/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given RelTRS could be proven: (0) RelTRS (1) RelTRS Reverse [EQUIVALENT, 0 ms] (2) RelTRS (3) FlatCCProof [EQUIVALENT, 0 ms] (4) RelTRS (5) RootLabelingProof [EQUIVALENT, 0 ms] (6) RelTRS (7) RelTRSRRRProof [EQUIVALENT, 677 ms] (8) RelTRS (9) RelTRSRRRProof [EQUIVALENT, 3 ms] (10) RelTRS (11) RIsEmptyProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a(b(b(x1))) -> c(b(a(x1))) b(a(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(c(b(x1))) b(c(a(x1))) -> b(b(a(x1))) The relative TRS consists of the following S rules: b(c(c(x1))) -> a(a(a(x1))) a(b(c(x1))) -> a(a(b(x1))) b(c(b(x1))) -> c(a(b(x1))) ---------------------------------------- (1) RelTRS Reverse (EQUIVALENT) We have reversed the following relative TRS [REVERSE]: The set of rules R is a(b(b(x1))) -> c(b(a(x1))) b(a(a(x1))) -> a(c(c(x1))) c(a(c(x1))) -> c(c(b(x1))) b(c(a(x1))) -> b(b(a(x1))) The set of rules S is b(c(c(x1))) -> a(a(a(x1))) a(b(c(x1))) -> a(a(b(x1))) b(c(b(x1))) -> c(a(b(x1))) We have obtained the following relative TRS: The set of rules R is b(b(a(x1))) -> a(b(c(x1))) a(a(b(x1))) -> c(c(a(x1))) c(a(c(x1))) -> b(c(c(x1))) a(c(b(x1))) -> a(b(b(x1))) The set of rules S is c(c(b(x1))) -> a(a(a(x1))) c(b(a(x1))) -> b(a(a(x1))) b(c(b(x1))) -> b(a(c(x1))) ---------------------------------------- (2) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b(b(a(x1))) -> a(b(c(x1))) a(a(b(x1))) -> c(c(a(x1))) c(a(c(x1))) -> b(c(c(x1))) a(c(b(x1))) -> a(b(b(x1))) The relative TRS consists of the following S rules: c(c(b(x1))) -> a(a(a(x1))) c(b(a(x1))) -> b(a(a(x1))) b(c(b(x1))) -> b(a(c(x1))) ---------------------------------------- (3) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] ---------------------------------------- (4) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a(c(b(x1))) -> a(b(b(x1))) b(b(b(a(x1)))) -> b(a(b(c(x1)))) a(b(b(a(x1)))) -> a(a(b(c(x1)))) c(b(b(a(x1)))) -> c(a(b(c(x1)))) b(a(a(b(x1)))) -> b(c(c(a(x1)))) a(a(a(b(x1)))) -> a(c(c(a(x1)))) c(a(a(b(x1)))) -> c(c(c(a(x1)))) b(c(a(c(x1)))) -> b(b(c(c(x1)))) a(c(a(c(x1)))) -> a(b(c(c(x1)))) c(c(a(c(x1)))) -> c(b(c(c(x1)))) The relative TRS consists of the following S rules: b(c(b(x1))) -> b(a(c(x1))) b(c(c(b(x1)))) -> b(a(a(a(x1)))) a(c(c(b(x1)))) -> a(a(a(a(x1)))) c(c(c(b(x1)))) -> c(a(a(a(x1)))) b(c(b(a(x1)))) -> b(b(a(a(x1)))) a(c(b(a(x1)))) -> a(b(a(a(x1)))) c(c(b(a(x1)))) -> c(b(a(a(x1)))) ---------------------------------------- (5) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled ---------------------------------------- (6) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1))) a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1))) a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) The relative TRS consists of the following S rules: b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1))) b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1))) b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1))) b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{c_1}(c_{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_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) ---------------------------------------- (7) 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)) = 1 + 2*x_1 POL(a_{c_1}(x_1)) = 1 + 4*x_1 POL(b_{a_1}(x_1)) = 3 + 2*x_1 POL(b_{b_1}(x_1)) = 2 + 2*x_1 POL(b_{c_1}(x_1)) = 1 + 4*x_1 POL(c_{a_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = 1 + 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: b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) Rules from S: b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1))) b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1))) b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{c_1}(c_{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_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) ---------------------------------------- (8) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1))) a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1))) a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1))) c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) The relative TRS consists of the following S rules: b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1))) ---------------------------------------- (9) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Knuth-Bendix order [KBO] with precedence:c_{a_1}_1 > b_{c_1}_1 > b_{b_1}_1 > c_{c_1}_1 > a_{c_1}_1 > a_{b_1}_1 > b_{a_1}_1 > c_{b_1}_1 and weight map: a_{c_1}_1=2 c_{b_1}_1=2 b_{a_1}_1=1 a_{b_1}_1=3 b_{b_1}_1=1 b_{c_1}_1=2 c_{c_1}_1=3 c_{a_1}_1=2 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1))) a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1))) a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1))) c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) Rules from S: b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1))) ---------------------------------------- (10) Obligation: Relative term rewrite system: R is empty. S is empty. ---------------------------------------- (11) RIsEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (12) YES