/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given RelTRS could be proven: (0) RelTRS (1) RelTRS S Cleaner [EQUIVALENT, 0 ms] (2) RelTRS (3) RelTRS Reverse [EQUIVALENT, 0 ms] (4) RelTRS (5) FlatCCProof [EQUIVALENT, 0 ms] (6) RelTRS (7) RootLabelingProof [EQUIVALENT, 21 ms] (8) RelTRS (9) RelTRSRRRProof [EQUIVALENT, 6079 ms] (10) RelTRS (11) RelTRSRRRProof [EQUIVALENT, 5 ms] (12) RelTRS (13) RIsEmptyProof [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a(a(b(x1))) -> a(c(c(x1))) c(c(b(x1))) -> b(b(c(x1))) c(a(a(x1))) -> c(a(c(x1))) The relative TRS consists of the following S rules: b(b(c(x1))) -> b(a(c(x1))) a(b(a(x1))) -> a(b(c(x1))) c(a(c(x1))) -> c(b(a(x1))) a(a(b(x1))) -> a(a(b(x1))) ---------------------------------------- (1) RelTRS S Cleaner (EQUIVALENT) We have deleted all rules from S that have the shape t -> t: a(a(b(x1))) -> a(a(b(x1))) ---------------------------------------- (2) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a(a(b(x1))) -> a(c(c(x1))) c(c(b(x1))) -> b(b(c(x1))) c(a(a(x1))) -> c(a(c(x1))) The relative TRS consists of the following S rules: b(b(c(x1))) -> b(a(c(x1))) a(b(a(x1))) -> a(b(c(x1))) c(a(c(x1))) -> c(b(a(x1))) ---------------------------------------- (3) RelTRS Reverse (EQUIVALENT) We have reversed the following relative TRS [REVERSE]: The set of rules R is a(a(b(x1))) -> a(c(c(x1))) c(c(b(x1))) -> b(b(c(x1))) c(a(a(x1))) -> c(a(c(x1))) The set of rules S is b(b(c(x1))) -> b(a(c(x1))) a(b(a(x1))) -> a(b(c(x1))) c(a(c(x1))) -> c(b(a(x1))) We have obtained the following relative TRS: The set of rules R is b(a(a(x1))) -> c(c(a(x1))) b(c(c(x1))) -> c(b(b(x1))) a(a(c(x1))) -> c(a(c(x1))) The set of rules S is c(b(b(x1))) -> c(a(b(x1))) a(b(a(x1))) -> c(b(a(x1))) c(a(c(x1))) -> a(b(c(x1))) ---------------------------------------- (4) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b(a(a(x1))) -> c(c(a(x1))) b(c(c(x1))) -> c(b(b(x1))) a(a(c(x1))) -> c(a(c(x1))) The relative TRS consists of the following S rules: c(b(b(x1))) -> c(a(b(x1))) a(b(a(x1))) -> c(b(a(x1))) c(a(c(x1))) -> a(b(c(x1))) ---------------------------------------- (5) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] ---------------------------------------- (6) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b(b(a(a(x1)))) -> b(c(c(a(x1)))) a(b(a(a(x1)))) -> a(c(c(a(x1)))) c(b(a(a(x1)))) -> c(c(c(a(x1)))) b(b(c(c(x1)))) -> b(c(b(b(x1)))) a(b(c(c(x1)))) -> a(c(b(b(x1)))) c(b(c(c(x1)))) -> c(c(b(b(x1)))) b(a(a(c(x1)))) -> b(c(a(c(x1)))) a(a(a(c(x1)))) -> a(c(a(c(x1)))) c(a(a(c(x1)))) -> c(c(a(c(x1)))) The relative TRS consists of the following S rules: c(b(b(x1))) -> c(a(b(x1))) b(a(b(a(x1)))) -> b(c(b(a(x1)))) a(a(b(a(x1)))) -> a(c(b(a(x1)))) c(a(b(a(x1)))) -> c(c(b(a(x1)))) b(c(a(c(x1)))) -> b(a(b(c(x1)))) a(c(a(c(x1)))) -> a(a(b(c(x1)))) c(c(a(c(x1)))) -> c(a(b(c(x1)))) ---------------------------------------- (7) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled ---------------------------------------- (8) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) The relative TRS consists of the following S rules: c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) ---------------------------------------- (9) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(b_{b_1}(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0], [2]] + [[1, 2], [0, 0]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0], [2]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[2], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(c_{c_1}(x_1)) = [[2], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(c_{a_1}(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[2], [2]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(c_{b_1}(x_1)) = [[0], [0]] + [[1, 2], [0, 0]] * x_1 >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) Rules from S: a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) ---------------------------------------- (10) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) The relative TRS consists of the following S rules: c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) ---------------------------------------- (11) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Knuth-Bendix order [KBO] with precedence:c_{b_1}_1 > b_{b_1}_1 > a_{b_1}_1 > a_{c_1}_1 > c_{a_1}_1 > a_{a_1}_1 > b_{a_1}_1 > c_{c_1}_1 > b_{c_1}_1 and weight map: b_{b_1}_1=3 b_{a_1}_1=4 a_{a_1}_1=3 a_{b_1}_1=2 b_{c_1}_1=5 c_{c_1}_1=3 c_{a_1}_1=2 a_{c_1}_1=4 c_{b_1}_1=1 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) Rules from S: c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) ---------------------------------------- (12) Obligation: Relative term rewrite system: R is empty. S is empty. ---------------------------------------- (13) RIsEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (14) YES