YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS (5) DependencyPairsProof [EQUIVALENT, 31 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 379 ms] (10) QDP (11) QDPOrderProof [EQUIVALENT, 735 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 279 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(a(x1))) -> a(b(b(b(x1)))) b(b(a(b(x1)))) -> a(a(a(x1))) Q is empty. ---------------------------------------- (1) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(a(x1))) -> a(b(b(b(x1)))) a(b(b(a(b(x1))))) -> a(a(a(a(x1)))) b(b(b(a(b(x1))))) -> b(a(a(a(x1)))) Q is empty. ---------------------------------------- (3) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled As Q is empty the root labeling was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 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}(x1)))) Q is empty. ---------------------------------------- (5) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 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}(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 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}(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1)) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1)) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{A_1}(x_1)) = 2 + 2*x_1 POL(A_{B_1}(x_1)) = 2 + 2*x_1 POL(B_{B_1}(x_1)) = 2*x_1 POL(a_{a_1}(x_1)) = 3 + x_1 POL(a_{b_1}(x_1)) = 3 + x_1 POL(b_{a_1}(x_1)) = 2 + x_1 POL(b_{b_1}(x_1)) = 2 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 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}(x1)))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 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}(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A_{B_1}(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [-I, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [1A, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(A_{A_1}(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 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}(x1)))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 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}(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A_{B_1}(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [-I], [1A]] + [[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(A_{A_1}(x_1)) = [[1A]] + [[-I, -I, -I]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 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}(x1)))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 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}(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (16) TRUE