1))) -> B_{A_1}^1(A_{b_1}(x1)) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{a_1}(x1))) -> B_{A_1}^1(A_{a_1}(x1)) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{b_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{b_1}(x1))) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{b_1}(x1))) -> A_{B_1}^2(B_{b_1}(x1)) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{a_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{a_1}(x1))) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{a_1}(x1))) -> A_{B_1}^2(B_{a_1}(x1)) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{B_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{B_1}(x1))) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{B_1}(x1))) -> A_{B_1}^2(B_{B_1}(x1)) 19.12/5.84 A_{B_1}^2(B_{a_1}(a_{B_1}(x1))) -> B_{B_1}^1(x1) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{b_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{b_1}(x1))) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{b_1}(x1))) -> A_{B_1}^2(B_{b_1}(x1)) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{a_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{a_1}(x1))) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{a_1}(x1))) -> A_{B_1}^2(B_{a_1}(x1)) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{B_1}(x1))) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> A_{B_1}^2(B_{B_1}(x1)) 19.12/5.84 A_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> B_{B_1}^1(x1) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{b_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{b_1}(x1))) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{b_1}(x1))) -> A_{B_1}^2(B_{b_1}(x1)) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{a_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{a_1}(x1))) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{a_1}(x1))) -> A_{B_1}^2(B_{a_1}(x1)) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> B_{A_1}^2(A_{B_1}(B_{B_1}(x1))) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> A_{B_1}^2(B_{B_1}(x1)) 19.12/5.84 B_{B_1}^1(B_{a_1}(a_{B_1}(x1))) -> B_{B_1}^1(x1) 19.12/5.84 B_{A_1}^1(A_{a_1}(a_{B_1}(x1))) -> B_{B_1}^1(x1) 19.12/5.84 A_{B_1}^2(B_{b_1}(b_{A_1}(x1))) -> A_{A_1}^1(x1) 19.12/5.84 The remaining pairs can at least be oriented weakly. 19.12/5.84 Used ordering: Polynomial interpretation [POLO]: 19.12/5.84 19.12/5.84 POL(A_{A_1}(x_1)) = 1 + x_1 19.12/5.84 POL(A_{A_1}^1(x_1)) = x_1 19.12/5.84 POL(A_{B_1}(x_1)) = x_1 19.12/5.84 POL(A_{B_1}^1(x_1)) = 1 + x_1 19.12/5.84 POL(A_{B_1}^2(x_1)) = x_1 19.12/5.84 POL(A_{a_1}(x_1)) = x_1 19.12/5.84 POL(A_{b_1}(x_1)) = 1 + x_1 19.12/5.84 POL(B_{A_1}(x_1)) = x_1 19.12/5.84 POL(B_{A_1}^1(x_1)) = x_1 19.12/5.84 POL(B_{A_1}^2(x_1)) = x_1 19.12/5.84 POL(B_{B_1}(x_1)) = 1 + x_1 19.12/5.84 POL(B_{B_1}^1(x_1)) = x_1 19.12/5.84 POL(B_{a_1}(x_1)) = 1 + x_1 19.12/5.84 POL(B_{b_1}(x_1)) = x_1 19.12/5.84 POL(a_{B_1}(x_1)) = 1 + x_1 19.12/5.84 POL(a_{a_1}(x_1)) = 1 + x_1 19.12/5.84 POL(a_{b_1}(x_1)) = x_1 19.12/5.84 POL(b_{A_1}(x_1)) = 1 + x_1 19.12/5.84 POL(b_{a_1}(x_1)) = x_1 19.12/5.84 POL(b_{b_1}(x_1)) = 1 + x_1 19.12/5.84 19.12/5.84 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 19.12/5.84 19.12/5.84 A_{A_1}(A_{b_1}(b_{A_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{b_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{a_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 A_{A_1}(A_{a_1}(a_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{b_1}(b_{A_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{b_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{a_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 B_{A_1}(A_{a_1}(a_{b_1}(x1))) -> B_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{B_1}(x1))) -> B_{B_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{A_1}(x1))) -> A_{A_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{a_1}(x1))) -> A_{a_1}(x1) 19.12/5.84 A_{B_1}(B_{a_1}(a_{b_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{a_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{B_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{b_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{a_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{B_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 B_{B_1}(B_{b_1}(b_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 a_{B_1}(B_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{B_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{A_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 a_{B_1}(B_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1) 19.12/5.84 19.12/5.84 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (12) 19.12/5.84 Obligation: 19.12/5.84 Q DP problem: 19.12/5.84 The TRS P consists of the following rules: 19.12/5.84 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{A_1}(x1))) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{b_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{b_1}(x1))) 19.12/5.84 A_{A_1}^1(A_{b_1}(b_{a_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{a_1}(x1))) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{A_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{A_1}(x1))) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{b_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{b_1}(x1))) 19.12/5.84 B_{A_1}^2(A_{b_1}(b_{a_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{a_1}(x1))) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{A_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{A_1}(x1))) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{b_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{b_1}(x1))) 19.12/5.84 B_{A_1}^1(A_{b_1}(b_{a_1}(x1))) -> A_{B_1}^1(B_{A_1}(A_{a_1}(x1))) 19.12/5.84 19.12/5.84 The TRS R consists of the following rules: 19.12/5.84 19.12/5.84 A_{A_1}(A_{b_1}(b_{A_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{b_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 A_{A_1}(A_{b_1}(b_{a_1}(x1))) -> A_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{A_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 b_{A_1}(A_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{A_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{A_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{b_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{b_1}(x1))))) 19.12/5.84 B_{A_1}(A_{b_1}(b_{a_1}(x1))) -> B_{b_1}(b_{a_1}(a_{B_1}(B_{A_1}(A_{a_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{b_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{a_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 A_{B_1}(B_{a_1}(a_{B_1}(x1))) -> A_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 a_{B_1}(B_{a_1}(a_{B_1}(x1))) -> a_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{b_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{b_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{a_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{a_1}(x1))))) 19.12/5.84 B_{B_1}(B_{a_1}(a_{B_1}(x1))) -> B_{a_1}(a_{b_1}(b_{A_1}(A_{B_1}(B_{B_1}(x1))))) 19.12/5.84 A_{A_1}(A_{a_1}(a_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 b_{A_1}(A_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{b_1}(x1))) -> B_{b_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 B_{A_1}(A_{a_1}(a_{B_1}(x1))) -> B_{B_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{A_1}(x1))) -> A_{A_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{b_1}(x1))) -> A_{b_1}(x1) 19.12/5.84 A_{B_1}(B_{b_1}(b_{a_1}(x1))) -> A_{a_1}(x1) 19.12/5.84 a_{B_1}(B_{b_1}(b_{a_1}(x1))) -> a_{a_1}(x1) 19.12/5.84 B_{B_1}(B_{b_1}(b_{a_1}(x1))) -> B_{a_1}(x1) 19.12/5.84 19.12/5.84 Q is empty. 19.12/5.84 We have to consider all minimal (P,Q,R)-chains. 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (13) DependencyGraphProof (EQUIVALENT) 19.12/5.84 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 9 less nodes. 19.12/5.84 ---------------------------------------- 19.12/5.84 19.12/5.84 (14) 19.12/5.84 TRUE 19.60/5.96 EOF