/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given RelTRS could be proven: (0) RelTRS (1) FlatCCProof [EQUIVALENT, 0 ms] (2) RelTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) RelTRS (5) RelTRSRRRProof [EQUIVALENT, 487 ms] (6) RelTRS (7) SIsEmptyProof [EQUIVALENT, 0 ms] (8) QTRS (9) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a(b(b(x1))) -> c(b(a(x1))) a(a(c(x1))) -> b(c(a(x1))) a(a(b(x1))) -> a(c(b(x1))) The relative TRS consists of the following S rules: b(a(a(x1))) -> c(a(a(x1))) c(b(b(x1))) -> b(a(a(x1))) c(b(a(x1))) -> b(c(c(x1))) ---------------------------------------- (1) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] ---------------------------------------- (2) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a(a(b(x1))) -> a(c(b(x1))) a(a(b(b(x1)))) -> a(c(b(a(x1)))) b(a(b(b(x1)))) -> b(c(b(a(x1)))) c(a(b(b(x1)))) -> c(c(b(a(x1)))) a(a(a(c(x1)))) -> a(b(c(a(x1)))) b(a(a(c(x1)))) -> b(b(c(a(x1)))) c(a(a(c(x1)))) -> c(b(c(a(x1)))) The relative TRS consists of the following S rules: a(b(a(a(x1)))) -> a(c(a(a(x1)))) b(b(a(a(x1)))) -> b(c(a(a(x1)))) c(b(a(a(x1)))) -> c(c(a(a(x1)))) a(c(b(b(x1)))) -> a(b(a(a(x1)))) b(c(b(b(x1)))) -> b(b(a(a(x1)))) c(c(b(b(x1)))) -> c(b(a(a(x1)))) a(c(b(a(x1)))) -> a(b(c(c(x1)))) b(c(b(a(x1)))) -> b(b(c(c(x1)))) c(c(b(a(x1)))) -> c(b(c(c(x1)))) ---------------------------------------- (3) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled ---------------------------------------- (4) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) The relative TRS consists of the following S rules: a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) ---------------------------------------- (5) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = 2*x_1 POL(a_{b_1}(x_1)) = 2*x_1 POL(a_{c_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = 4 + 2*x_1 POL(b_{b_1}(x_1)) = 4 + 2*x_1 POL(b_{c_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = 4*x_1 POL(c_{b_1}(x_1)) = 4*x_1 POL(c_{c_1}(x_1)) = 2*x_1 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) Rules from S: a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) ---------------------------------------- (6) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) S is empty. ---------------------------------------- (7) SIsEmptyProof (EQUIVALENT) The TRS S is empty. Hence, termination of R/S is equivalent to termination of R. ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) Q is empty. ---------------------------------------- (9) RFCMatchBoundsTRSProof (EQUIVALENT) Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R. The following rules were used to construct the certificate: a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{b_1}(x1))) a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{c_1}(x1))) a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174 Node 163 is start node and node 164 is final node. Those nodes are connected through the following edges: * 163 to 165 labelled a_{c_1}_1(0), c_{c_1}_1(0)* 163 to 167 labelled a_{b_1}_1(0), b_{b_1}_1(0), c_{b_1}_1(0)* 164 to 164 labelled #_1(0), a_{c_1}_1(0), a_{c_1}_1(1)* 165 to 166 labelled c_{b_1}_1(0)* 166 to 164 labelled b_{a_1}_1(0), b_{b_1}_1(0), b_{c_1}_1(0)* 166 to 170 labelled b_{b_1}_1(1)* 167 to 168 labelled b_{c_1}_1(0)* 168 to 169 labelled c_{a_1}_1(0)* 168 to 170 labelled c_{b_1}_1(1)* 168 to 173 labelled c_{c_1}_1(1)* 169 to 164 labelled a_{a_1}_1(0), a_{b_1}_1(0), a_{c_1}_1(0)* 169 to 173 labelled a_{c_1}_1(1)* 169 to 170 labelled a_{b_1}_1(1)* 170 to 171 labelled b_{c_1}_1(1)* 171 to 172 labelled c_{a_1}_1(1)* 171 to 170 labelled c_{b_1}_1(1)* 171 to 173 labelled c_{c_1}_1(1)* 172 to 164 labelled a_{a_1}_1(1), a_{b_1}_1(1), a_{c_1}_1(1)* 172 to 173 labelled a_{c_1}_1(1)* 172 to 170 labelled a_{b_1}_1(1)* 173 to 174 labelled c_{b_1}_1(1)* 174 to 164 labelled b_{a_1}_1(1), b_{b_1}_1(1), b_{c_1}_1(1)* 174 to 170 labelled b_{b_1}_1(1) ---------------------------------------- (10) YES