4.50/1.92 YES 4.50/1.93 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 4.50/1.93 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.50/1.93 4.50/1.93 4.50/1.93 Termination of the given RelTRS could be proven: 4.50/1.93 4.50/1.93 (0) RelTRS 4.50/1.93 (1) FlatCCProof [EQUIVALENT, 0 ms] 4.50/1.93 (2) RelTRS 4.50/1.93 (3) RootLabelingProof [EQUIVALENT, 16 ms] 4.50/1.93 (4) RelTRS 4.50/1.93 (5) RelTRSRRRProof [EQUIVALENT, 18 ms] 4.50/1.93 (6) RelTRS 4.50/1.93 (7) RIsEmptyProof [EQUIVALENT, 0 ms] 4.50/1.93 (8) YES 4.50/1.93 4.50/1.93 4.50/1.93 ---------------------------------------- 4.50/1.93 4.50/1.93 (0) 4.50/1.93 Obligation: 4.50/1.93 Relative term rewrite system: 4.50/1.93 The relative TRS consists of the following R rules: 4.50/1.93 4.50/1.93 a(a(c(x1))) -> b(b(c(x1))) 4.50/1.93 b(a(c(x1))) -> a(a(b(x1))) 4.50/1.93 4.50/1.93 The relative TRS consists of the following S rules: 4.50/1.93 4.50/1.93 a(c(a(x1))) -> a(b(b(x1))) 4.50/1.93 b(b(b(x1))) -> c(c(b(x1))) 4.50/1.93 b(a(c(x1))) -> a(a(c(x1))) 4.50/1.93 c(a(a(x1))) -> a(a(a(x1))) 4.50/1.93 4.50/1.93 4.50/1.93 ---------------------------------------- 4.50/1.93 4.50/1.93 (1) FlatCCProof (EQUIVALENT) 4.50/1.93 We used flat context closure [ROOTLAB] 4.50/1.93 4.50/1.93 ---------------------------------------- 4.50/1.93 4.50/1.93 (2) 4.50/1.93 Obligation: 4.50/1.93 Relative term rewrite system: 4.50/1.93 The relative TRS consists of the following R rules: 4.50/1.93 4.50/1.93 a(a(a(c(x1)))) -> a(b(b(c(x1)))) 4.50/1.93 c(a(a(c(x1)))) -> c(b(b(c(x1)))) 4.50/1.93 b(a(a(c(x1)))) -> b(b(b(c(x1)))) 4.50/1.93 a(b(a(c(x1)))) -> a(a(a(b(x1)))) 4.50/1.93 c(b(a(c(x1)))) -> c(a(a(b(x1)))) 4.50/1.93 b(b(a(c(x1)))) -> b(a(a(b(x1)))) 4.50/1.93 4.50/1.93 The relative TRS consists of the following S rules: 4.50/1.93 4.50/1.93 a(c(a(x1))) -> a(b(b(x1))) 4.50/1.93 a(b(b(b(x1)))) -> a(c(c(b(x1)))) 4.50/1.93 c(b(b(b(x1)))) -> c(c(c(b(x1)))) 4.50/1.93 b(b(b(b(x1)))) -> b(c(c(b(x1)))) 4.50/1.93 a(b(a(c(x1)))) -> a(a(a(c(x1)))) 4.50/1.93 c(b(a(c(x1)))) -> c(a(a(c(x1)))) 4.50/1.93 b(b(a(c(x1)))) -> b(a(a(c(x1)))) 4.50/1.93 a(c(a(a(x1)))) -> a(a(a(a(x1)))) 4.50/1.93 c(c(a(a(x1)))) -> c(a(a(a(x1)))) 4.50/1.93 b(c(a(a(x1)))) -> b(a(a(a(x1)))) 4.50/1.93 4.50/1.93 4.50/1.93 ---------------------------------------- 4.50/1.93 4.50/1.93 (3) RootLabelingProof (EQUIVALENT) 4.50/1.93 We used plain root labeling [ROOTLAB] with the following heuristic: 4.50/1.93 LabelAll: All function symbols get labeled 4.50/1.93 4.50/1.93 4.50/1.93 ---------------------------------------- 4.50/1.93 4.50/1.93 (4) 4.50/1.93 Obligation: 4.50/1.93 Relative term rewrite system: 4.50/1.93 The relative TRS consists of the following R rules: 4.50/1.93 4.50/1.93 a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.50/1.93 a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.50/1.93 a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.50/1.93 c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.50/1.93 c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.50/1.93 c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.50/1.93 b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.50/1.93 b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.50/1.93 b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.50/1.93 4.50/1.93 The relative TRS consists of the following S rules: 4.50/1.93 4.50/1.93 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1))) 4.50/1.93 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 4.50/1.93 a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 4.50/1.93 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 4.50/1.93 c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 4.50/1.93 c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 4.50/1.93 c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 4.50/1.93 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 4.50/1.93 b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 4.50/1.93 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.50/1.93 c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.50/1.93 c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.50/1.93 c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.50/1.93 b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.50/1.93 b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.50/1.93 b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.50/1.93 4.50/1.93 4.50/1.93 ---------------------------------------- 4.50/1.93 4.50/1.93 (5) RelTRSRRRProof (EQUIVALENT) 4.50/1.93 We used the following monotonic ordering for rule removal: 4.50/1.93 Knuth-Bendix order [KBO] with precedence:c_{b_1}_1 > a_{c_1}_1 > b_{b_1}_1 > a_{a_1}_1 > b_{a_1}_1 > c_{c_1}_1 > c_{a_1}_1 > b_{c_1}_1 > a_{b_1}_1 4.50/1.93 4.50/1.93 and weight map: 4.50/1.93 4.50/1.93 a_{a_1}_1=13 4.50/1.93 a_{c_1}_1=26 4.50/1.93 c_{a_1}_1=12 4.50/1.93 a_{b_1}_1=6 4.50/1.93 b_{b_1}_1=18 4.50/1.93 b_{c_1}_1=27 4.50/1.93 c_{c_1}_1=14 4.50/1.93 c_{b_1}_1=1 4.50/1.93 b_{a_1}_1=25 4.50/1.93 4.50/1.93 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 4.50/1.93 Rules from R: 4.50/1.93 4.50/1.93 a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.50/1.93 a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.50/1.93 a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.50/1.93 c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.50/1.93 c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.50/1.93 c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.50/1.93 b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.50/1.93 b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.50/1.93 b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.50/1.93 Rules from S: 4.50/1.93 4.50/1.93 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(x1))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{b_1}(b_{c_1}(x1))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(x1))) 4.50/1.93 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 4.50/1.93 a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 4.50/1.93 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 4.50/1.93 c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 4.50/1.93 c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 4.50/1.93 c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 4.50/1.93 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) 4.50/1.93 b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) 4.50/1.93 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) 4.50/1.93 a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) 4.50/1.93 c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) 4.50/1.93 b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.50/1.93 a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.50/1.93 c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.50/1.93 c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.50/1.93 c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.50/1.93 b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 4.50/1.93 b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 4.50/1.93 b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 4.50/1.93 4.50/1.93 4.50/1.93 4.50/1.93 4.50/1.93 ---------------------------------------- 4.50/1.93 4.50/1.93 (6) 4.50/1.93 Obligation: 4.50/1.93 Relative term rewrite system: 4.50/1.93 R is empty. 4.50/1.93 S is empty. 4.50/1.93 4.50/1.93 ---------------------------------------- 4.50/1.93 4.50/1.93 (7) RIsEmptyProof (EQUIVALENT) 4.50/1.93 The TRS R is empty. Hence, termination is trivially proven. 4.50/1.93 ---------------------------------------- 4.50/1.93 4.50/1.93 (8) 4.50/1.93 YES 4.76/1.99 EOF