/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4)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) QTRSRRRProof [EQUIVALENT, 63 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 0 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) AND (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) QDP (15) QDPOrderProof [EQUIVALENT, 0 ms] (16) QDP (17) DependencyGraphProof [EQUIVALENT, 0 ms] (18) AND (19) QDP (20) QDPOrderProof [EQUIVALENT, 0 ms] (21) QDP (22) PisEmptyProof [EQUIVALENT, 0 ms] (23) YES (24) QDP (25) QDPOrderProof [EQUIVALENT, 16 ms] (26) QDP (27) PisEmptyProof [EQUIVALENT, 0 ms] (28) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(a(a(x1)))) -> b(a(a(a(x1)))) a(b(b(b(x1)))) -> b(a(b(a(x1)))) b(b(a(a(x1)))) -> a(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(a(a(a(x1))))) -> a(b(a(a(a(x1))))) b(a(a(a(a(x1))))) -> b(b(a(a(a(x1))))) a(a(b(b(b(x1))))) -> a(b(a(b(a(x1))))) b(a(b(b(b(x1))))) -> b(b(a(b(a(x1))))) a(b(b(a(a(x1))))) -> a(a(b(b(a(x1))))) b(b(b(a(a(x1))))) -> b(a(b(b(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}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 1 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. ---------------------------------------- (7) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1)) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1)) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) The graph contains the following edges 1 > 1 *B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1)) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{B_1}(x_1)) = x_1 POL(B_{A_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1)) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (18) Complex Obligation (AND) ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(B_{A_1}(x_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)) = 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_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) ---------------------------------------- (21) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (23) YES ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) 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_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{B_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = x_1 POL(b_{a_1}(x_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: a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) ---------------------------------------- (26) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (28) YES