40.11/11.04 YES 40.11/11.05 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 40.11/11.05 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 40.11/11.05 40.11/11.05 40.11/11.05 Termination of the given RelTRS could be proven: 40.11/11.05 40.11/11.05 (0) RelTRS 40.11/11.05 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 40.11/11.05 (2) RelTRS 40.11/11.05 (3) FlatCCProof [EQUIVALENT, 0 ms] 40.11/11.05 (4) RelTRS 40.11/11.05 (5) RootLabelingProof [EQUIVALENT, 0 ms] 40.11/11.05 (6) RelTRS 40.11/11.05 (7) RelTRSRRRProof [EQUIVALENT, 45 ms] 40.11/11.05 (8) RelTRS 40.11/11.05 (9) RelTRSRRRProof [EQUIVALENT, 3573 ms] 40.11/11.05 (10) RelTRS 40.11/11.05 (11) RelTRSRRRProof [EQUIVALENT, 4 ms] 40.11/11.05 (12) RelTRS 40.11/11.05 (13) RIsEmptyProof [EQUIVALENT, 0 ms] 40.11/11.05 (14) YES 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (0) 40.11/11.05 Obligation: 40.11/11.05 Relative term rewrite system: 40.11/11.05 The relative TRS consists of the following R rules: 40.11/11.05 40.11/11.05 a(b(b(x1))) -> c(b(c(x1))) 40.11/11.05 c(a(b(x1))) -> a(c(b(x1))) 40.11/11.05 c(c(a(x1))) -> c(b(b(x1))) 40.11/11.05 40.11/11.05 The relative TRS consists of the following S rules: 40.11/11.05 40.11/11.05 b(c(c(x1))) -> b(a(c(x1))) 40.11/11.05 a(c(b(x1))) -> c(b(a(x1))) 40.11/11.05 c(c(b(x1))) -> c(a(c(x1))) 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (1) RelTRS Reverse (EQUIVALENT) 40.11/11.05 We have reversed the following relative TRS [REVERSE]: 40.11/11.05 The set of rules R is 40.11/11.05 a(b(b(x1))) -> c(b(c(x1))) 40.11/11.05 c(a(b(x1))) -> a(c(b(x1))) 40.11/11.05 c(c(a(x1))) -> c(b(b(x1))) 40.11/11.05 40.11/11.05 The set of rules S is 40.11/11.05 b(c(c(x1))) -> b(a(c(x1))) 40.11/11.05 a(c(b(x1))) -> c(b(a(x1))) 40.11/11.05 c(c(b(x1))) -> c(a(c(x1))) 40.11/11.05 40.11/11.05 We have obtained the following relative TRS: 40.11/11.05 The set of rules R is 40.11/11.05 b(b(a(x1))) -> c(b(c(x1))) 40.11/11.05 b(a(c(x1))) -> b(c(a(x1))) 40.11/11.05 a(c(c(x1))) -> b(b(c(x1))) 40.11/11.05 40.11/11.05 The set of rules S is 40.11/11.05 c(c(b(x1))) -> c(a(b(x1))) 40.11/11.05 b(c(a(x1))) -> a(b(c(x1))) 40.11/11.05 b(c(c(x1))) -> c(a(c(x1))) 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (2) 40.11/11.05 Obligation: 40.11/11.05 Relative term rewrite system: 40.11/11.05 The relative TRS consists of the following R rules: 40.11/11.05 40.11/11.05 b(b(a(x1))) -> c(b(c(x1))) 40.11/11.05 b(a(c(x1))) -> b(c(a(x1))) 40.11/11.05 a(c(c(x1))) -> b(b(c(x1))) 40.11/11.05 40.11/11.05 The relative TRS consists of the following S rules: 40.11/11.05 40.11/11.05 c(c(b(x1))) -> c(a(b(x1))) 40.11/11.05 b(c(a(x1))) -> a(b(c(x1))) 40.11/11.05 b(c(c(x1))) -> c(a(c(x1))) 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (3) FlatCCProof (EQUIVALENT) 40.11/11.05 We used flat context closure [ROOTLAB] 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (4) 40.11/11.05 Obligation: 40.11/11.05 Relative term rewrite system: 40.11/11.05 The relative TRS consists of the following R rules: 40.11/11.05 40.11/11.05 b(a(c(x1))) -> b(c(a(x1))) 40.11/11.05 b(b(b(a(x1)))) -> b(c(b(c(x1)))) 40.11/11.05 a(b(b(a(x1)))) -> a(c(b(c(x1)))) 40.11/11.05 c(b(b(a(x1)))) -> c(c(b(c(x1)))) 40.11/11.05 b(a(c(c(x1)))) -> b(b(b(c(x1)))) 40.11/11.05 a(a(c(c(x1)))) -> a(b(b(c(x1)))) 40.11/11.05 c(a(c(c(x1)))) -> c(b(b(c(x1)))) 40.11/11.05 40.11/11.05 The relative TRS consists of the following S rules: 40.11/11.05 40.11/11.05 c(c(b(x1))) -> c(a(b(x1))) 40.11/11.05 b(b(c(a(x1)))) -> b(a(b(c(x1)))) 40.11/11.05 a(b(c(a(x1)))) -> a(a(b(c(x1)))) 40.11/11.05 c(b(c(a(x1)))) -> c(a(b(c(x1)))) 40.11/11.05 b(b(c(c(x1)))) -> b(c(a(c(x1)))) 40.11/11.05 a(b(c(c(x1)))) -> a(c(a(c(x1)))) 40.11/11.05 c(b(c(c(x1)))) -> c(c(a(c(x1)))) 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (5) RootLabelingProof (EQUIVALENT) 40.11/11.05 We used plain root labeling [ROOTLAB] with the following heuristic: 40.11/11.05 LabelAll: All function symbols get labeled 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (6) 40.11/11.05 Obligation: 40.11/11.05 Relative term rewrite system: 40.11/11.05 The relative TRS consists of the following R rules: 40.11/11.05 40.11/11.05 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 40.11/11.05 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 40.11/11.05 The relative TRS consists of the following S rules: 40.11/11.05 40.11/11.05 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) 40.11/11.05 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) 40.11/11.05 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (7) RelTRSRRRProof (EQUIVALENT) 40.11/11.05 We used the following monotonic ordering for rule removal: 40.11/11.05 Polynomial interpretation [POLO]: 40.11/11.05 40.11/11.05 POL(a_{a_1}(x_1)) = x_1 40.11/11.05 POL(a_{b_1}(x_1)) = 1 + x_1 40.11/11.05 POL(a_{c_1}(x_1)) = 1 + x_1 40.11/11.05 POL(b_{a_1}(x_1)) = x_1 40.11/11.05 POL(b_{b_1}(x_1)) = 1 + x_1 40.11/11.05 POL(b_{c_1}(x_1)) = x_1 40.11/11.05 POL(c_{a_1}(x_1)) = x_1 40.11/11.05 POL(c_{b_1}(x_1)) = 1 + x_1 40.11/11.05 POL(c_{c_1}(x_1)) = 1 + x_1 40.11/11.05 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 40.11/11.05 Rules from R: 40.11/11.05 40.11/11.05 b_{a_1}(a_{c_1}(c_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 40.11/11.05 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 Rules from S: 40.11/11.05 40.11/11.05 c_{c_1}(c_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) 40.11/11.05 c_{c_1}(c_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) 40.11/11.05 c_{c_1}(c_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 40.11/11.05 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (8) 40.11/11.05 Obligation: 40.11/11.05 Relative term rewrite system: 40.11/11.05 The relative TRS consists of the following R rules: 40.11/11.05 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 40.11/11.05 The relative TRS consists of the following S rules: 40.11/11.05 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (9) RelTRSRRRProof (EQUIVALENT) 40.11/11.05 We used the following monotonic ordering for rule removal: 40.11/11.05 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 40.11/11.05 40.11/11.05 <<< 40.11/11.05 POL(a_{b_1}(x_1)) = [[0], [1]] + [[2, 0], [1, 0]] * x_1 40.11/11.05 >>> 40.11/11.05 40.11/11.05 <<< 40.11/11.05 POL(b_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 40.11/11.05 >>> 40.11/11.05 40.11/11.05 <<< 40.11/11.05 POL(b_{a_1}(x_1)) = [[1], [0]] + [[1, 1], [0, 0]] * x_1 40.11/11.05 >>> 40.11/11.05 40.11/11.05 <<< 40.11/11.05 POL(a_{c_1}(x_1)) = [[0], [1]] + [[2, 0], [1, 0]] * x_1 40.11/11.05 >>> 40.11/11.05 40.11/11.05 <<< 40.11/11.05 POL(c_{b_1}(x_1)) = [[1], [0]] + [[1, 0], [1, 0]] * x_1 40.11/11.05 >>> 40.11/11.05 40.11/11.05 <<< 40.11/11.05 POL(b_{c_1}(x_1)) = [[0], [0]] + [[1, 2], [0, 0]] * x_1 40.11/11.05 >>> 40.11/11.05 40.11/11.05 <<< 40.11/11.05 POL(a_{a_1}(x_1)) = [[1], [1]] + [[2, 2], [1, 1]] * x_1 40.11/11.05 >>> 40.11/11.05 40.11/11.05 <<< 40.11/11.05 POL(c_{a_1}(x_1)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 40.11/11.05 >>> 40.11/11.05 40.11/11.05 <<< 40.11/11.05 POL(c_{c_1}(x_1)) = [[1], [0]] + [[1, 0], [1, 0]] * x_1 40.11/11.05 >>> 40.11/11.05 40.11/11.05 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 40.11/11.05 Rules from R: 40.11/11.05 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 Rules from S: 40.11/11.05 40.11/11.05 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 40.11/11.05 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (10) 40.11/11.05 Obligation: 40.11/11.05 Relative term rewrite system: 40.11/11.05 The relative TRS consists of the following R rules: 40.11/11.05 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 40.11/11.05 The relative TRS consists of the following S rules: 40.11/11.05 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (11) RelTRSRRRProof (EQUIVALENT) 40.11/11.05 We used the following monotonic ordering for rule removal: 40.11/11.05 Knuth-Bendix order [KBO] with precedence:a_{b_1}_1 > b_{c_1}_1 > a_{c_1}_1 > c_{a_1}_1 > c_{b_1}_1 > c_{c_1}_1 > b_{b_1}_1 > b_{a_1}_1 > a_{a_1}_1 40.11/11.05 40.11/11.05 and weight map: 40.11/11.05 40.11/11.05 a_{b_1}_1=7 40.11/11.05 b_{b_1}_1=12 40.11/11.05 b_{a_1}_1=7 40.11/11.05 a_{c_1}_1=3 40.11/11.05 c_{b_1}_1=3 40.11/11.05 b_{c_1}_1=5 40.11/11.05 a_{a_1}_1=3 40.11/11.05 c_{a_1}_1=4 40.11/11.05 c_{c_1}_1=1 40.11/11.05 40.11/11.05 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 40.11/11.05 Rules from R: 40.11/11.05 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 Rules from S: 40.11/11.05 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 40.11/11.05 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 40.11/11.05 c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 40.11/11.05 40.11/11.05 40.11/11.05 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (12) 40.11/11.05 Obligation: 40.11/11.05 Relative term rewrite system: 40.11/11.05 R is empty. 40.11/11.05 S is empty. 40.11/11.05 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (13) RIsEmptyProof (EQUIVALENT) 40.11/11.05 The TRS R is empty. Hence, termination is trivially proven. 40.11/11.05 ---------------------------------------- 40.11/11.05 40.11/11.05 (14) 40.11/11.05 YES 40.49/11.12 EOF