4.65/2.10 YES 4.65/2.15 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.65/2.15 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.65/2.15 4.65/2.15 4.65/2.15 Termination of the given RelTRS could be proven: 4.65/2.15 4.65/2.15 (0) RelTRS 4.65/2.15 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 4.65/2.15 (2) RelTRS 4.65/2.15 (3) FlatCCProof [EQUIVALENT, 0 ms] 4.65/2.15 (4) RelTRS 4.65/2.15 (5) RootLabelingProof [EQUIVALENT, 21 ms] 4.65/2.15 (6) RelTRS 4.65/2.15 (7) RelTRSRRRProof [EQUIVALENT, 17 ms] 4.65/2.15 (8) RelTRS 4.65/2.15 (9) RIsEmptyProof [EQUIVALENT, 1 ms] 4.65/2.15 (10) YES 4.65/2.15 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (0) 4.65/2.15 Obligation: 4.65/2.15 Relative term rewrite system: 4.65/2.15 The relative TRS consists of the following R rules: 4.65/2.15 4.65/2.15 b(b(b(x1))) -> b(a(c(x1))) 4.65/2.15 a(a(c(x1))) -> b(b(a(x1))) 4.65/2.15 4.65/2.15 The relative TRS consists of the following S rules: 4.65/2.15 4.65/2.15 a(b(c(x1))) -> c(b(a(x1))) 4.65/2.15 b(b(a(x1))) -> c(b(c(x1))) 4.65/2.15 c(a(c(x1))) -> a(b(c(x1))) 4.65/2.15 a(b(b(x1))) -> c(b(a(x1))) 4.65/2.15 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (1) RelTRS Reverse (EQUIVALENT) 4.65/2.15 We have reversed the following relative TRS [REVERSE]: 4.65/2.15 The set of rules R is 4.65/2.15 b(b(b(x1))) -> b(a(c(x1))) 4.65/2.15 a(a(c(x1))) -> b(b(a(x1))) 4.65/2.15 4.65/2.15 The set of rules S is 4.65/2.15 a(b(c(x1))) -> c(b(a(x1))) 4.65/2.15 b(b(a(x1))) -> c(b(c(x1))) 4.65/2.15 c(a(c(x1))) -> a(b(c(x1))) 4.65/2.15 a(b(b(x1))) -> c(b(a(x1))) 4.65/2.15 4.65/2.15 We have obtained the following relative TRS: 4.65/2.15 The set of rules R is 4.65/2.15 b(b(b(x1))) -> c(a(b(x1))) 4.65/2.15 c(a(a(x1))) -> a(b(b(x1))) 4.65/2.15 4.65/2.15 The set of rules S is 4.65/2.15 c(b(a(x1))) -> a(b(c(x1))) 4.65/2.15 a(b(b(x1))) -> c(b(c(x1))) 4.65/2.15 c(a(c(x1))) -> c(b(a(x1))) 4.65/2.15 b(b(a(x1))) -> a(b(c(x1))) 4.65/2.15 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (2) 4.65/2.15 Obligation: 4.65/2.15 Relative term rewrite system: 4.65/2.15 The relative TRS consists of the following R rules: 4.65/2.15 4.65/2.15 b(b(b(x1))) -> c(a(b(x1))) 4.65/2.15 c(a(a(x1))) -> a(b(b(x1))) 4.65/2.15 4.65/2.15 The relative TRS consists of the following S rules: 4.65/2.15 4.65/2.15 c(b(a(x1))) -> a(b(c(x1))) 4.65/2.15 a(b(b(x1))) -> c(b(c(x1))) 4.65/2.15 c(a(c(x1))) -> c(b(a(x1))) 4.65/2.15 b(b(a(x1))) -> a(b(c(x1))) 4.65/2.15 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (3) FlatCCProof (EQUIVALENT) 4.65/2.15 We used flat context closure [ROOTLAB] 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (4) 4.65/2.15 Obligation: 4.65/2.15 Relative term rewrite system: 4.65/2.15 The relative TRS consists of the following R rules: 4.65/2.15 4.65/2.15 b(b(b(b(x1)))) -> b(c(a(b(x1)))) 4.65/2.15 c(b(b(b(x1)))) -> c(c(a(b(x1)))) 4.65/2.15 a(b(b(b(x1)))) -> a(c(a(b(x1)))) 4.65/2.15 b(c(a(a(x1)))) -> b(a(b(b(x1)))) 4.65/2.15 c(c(a(a(x1)))) -> c(a(b(b(x1)))) 4.65/2.15 a(c(a(a(x1)))) -> a(a(b(b(x1)))) 4.65/2.15 4.65/2.15 The relative TRS consists of the following S rules: 4.65/2.15 4.65/2.15 c(a(c(x1))) -> c(b(a(x1))) 4.65/2.15 b(c(b(a(x1)))) -> b(a(b(c(x1)))) 4.65/2.15 c(c(b(a(x1)))) -> c(a(b(c(x1)))) 4.65/2.15 a(c(b(a(x1)))) -> a(a(b(c(x1)))) 4.65/2.15 b(a(b(b(x1)))) -> b(c(b(c(x1)))) 4.65/2.15 c(a(b(b(x1)))) -> c(c(b(c(x1)))) 4.65/2.15 a(a(b(b(x1)))) -> a(c(b(c(x1)))) 4.65/2.15 b(b(b(a(x1)))) -> b(a(b(c(x1)))) 4.65/2.15 c(b(b(a(x1)))) -> c(a(b(c(x1)))) 4.65/2.15 a(b(b(a(x1)))) -> a(a(b(c(x1)))) 4.65/2.15 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (5) RootLabelingProof (EQUIVALENT) 4.65/2.15 We used plain root labeling [ROOTLAB] with the following heuristic: 4.65/2.15 LabelAll: All function symbols get labeled 4.65/2.15 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (6) 4.65/2.15 Obligation: 4.65/2.15 Relative term rewrite system: 4.65/2.15 The relative TRS consists of the following R rules: 4.65/2.15 4.65/2.15 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.65/2.15 b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.65/2.15 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.65/2.15 c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.65/2.15 c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.65/2.15 c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.65/2.15 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.65/2.15 a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.65/2.15 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.65/2.15 b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.65/2.15 b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.65/2.15 b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.65/2.15 c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.65/2.15 c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.65/2.15 c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.65/2.15 a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.65/2.15 a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.65/2.15 a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.65/2.15 4.65/2.15 The relative TRS consists of the following S rules: 4.65/2.15 4.65/2.15 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1))) 4.65/2.15 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1))) 4.65/2.15 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1))) 4.65/2.15 b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (7) RelTRSRRRProof (EQUIVALENT) 4.65/2.15 We used the following monotonic ordering for rule removal: 4.65/2.15 Knuth-Bendix order [KBO] with precedence:b_{c_1}_1 > b_{b_1}_1 > c_{b_1}_1 > c_{a_1}_1 > b_{a_1}_1 > a_{b_1}_1 > a_{c_1}_1 > a_{a_1}_1 > c_{c_1}_1 4.65/2.15 4.65/2.15 and weight map: 4.65/2.15 4.65/2.15 b_{b_1}_1=15 4.65/2.15 b_{c_1}_1=20 4.65/2.15 c_{a_1}_1=3 4.65/2.15 a_{b_1}_1=14 4.65/2.15 b_{a_1}_1=11 4.65/2.15 c_{b_1}_1=1 4.65/2.15 c_{c_1}_1=13 4.65/2.15 a_{c_1}_1=27 4.65/2.15 a_{a_1}_1=18 4.65/2.15 4.65/2.15 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 4.65/2.15 Rules from R: 4.65/2.15 4.65/2.15 b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.65/2.15 b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.65/2.15 b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.65/2.15 c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.65/2.15 c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.65/2.15 c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.65/2.15 a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 4.65/2.15 a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 4.65/2.15 a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 4.65/2.15 b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.65/2.15 b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.65/2.15 b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.65/2.15 c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.65/2.15 c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.65/2.15 c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.65/2.15 a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) 4.65/2.15 a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) 4.65/2.15 a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) 4.65/2.15 Rules from S: 4.65/2.15 4.65/2.15 c_{a_1}(a_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1))) 4.65/2.15 c_{a_1}(a_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1))) 4.65/2.15 c_{a_1}(a_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1))) 4.65/2.15 b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 4.65/2.15 a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 4.65/2.15 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 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)))) 4.65/2.15 4.65/2.15 4.65/2.15 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (8) 4.65/2.15 Obligation: 4.65/2.15 Relative term rewrite system: 4.65/2.15 R is empty. 4.65/2.15 S is empty. 4.65/2.15 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (9) RIsEmptyProof (EQUIVALENT) 4.65/2.15 The TRS R is empty. Hence, termination is trivially proven. 4.65/2.15 ---------------------------------------- 4.65/2.15 4.65/2.15 (10) 4.65/2.15 YES 5.03/2.19 EOF