30.55/8.67 YES 31.02/8.68 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 31.02/8.68 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 31.02/8.68 31.02/8.68 31.02/8.68 Termination of the given RelTRS could be proven: 31.02/8.68 31.02/8.68 (0) RelTRS 31.02/8.68 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 31.02/8.68 (2) RelTRS 31.02/8.68 (3) FlatCCProof [EQUIVALENT, 0 ms] 31.02/8.68 (4) RelTRS 31.02/8.68 (5) RootLabelingProof [EQUIVALENT, 5 ms] 31.02/8.68 (6) RelTRS 31.02/8.68 (7) RelTRSRRRProof [EQUIVALENT, 51 ms] 31.02/8.68 (8) RelTRS 31.02/8.68 (9) RelTRSRRRProof [EQUIVALENT, 851 ms] 31.02/8.68 (10) RelTRS 31.02/8.68 (11) RelTRSRRRProof [EQUIVALENT, 448 ms] 31.02/8.68 (12) RelTRS 31.02/8.68 (13) RelTRSRRRProof [EQUIVALENT, 3 ms] 31.02/8.68 (14) RelTRS 31.02/8.68 (15) RIsEmptyProof [EQUIVALENT, 0 ms] 31.02/8.68 (16) YES 31.02/8.68 31.02/8.68 31.02/8.68 ---------------------------------------- 31.02/8.68 31.02/8.68 (0) 31.02/8.68 Obligation: 31.02/8.68 Relative term rewrite system: 31.02/8.68 The relative TRS consists of the following R rules: 31.02/8.68 31.02/8.68 c(b(b(x1))) -> a(c(a(x1))) 31.02/8.68 a(c(c(x1))) -> a(a(b(x1))) 31.02/8.68 a(a(c(x1))) -> b(b(a(x1))) 31.02/8.68 31.02/8.68 The relative TRS consists of the following S rules: 31.02/8.68 31.02/8.68 b(c(b(x1))) -> a(b(b(x1))) 31.02/8.68 b(c(c(x1))) -> c(b(b(x1))) 31.02/8.68 a(b(b(x1))) -> a(c(b(x1))) 31.02/8.68 c(c(a(x1))) -> a(b(c(x1))) 31.02/8.68 31.02/8.68 31.02/8.68 ---------------------------------------- 31.02/8.68 31.02/8.68 (1) RelTRS Reverse (EQUIVALENT) 31.02/8.68 We have reversed the following relative TRS [REVERSE]: 31.02/8.68 The set of rules R is 31.02/8.68 c(b(b(x1))) -> a(c(a(x1))) 31.02/8.68 a(c(c(x1))) -> a(a(b(x1))) 31.02/8.68 a(a(c(x1))) -> b(b(a(x1))) 31.02/8.68 31.02/8.68 The set of rules S is 31.02/8.68 b(c(b(x1))) -> a(b(b(x1))) 31.02/8.68 b(c(c(x1))) -> c(b(b(x1))) 31.02/8.68 a(b(b(x1))) -> a(c(b(x1))) 31.02/8.68 c(c(a(x1))) -> a(b(c(x1))) 31.02/8.68 31.02/8.68 We have obtained the following relative TRS: 31.02/8.68 The set of rules R is 31.02/8.68 b(b(c(x1))) -> a(c(a(x1))) 31.02/8.68 c(c(a(x1))) -> b(a(a(x1))) 31.02/8.68 c(a(a(x1))) -> a(b(b(x1))) 31.02/8.68 31.02/8.68 The set of rules S is 31.02/8.68 b(c(b(x1))) -> b(b(a(x1))) 31.02/8.68 c(c(b(x1))) -> b(b(c(x1))) 31.02/8.68 b(b(a(x1))) -> b(c(a(x1))) 31.02/8.68 a(c(c(x1))) -> c(b(a(x1))) 31.02/8.68 31.02/8.68 31.02/8.68 ---------------------------------------- 31.02/8.68 31.02/8.68 (2) 31.02/8.68 Obligation: 31.02/8.68 Relative term rewrite system: 31.02/8.68 The relative TRS consists of the following R rules: 31.02/8.68 31.02/8.68 b(b(c(x1))) -> a(c(a(x1))) 31.02/8.68 c(c(a(x1))) -> b(a(a(x1))) 31.02/8.68 c(a(a(x1))) -> a(b(b(x1))) 31.02/8.68 31.02/8.68 The relative TRS consists of the following S rules: 31.02/8.68 31.02/8.68 b(c(b(x1))) -> b(b(a(x1))) 31.02/8.68 c(c(b(x1))) -> b(b(c(x1))) 31.02/8.68 b(b(a(x1))) -> b(c(a(x1))) 31.02/8.68 a(c(c(x1))) -> c(b(a(x1))) 31.02/8.68 31.02/8.68 31.02/8.68 ---------------------------------------- 31.02/8.68 31.02/8.68 (3) FlatCCProof (EQUIVALENT) 31.02/8.68 We used flat context closure [ROOTLAB] 31.02/8.68 31.02/8.68 ---------------------------------------- 31.02/8.68 31.02/8.68 (4) 31.02/8.68 Obligation: 31.02/8.68 Relative term rewrite system: 31.02/8.68 The relative TRS consists of the following R rules: 31.02/8.68 31.02/8.68 b(b(b(c(x1)))) -> b(a(c(a(x1)))) 31.02/8.68 c(b(b(c(x1)))) -> c(a(c(a(x1)))) 31.02/8.68 a(b(b(c(x1)))) -> a(a(c(a(x1)))) 31.02/8.68 b(c(c(a(x1)))) -> b(b(a(a(x1)))) 31.02/8.68 c(c(c(a(x1)))) -> c(b(a(a(x1)))) 31.02/8.68 a(c(c(a(x1)))) -> a(b(a(a(x1)))) 31.02/8.68 b(c(a(a(x1)))) -> b(a(b(b(x1)))) 31.02/8.68 c(c(a(a(x1)))) -> c(a(b(b(x1)))) 31.02/8.68 a(c(a(a(x1)))) -> a(a(b(b(x1)))) 31.02/8.68 31.02/8.68 The relative TRS consists of the following S rules: 31.02/8.68 31.02/8.68 b(c(b(x1))) -> b(b(a(x1))) 31.02/8.68 b(b(a(x1))) -> b(c(a(x1))) 31.02/8.68 b(c(c(b(x1)))) -> b(b(b(c(x1)))) 31.02/8.68 c(c(c(b(x1)))) -> c(b(b(c(x1)))) 31.02/8.68 a(c(c(b(x1)))) -> a(b(b(c(x1)))) 31.02/8.68 b(a(c(c(x1)))) -> b(c(b(a(x1)))) 31.02/8.68 c(a(c(c(x1)))) -> c(c(b(a(x1)))) 31.02/8.68 a(a(c(c(x1)))) -> a(c(b(a(x1)))) 31.02/8.68 31.02/8.68 31.02/8.68 ---------------------------------------- 31.02/8.68 31.02/8.68 (5) RootLabelingProof (EQUIVALENT) 31.02/8.68 We used plain root labeling [ROOTLAB] with the following heuristic: 31.02/8.68 LabelAll: All function symbols get labeled 31.02/8.68 31.02/8.68 31.02/8.68 ---------------------------------------- 31.02/8.68 31.02/8.68 (6) 31.02/8.68 Obligation: 31.02/8.68 Relative term rewrite system: 31.02/8.68 The relative TRS consists of the following R rules: 31.02/8.68 31.02/8.68 b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.68 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 31.02/8.68 b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.68 c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.68 c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 31.02/8.68 c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.68 a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.68 a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 31.02/8.68 a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.68 b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 31.02/8.68 b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 31.02/8.68 b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 31.02/8.68 c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 31.02/8.68 c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 31.02/8.68 c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 31.02/8.68 a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 31.02/8.68 a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 31.02/8.68 a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 31.02/8.68 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)))) 31.02/8.68 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)))) 31.02/8.68 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)))) 31.02/8.68 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)))) 31.02/8.68 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)))) 31.02/8.68 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)))) 31.02/8.68 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)))) 31.02/8.68 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)))) 31.02/8.68 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)))) 31.02/8.68 31.02/8.68 The relative TRS consists of the following S rules: 31.02/8.68 31.02/8.68 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1))) 31.02/8.68 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1))) 31.02/8.68 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1))) 31.02/8.68 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 31.02/8.68 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 31.02/8.68 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 31.02/8.68 b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 31.02/8.68 b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 31.02/8.68 b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 31.02/8.68 c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 31.02/8.68 c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 31.02/8.68 c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 31.02/8.68 a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 31.02/8.68 a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 31.02/8.68 a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 31.02/8.68 b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 31.02/8.68 b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 31.02/8.68 b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 31.02/8.68 c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 31.02/8.68 c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 31.02/8.68 c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 31.02/8.68 a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 31.02/8.68 a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 31.02/8.68 a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 31.02/8.68 31.02/8.68 31.02/8.68 ---------------------------------------- 31.02/8.68 31.02/8.68 (7) RelTRSRRRProof (EQUIVALENT) 31.02/8.68 We used the following monotonic ordering for rule removal: 31.02/8.68 Polynomial interpretation [POLO]: 31.02/8.68 31.02/8.68 POL(a_{a_1}(x_1)) = x_1 31.02/8.68 POL(a_{b_1}(x_1)) = x_1 31.02/8.68 POL(a_{c_1}(x_1)) = x_1 31.02/8.68 POL(b_{a_1}(x_1)) = x_1 31.02/8.68 POL(b_{b_1}(x_1)) = x_1 31.02/8.68 POL(b_{c_1}(x_1)) = x_1 31.02/8.68 POL(c_{a_1}(x_1)) = x_1 31.02/8.68 POL(c_{b_1}(x_1)) = x_1 31.02/8.68 POL(c_{c_1}(x_1)) = 1 + x_1 31.02/8.68 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 31.02/8.68 Rules from R: 31.02/8.68 31.02/8.68 b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 31.02/8.69 c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 31.02/8.69 a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 31.02/8.69 b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 31.02/8.69 b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 31.02/8.69 b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 31.02/8.69 c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 31.02/8.69 c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 31.02/8.69 c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 31.02/8.69 a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) 31.02/8.69 a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) 31.02/8.69 a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 Rules from S: 31.02/8.69 31.02/8.69 b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 31.02/8.69 b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 31.02/8.69 c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 31.02/8.69 c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 31.02/8.69 c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 31.02/8.69 a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) 31.02/8.69 a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) 31.02/8.69 b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 31.02/8.69 b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 31.02/8.69 b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 31.02/8.69 c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 31.02/8.69 a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 31.02/8.69 a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 31.02/8.69 a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 31.02/8.69 31.02/8.69 31.02/8.69 31.02/8.69 31.02/8.69 ---------------------------------------- 31.02/8.69 31.02/8.69 (8) 31.02/8.69 Obligation: 31.02/8.69 Relative term rewrite system: 31.02/8.69 The relative TRS consists of the following R rules: 31.02/8.69 31.02/8.69 b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 31.02/8.69 The relative TRS consists of the following S rules: 31.02/8.69 31.02/8.69 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1))) 31.02/8.69 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1))) 31.02/8.69 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 31.02/8.69 b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 31.02/8.69 a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 31.02/8.69 c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 31.02/8.69 c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 31.02/8.69 31.02/8.69 31.02/8.69 ---------------------------------------- 31.02/8.69 31.02/8.69 (9) RelTRSRRRProof (EQUIVALENT) 31.02/8.69 We used the following monotonic ordering for rule removal: 31.02/8.69 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(b_{b_1}(x_1)) = [[0], [1]] + [[2, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(b_{c_1}(x_1)) = [[0], [1]] + [[2, 0], [0, 2]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(c_{b_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 2]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(b_{a_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(a_{c_1}(x_1)) = [[0], [1]] + [[2, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(c_{a_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(a_{b_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(a_{a_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(c_{c_1}(x_1)) = [[0], [0]] + [[1, 2], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 31.02/8.69 Rules from R: 31.02/8.69 none 31.02/8.69 Rules from S: 31.02/8.69 31.02/8.69 b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 31.02/8.69 a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) 31.02/8.69 31.02/8.69 31.02/8.69 31.02/8.69 31.02/8.69 ---------------------------------------- 31.02/8.69 31.02/8.69 (10) 31.02/8.69 Obligation: 31.02/8.69 Relative term rewrite system: 31.02/8.69 The relative TRS consists of the following R rules: 31.02/8.69 31.02/8.69 b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 31.02/8.69 The relative TRS consists of the following S rules: 31.02/8.69 31.02/8.69 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1))) 31.02/8.69 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1))) 31.02/8.69 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 31.02/8.69 c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 31.02/8.69 c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 31.02/8.69 31.02/8.69 31.02/8.69 ---------------------------------------- 31.02/8.69 31.02/8.69 (11) RelTRSRRRProof (EQUIVALENT) 31.02/8.69 We used the following monotonic ordering for rule removal: 31.02/8.69 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(b_{b_1}(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(b_{c_1}(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(c_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(b_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(a_{c_1}(x_1)) = [[0], [0]] + [[1, 0], [2, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(c_{a_1}(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(a_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(a_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 <<< 31.02/8.69 POL(c_{c_1}(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 31.02/8.69 >>> 31.02/8.69 31.02/8.69 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 31.02/8.69 Rules from R: 31.02/8.69 none 31.02/8.69 Rules from S: 31.02/8.69 31.02/8.69 c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 31.02/8.69 31.02/8.69 31.02/8.69 31.02/8.69 31.02/8.69 ---------------------------------------- 31.02/8.69 31.02/8.69 (12) 31.02/8.69 Obligation: 31.02/8.69 Relative term rewrite system: 31.02/8.69 The relative TRS consists of the following R rules: 31.02/8.69 31.02/8.69 b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 31.02/8.69 The relative TRS consists of the following S rules: 31.02/8.69 31.02/8.69 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1))) 31.02/8.69 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1))) 31.02/8.69 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 31.02/8.69 c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 31.02/8.69 31.02/8.69 31.02/8.69 ---------------------------------------- 31.02/8.69 31.02/8.69 (13) RelTRSRRRProof (EQUIVALENT) 31.02/8.69 We used the following monotonic ordering for rule removal: 31.02/8.69 Knuth-Bendix order [KBO] with precedence:b_{b_1}_1 > b_{c_1}_1 > b_{a_1}_1 > a_{b_1}_1 > a_{c_1}_1 > a_{a_1}_1 > c_{a_1}_1 > c_{c_1}_1 > c_{b_1}_1 31.02/8.69 31.02/8.69 and weight map: 31.02/8.69 31.02/8.69 b_{b_1}_1=7 31.02/8.69 b_{c_1}_1=4 31.02/8.69 c_{b_1}_1=12 31.02/8.69 b_{a_1}_1=3 31.02/8.69 a_{c_1}_1=8 31.02/8.69 c_{a_1}_1=6 31.02/8.69 a_{b_1}_1=11 31.02/8.69 a_{a_1}_1=7 31.02/8.69 c_{c_1}_1=1 31.02/8.69 31.02/8.69 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 31.02/8.69 Rules from R: 31.02/8.69 31.02/8.69 b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 31.02/8.69 a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 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)))) 31.02/8.69 Rules from S: 31.02/8.69 31.02/8.69 b_{c_1}(c_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(x1))) 31.02/8.69 b_{c_1}(c_{b_1}(b_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{c_1}(x1))) 31.02/8.69 b_{c_1}(c_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 31.02/8.69 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 31.02/8.69 c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 31.02/8.69 31.02/8.69 31.02/8.69 31.02/8.69 31.02/8.69 ---------------------------------------- 31.02/8.69 31.02/8.69 (14) 31.02/8.69 Obligation: 31.02/8.69 Relative term rewrite system: 31.02/8.69 R is empty. 31.02/8.69 S is empty. 31.02/8.69 31.02/8.69 ---------------------------------------- 31.02/8.69 31.02/8.69 (15) RIsEmptyProof (EQUIVALENT) 31.02/8.69 The TRS R is empty. Hence, termination is trivially proven. 31.02/8.69 ---------------------------------------- 31.02/8.69 31.02/8.69 (16) 31.02/8.69 YES 31.08/8.77 EOF