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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) FlatCCProof [EQUIVALENT, 0 ms] (4) QTRS (5) RootLabelingProof [EQUIVALENT, 0 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 38 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 2 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 40 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) QDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(x1) -> b(c(x1)) b(a(b(x1))) -> a(a(a(x1))) c(c(x1)) -> a(x1) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(x1) -> c(b(x1)) b(a(b(x1))) -> a(a(a(x1))) c(c(x1)) -> a(x1) Q is empty. ---------------------------------------- (3) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(x1)) -> a(c(b(x1))) c(a(x1)) -> c(c(b(x1))) b(a(x1)) -> b(c(b(x1))) a(b(a(b(x1)))) -> a(a(a(a(x1)))) c(b(a(b(x1)))) -> c(a(a(a(x1)))) b(b(a(b(x1)))) -> b(a(a(a(x1)))) a(c(c(x1))) -> a(a(x1)) c(c(c(x1))) -> c(a(x1)) b(c(c(x1))) -> b(a(x1)) Q is empty. ---------------------------------------- (5) 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. ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1))) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1))) a_{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_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{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)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 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_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 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)))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(a_{c_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 1 + x_1 POL(b_{c_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = 1 + x_1 POL(c_{b_1}(x_1)) = x_1 POL(c_{c_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_{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_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{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_{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_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 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)))) a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{a_1}(x1)) a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(x1)) c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{a_1}(x1)) c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{c_1}(x1)) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(x1)) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{c_1}(x1)) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1))) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) Q is empty. ---------------------------------------- (9) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1))) A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{b_1}(b_{c_1}(x1))) A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1) A_{A_1}(a_{b_1}(x1)) -> A_{C_1}(c_{b_1}(b_{b_1}(x1))) A_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1)) C_{A_1}(a_{a_1}(x1)) -> C_{C_1}(c_{b_1}(b_{a_1}(x1))) C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) C_{A_1}(a_{c_1}(x1)) -> C_{C_1}(c_{b_1}(b_{c_1}(x1))) C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1) C_{A_1}(a_{b_1}(x1)) -> C_{C_1}(c_{b_1}(b_{b_1}(x1))) C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1)) B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1))) B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1))) B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1) B_{A_1}(a_{b_1}(x1)) -> B_{C_1}(c_{b_1}(b_{b_1}(x1))) B_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{c_1}(x1))) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{C_1}(x1) C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1)) A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1)) C_{C_1}(c_{c_1}(c_{b_1}(x1))) -> C_{A_1}(a_{b_1}(x1)) B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{b_1}(x1)) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1))) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{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 1 SCC with 20 less nodes. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) C_{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}(x1)) -> B_{A_1}(x1) B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1)) A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{c_1}(x1))) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1))) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{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}(x1)) -> C_{B_1}(b_{a_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> C_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{A_1}(x_1)) = 1 + x_1 POL(B_{A_1}(x_1)) = x_1 POL(C_{A_1}(x_1)) = x_1 POL(C_{B_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(a_{c_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 1 + x_1 POL(b_{c_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = 1 + x_1 POL(c_{b_1}(x_1)) = x_1 POL(c_{c_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}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1))) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1))) a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1))) B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{c_1}(x1))) C_{B_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1)) C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1)) C_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1))) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1))) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{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 1 SCC with 11 less nodes. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1))) b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(x1)) c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{b_1}(x1)) b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{b_1}(x1)) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) 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. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) 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}(x1)) -> B_{A_1}(x1) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES