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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) QTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4)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) QTRSRRRProof [EQUIVALENT, 72 ms] (6) QTRS (7) DependencyPairsProof [EQUIVALENT, 11 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) AND (11) QDP (12) UsableRulesProof [EQUIVALENT, 0 ms] (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) QDP (17) QDPOrderProof [EQUIVALENT, 28 ms] (18) QDP (19) PisEmptyProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(x1) -> b(x1) a(a(x1)) -> a(b(a(x1))) a(b(x1)) -> b(b(b(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(a(b(x1))) -> a(b(b(a(b(x1))))) a(b(a(x1))) -> b(a(b(b(a(x1))))) a(b(b(x1))) -> b(b(b(b(b(x1))))) b(a(x1)) -> b(b(b(x1))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) b(b(a(x1))) -> b(b(b(b(b(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(x1)) -> a(b(a(x1))) a(a(a(x1))) -> a(a(b(a(a(x1))))) a(a(b(x1))) -> a(b(b(a(b(x1))))) b(a(x1)) -> b(b(b(x1))) a(b(a(x1))) -> a(b(b(a(b(x1))))) b(a(a(x1))) -> b(a(b(b(a(x1))))) b(b(a(x1))) -> b(b(b(b(b(x1))))) a(a(x1)) -> a(b(x1)) b(a(x1)) -> b(b(x1)) a(a(b(x1))) -> a(b(b(b(x1)))) b(a(b(x1))) -> b(b(b(b(x1)))) a(a(b(a(x1)))) -> a(b(a(b(b(a(x1)))))) b(a(b(a(x1)))) -> b(b(a(b(b(a(x1)))))) a(a(b(b(x1)))) -> a(b(b(b(b(b(x1)))))) b(a(b(b(x1)))) -> b(b(b(b(b(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_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) a_{a_1}(a_{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_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> 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}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) 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_{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))))) b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1)) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{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)) = x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_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}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) a_{a_1}(a_{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_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{a_1}(a_{a_1}(x1)) -> 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_{b_1}(b_{a_1}(x1))))) 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_{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))))) b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1)) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1)) b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{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}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{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_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{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: A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1))) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1))) B_{A_1}(a_{b_1}(x1)) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{A_1}(a_{b_1}(x1)) -> B_{B_1}(b_{b_1}(x1)) B_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1) A_{B_1}(b_{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}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1)) A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1)) B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1)) B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{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_{a_1}(a_{a_1}(x1))))) 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}(x1)))) B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1))) B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))) B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1))) B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1))) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1))) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{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_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{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 38 less nodes. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: 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_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{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_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) 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_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{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_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) 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_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_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_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = 1 POL(b_{a_1}(x_1)) = 0 POL(b_{b_1}(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) ---------------------------------------- (18) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1)) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{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_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (20) YES