/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) QTRS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, 30 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 81 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(b(a(x1)))) -> a(a(a(b(x1)))) b(b(b(a(x1)))) -> a(a(a(b(x1)))) b(a(a(b(x1)))) -> b(b(b(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(b(a(x1)))) -> a(a(a(b(x1)))) b(a(a(b(x1)))) -> b(b(b(a(x1)))) a(b(b(b(a(x1))))) -> a(a(a(a(b(x1))))) b(b(b(b(a(x1))))) -> b(a(a(a(b(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_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{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_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1)) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{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 11 less nodes. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) B_{A_1}(a_{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_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{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_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1)) B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) B_{A_1}(a_{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)) = x_1 POL(A_{B_1}(x_1)) = x_1 POL(B_{A_1}(x_1)) = 1 + x_1 POL(B_{B_1}(x_1)) = 1 + x_1 POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = 1 + x_1 POL(b_{b_1}(x_1)) = 1 + 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_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (12) TRUE