/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given RelTRS could be proven: (0) RelTRS (1) RelTRS Reverse [EQUIVALENT, 0 ms] (2) RelTRS (3) FlatCCProof [EQUIVALENT, 0 ms] (4)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, 13 ms] (6) RelTRS (7) RelTRSRRRProof [EQUIVALENT, 163 ms] (8) RelTRS (9) SIsEmptyProof [EQUIVALENT, 0 ms] (10) QTRS (11) QTRSRRRProof [EQUIVALENT, 12 ms] (12) QTRS (13) DependencyPairsProof [EQUIVALENT, 0 ms] (14) QDP (15) DependencyGraphProof [EQUIVALENT, 0 ms] (16) QDP (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] (18) YES ---------------------------------------- (0) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: c(c(b(x1))) -> b(b(b(x1))) c(a(c(x1))) -> b(a(a(x1))) c(c(c(x1))) -> a(a(b(x1))) c(b(c(x1))) -> c(b(b(x1))) c(a(c(x1))) -> b(b(c(x1))) c(b(c(x1))) -> c(c(b(x1))) The relative TRS consists of the following S rules: a(b(b(x1))) -> b(a(c(x1))) ---------------------------------------- (1) RelTRS Reverse (EQUIVALENT) We have reversed the following relative TRS [REVERSE]: The set of rules R is c(c(b(x1))) -> b(b(b(x1))) c(a(c(x1))) -> b(a(a(x1))) c(c(c(x1))) -> a(a(b(x1))) c(b(c(x1))) -> c(b(b(x1))) c(a(c(x1))) -> b(b(c(x1))) c(b(c(x1))) -> c(c(b(x1))) The set of rules S is a(b(b(x1))) -> b(a(c(x1))) We have obtained the following relative TRS: The set of rules R is b(c(c(x1))) -> b(b(b(x1))) c(a(c(x1))) -> a(a(b(x1))) c(c(c(x1))) -> b(a(a(x1))) c(b(c(x1))) -> b(b(c(x1))) c(a(c(x1))) -> c(b(b(x1))) c(b(c(x1))) -> b(c(c(x1))) The set of rules S is b(b(a(x1))) -> c(a(b(x1))) ---------------------------------------- (2) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b(c(c(x1))) -> b(b(b(x1))) c(a(c(x1))) -> a(a(b(x1))) c(c(c(x1))) -> b(a(a(x1))) c(b(c(x1))) -> b(b(c(x1))) c(a(c(x1))) -> c(b(b(x1))) c(b(c(x1))) -> b(c(c(x1))) The relative TRS consists of the following S rules: b(b(a(x1))) -> c(a(b(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: b(c(c(x1))) -> b(b(b(x1))) c(a(c(x1))) -> c(b(b(x1))) b(c(a(c(x1)))) -> b(a(a(b(x1)))) c(c(a(c(x1)))) -> c(a(a(b(x1)))) a(c(a(c(x1)))) -> a(a(a(b(x1)))) b(c(c(c(x1)))) -> b(b(a(a(x1)))) c(c(c(c(x1)))) -> c(b(a(a(x1)))) a(c(c(c(x1)))) -> a(b(a(a(x1)))) b(c(b(c(x1)))) -> b(b(b(c(x1)))) c(c(b(c(x1)))) -> c(b(b(c(x1)))) a(c(b(c(x1)))) -> a(b(b(c(x1)))) b(c(b(c(x1)))) -> b(b(c(c(x1)))) c(c(b(c(x1)))) -> c(b(c(c(x1)))) a(c(b(c(x1)))) -> a(b(c(c(x1)))) The relative TRS consists of the following S rules: b(b(b(a(x1)))) -> b(c(a(b(x1)))) c(b(b(a(x1)))) -> c(c(a(b(x1)))) a(b(b(a(x1)))) -> a(c(a(b(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: b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1))) c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1))) c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1))) b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) The relative TRS consists of the following S rules: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) ---------------------------------------- (7) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = 1 + 2*x_1 POL(a_{b_1}(x_1)) = 2*x_1 POL(a_{c_1}(x_1)) = 3 + x_1 POL(b_{a_1}(x_1)) = 1 + 2*x_1 POL(b_{b_1}(x_1)) = 2*x_1 POL(b_{c_1}(x_1)) = 3 + 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)) = 4 + 2*x_1 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1))) b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1))) c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1))) c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1))) b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) Rules from S: b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) ---------------------------------------- (8) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) S is empty. ---------------------------------------- (9) SIsEmptyProof (EQUIVALENT) The TRS S is empty. Hence, termination of R/S is equivalent to termination of R. ---------------------------------------- (10) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) Q is empty. ---------------------------------------- (11) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{c_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{c_1}(x_1)) = x_1 POL(c_{a_1}(x_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: c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) ---------------------------------------- (12) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) Q is empty. ---------------------------------------- (13) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1)) C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{C_1}(c_{c_1}(x1)) C_{C_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> C_{C_1}(c_{a_1}(x1)) The TRS R consists of the following rules: c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_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 1 less node. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{C_1}(c_{c_1}(x1)) C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1)) The TRS R consists of the following rules: c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) 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: *C_{C_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{C_1}(c_{c_1}(x1)) The graph contains the following edges 1 > 1 *C_{C_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(x1)) The graph contains the following edges 1 > 1 ---------------------------------------- (18) YES