53.73/14.46 YES 53.73/14.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 53.73/14.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 53.73/14.49 53.73/14.49 53.73/14.49 Termination of the given RelTRS could be proven: 53.73/14.49 53.73/14.49 (0) RelTRS 53.73/14.49 (1) RelTRS S Cleaner [EQUIVALENT, 0 ms] 53.73/14.49 (2) RelTRS 53.73/14.49 (3) RelTRS Reverse [EQUIVALENT, 0 ms] 53.73/14.49 (4) RelTRS 53.73/14.49 (5) FlatCCProof [EQUIVALENT, 0 ms] 53.73/14.49 (6) RelTRS 53.73/14.49 (7) RootLabelingProof [EQUIVALENT, 7 ms] 53.73/14.49 (8) RelTRS 53.73/14.49 (9) RelTRSRRRProof [EQUIVALENT, 6153 ms] 53.73/14.49 (10) RelTRS 53.73/14.49 (11) RelTRSRRRProof [EQUIVALENT, 14 ms] 53.73/14.49 (12) RelTRS 53.73/14.49 (13) RIsEmptyProof [EQUIVALENT, 0 ms] 53.73/14.49 (14) YES 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (0) 53.73/14.49 Obligation: 53.73/14.49 Relative term rewrite system: 53.73/14.49 The relative TRS consists of the following R rules: 53.73/14.49 53.73/14.49 a(a(b(x1))) -> a(c(c(x1))) 53.73/14.49 c(c(b(x1))) -> b(b(c(x1))) 53.73/14.49 c(a(a(x1))) -> c(a(c(x1))) 53.73/14.49 53.73/14.49 The relative TRS consists of the following S rules: 53.73/14.49 53.73/14.49 b(b(c(x1))) -> b(a(c(x1))) 53.73/14.49 a(b(a(x1))) -> a(b(c(x1))) 53.73/14.49 c(a(c(x1))) -> c(b(a(x1))) 53.73/14.49 a(a(b(x1))) -> a(a(b(x1))) 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (1) RelTRS S Cleaner (EQUIVALENT) 53.73/14.49 We have deleted all rules from S that have the shape t -> t: 53.73/14.49 53.73/14.49 a(a(b(x1))) -> a(a(b(x1))) 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (2) 53.73/14.49 Obligation: 53.73/14.49 Relative term rewrite system: 53.73/14.49 The relative TRS consists of the following R rules: 53.73/14.49 53.73/14.49 a(a(b(x1))) -> a(c(c(x1))) 53.73/14.49 c(c(b(x1))) -> b(b(c(x1))) 53.73/14.49 c(a(a(x1))) -> c(a(c(x1))) 53.73/14.49 53.73/14.49 The relative TRS consists of the following S rules: 53.73/14.49 53.73/14.49 b(b(c(x1))) -> b(a(c(x1))) 53.73/14.49 a(b(a(x1))) -> a(b(c(x1))) 53.73/14.49 c(a(c(x1))) -> c(b(a(x1))) 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (3) RelTRS Reverse (EQUIVALENT) 53.73/14.49 We have reversed the following relative TRS [REVERSE]: 53.73/14.49 The set of rules R is 53.73/14.49 a(a(b(x1))) -> a(c(c(x1))) 53.73/14.49 c(c(b(x1))) -> b(b(c(x1))) 53.73/14.49 c(a(a(x1))) -> c(a(c(x1))) 53.73/14.49 53.73/14.49 The set of rules S is 53.73/14.49 b(b(c(x1))) -> b(a(c(x1))) 53.73/14.49 a(b(a(x1))) -> a(b(c(x1))) 53.73/14.49 c(a(c(x1))) -> c(b(a(x1))) 53.73/14.49 53.73/14.49 We have obtained the following relative TRS: 53.73/14.49 The set of rules R is 53.73/14.49 b(a(a(x1))) -> c(c(a(x1))) 53.73/14.49 b(c(c(x1))) -> c(b(b(x1))) 53.73/14.49 a(a(c(x1))) -> c(a(c(x1))) 53.73/14.49 53.73/14.49 The set of rules S is 53.73/14.49 c(b(b(x1))) -> c(a(b(x1))) 53.73/14.49 a(b(a(x1))) -> c(b(a(x1))) 53.73/14.49 c(a(c(x1))) -> a(b(c(x1))) 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (4) 53.73/14.49 Obligation: 53.73/14.49 Relative term rewrite system: 53.73/14.49 The relative TRS consists of the following R rules: 53.73/14.49 53.73/14.49 b(a(a(x1))) -> c(c(a(x1))) 53.73/14.49 b(c(c(x1))) -> c(b(b(x1))) 53.73/14.49 a(a(c(x1))) -> c(a(c(x1))) 53.73/14.49 53.73/14.49 The relative TRS consists of the following S rules: 53.73/14.49 53.73/14.49 c(b(b(x1))) -> c(a(b(x1))) 53.73/14.49 a(b(a(x1))) -> c(b(a(x1))) 53.73/14.49 c(a(c(x1))) -> a(b(c(x1))) 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (5) FlatCCProof (EQUIVALENT) 53.73/14.49 We used flat context closure [ROOTLAB] 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (6) 53.73/14.49 Obligation: 53.73/14.49 Relative term rewrite system: 53.73/14.49 The relative TRS consists of the following R rules: 53.73/14.49 53.73/14.49 b(b(a(a(x1)))) -> b(c(c(a(x1)))) 53.73/14.49 a(b(a(a(x1)))) -> a(c(c(a(x1)))) 53.73/14.49 c(b(a(a(x1)))) -> c(c(c(a(x1)))) 53.73/14.49 b(b(c(c(x1)))) -> b(c(b(b(x1)))) 53.73/14.49 a(b(c(c(x1)))) -> a(c(b(b(x1)))) 53.73/14.49 c(b(c(c(x1)))) -> c(c(b(b(x1)))) 53.73/14.49 b(a(a(c(x1)))) -> b(c(a(c(x1)))) 53.73/14.49 a(a(a(c(x1)))) -> a(c(a(c(x1)))) 53.73/14.49 c(a(a(c(x1)))) -> c(c(a(c(x1)))) 53.73/14.49 53.73/14.49 The relative TRS consists of the following S rules: 53.73/14.49 53.73/14.49 c(b(b(x1))) -> c(a(b(x1))) 53.73/14.49 b(a(b(a(x1)))) -> b(c(b(a(x1)))) 53.73/14.49 a(a(b(a(x1)))) -> a(c(b(a(x1)))) 53.73/14.49 c(a(b(a(x1)))) -> c(c(b(a(x1)))) 53.73/14.49 b(c(a(c(x1)))) -> b(a(b(c(x1)))) 53.73/14.49 a(c(a(c(x1)))) -> a(a(b(c(x1)))) 53.73/14.49 c(c(a(c(x1)))) -> c(a(b(c(x1)))) 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (7) RootLabelingProof (EQUIVALENT) 53.73/14.49 We used plain root labeling [ROOTLAB] with the following heuristic: 53.73/14.49 LabelAll: All function symbols get labeled 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (8) 53.73/14.49 Obligation: 53.73/14.49 Relative term rewrite system: 53.73/14.49 The relative TRS consists of the following R rules: 53.73/14.49 53.73/14.49 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.73/14.49 b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.73/14.49 b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.73/14.49 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.73/14.49 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.73/14.49 a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.73/14.49 c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.73/14.49 c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.73/14.49 c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.73/14.49 b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 53.73/14.49 b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 53.73/14.49 b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 53.73/14.49 a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 53.73/14.49 a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 53.73/14.49 a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 53.73/14.49 c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 53.73/14.49 c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 53.73/14.49 c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 53.73/14.49 b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 53.73/14.49 b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 53.73/14.49 b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 53.73/14.49 a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 53.73/14.49 a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 53.73/14.49 a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 53.73/14.49 c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 53.73/14.49 c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 53.73/14.49 c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 53.73/14.49 53.73/14.49 The relative TRS consists of the following S rules: 53.73/14.49 53.73/14.49 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) 53.73/14.49 c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) 53.73/14.49 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) 53.73/14.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 53.73/14.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 53.73/14.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 53.73/14.49 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 53.73/14.49 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 53.73/14.49 a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 53.73/14.49 c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 53.73/14.49 c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 53.73/14.49 c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 53.73/14.49 b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 53.73/14.49 b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 53.73/14.49 b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 53.73/14.49 a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 53.73/14.49 a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 53.73/14.49 a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 53.73/14.49 c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 53.73/14.49 c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 53.73/14.49 c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (9) RelTRSRRRProof (EQUIVALENT) 53.73/14.49 We used the following monotonic ordering for rule removal: 53.73/14.49 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 53.73/14.49 53.73/14.49 <<< 53.73/14.49 POL(b_{b_1}(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 53.73/14.49 >>> 53.73/14.49 53.73/14.49 <<< 53.73/14.49 POL(b_{a_1}(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 53.73/14.49 >>> 53.73/14.49 53.73/14.49 <<< 53.73/14.49 POL(a_{a_1}(x_1)) = [[0], [2]] + [[1, 2], [0, 0]] * x_1 53.73/14.49 >>> 53.73/14.49 53.73/14.49 <<< 53.73/14.49 POL(a_{b_1}(x_1)) = [[0], [2]] + [[1, 0], [0, 0]] * x_1 53.73/14.49 >>> 53.73/14.49 53.73/14.49 <<< 53.73/14.49 POL(b_{c_1}(x_1)) = [[2], [0]] + [[1, 0], [0, 0]] * x_1 53.73/14.49 >>> 53.73/14.49 53.73/14.49 <<< 53.73/14.49 POL(c_{c_1}(x_1)) = [[2], [0]] + [[1, 0], [0, 0]] * x_1 53.73/14.49 >>> 53.73/14.49 53.73/14.49 <<< 53.73/14.49 POL(c_{a_1}(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 53.73/14.49 >>> 53.73/14.49 53.73/14.49 <<< 53.73/14.49 POL(a_{c_1}(x_1)) = [[2], [2]] + [[1, 0], [0, 0]] * x_1 53.73/14.49 >>> 53.73/14.49 53.73/14.49 <<< 53.73/14.49 POL(c_{b_1}(x_1)) = [[0], [0]] + [[1, 2], [0, 0]] * x_1 53.73/14.49 >>> 53.73/14.49 53.73/14.49 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 53.73/14.49 Rules from R: 53.73/14.49 53.73/14.49 b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 53.73/14.49 b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 53.73/14.49 b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 53.73/14.49 a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 53.73/14.49 a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 53.73/14.49 a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 53.73/14.49 c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) 53.73/14.49 c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) 53.73/14.49 c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) 53.73/14.49 Rules from S: 53.73/14.49 53.73/14.49 a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 53.73/14.49 a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 53.73/14.49 a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 53.73/14.49 b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 53.73/14.49 b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 53.73/14.49 b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 53.73/14.49 c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 53.73/14.49 c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 53.73/14.49 c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 53.73/14.49 53.73/14.49 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (10) 53.73/14.49 Obligation: 53.73/14.49 Relative term rewrite system: 53.73/14.49 The relative TRS consists of the following R rules: 53.73/14.49 53.73/14.49 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.73/14.49 b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.73/14.49 b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.73/14.49 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.73/14.49 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.73/14.49 a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.73/14.49 c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.73/14.49 c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.73/14.49 c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.73/14.49 b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 53.73/14.49 b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 53.73/14.49 b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 53.73/14.49 a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 53.73/14.49 a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 53.73/14.49 a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 53.73/14.49 c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 53.73/14.49 c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 53.73/14.49 c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 53.73/14.49 53.73/14.49 The relative TRS consists of the following S rules: 53.73/14.49 53.73/14.49 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) 53.73/14.49 c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) 53.73/14.49 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) 53.73/14.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 53.73/14.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 53.73/14.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 53.73/14.49 c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 53.73/14.49 c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 53.73/14.49 c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 53.73/14.49 a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 53.73/14.49 a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 53.73/14.49 a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (11) RelTRSRRRProof (EQUIVALENT) 53.73/14.49 We used the following monotonic ordering for rule removal: 53.73/14.49 Knuth-Bendix order [KBO] with precedence:c_{b_1}_1 > b_{b_1}_1 > a_{b_1}_1 > a_{c_1}_1 > c_{a_1}_1 > a_{a_1}_1 > b_{a_1}_1 > c_{c_1}_1 > b_{c_1}_1 53.73/14.49 53.73/14.49 and weight map: 53.73/14.49 53.73/14.49 b_{b_1}_1=3 53.73/14.49 b_{a_1}_1=4 53.73/14.49 a_{a_1}_1=3 53.73/14.49 a_{b_1}_1=2 53.73/14.49 b_{c_1}_1=5 53.73/14.49 c_{c_1}_1=3 53.73/14.49 c_{a_1}_1=2 53.73/14.49 a_{c_1}_1=4 53.73/14.49 c_{b_1}_1=1 53.73/14.49 53.73/14.49 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 53.73/14.49 Rules from R: 53.73/14.49 53.73/14.49 b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.73/14.49 b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.73/14.49 b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.73/14.49 a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.73/14.49 a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.73/14.49 a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.73/14.49 c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.73/14.49 c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.73/14.49 c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.73/14.49 b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 53.73/14.49 b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 53.73/14.49 b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 53.73/14.49 a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 53.73/14.49 a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 53.73/14.49 a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 53.73/14.49 c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) 53.73/14.49 c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) 53.73/14.49 c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) 53.73/14.49 Rules from S: 53.73/14.49 53.73/14.49 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1))) 53.73/14.49 c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1))) 53.73/14.49 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1))) 53.73/14.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 53.73/14.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 53.73/14.49 b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 53.73/14.49 c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) 53.73/14.49 c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) 53.73/14.49 c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) 53.73/14.49 a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) 53.73/14.49 a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) 53.73/14.49 a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) 53.73/14.49 53.73/14.49 53.73/14.49 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (12) 53.73/14.49 Obligation: 53.73/14.49 Relative term rewrite system: 53.73/14.49 R is empty. 53.73/14.49 S is empty. 53.73/14.49 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (13) RIsEmptyProof (EQUIVALENT) 53.73/14.49 The TRS R is empty. Hence, termination is trivially proven. 53.73/14.49 ---------------------------------------- 53.73/14.49 53.73/14.49 (14) 53.73/14.49 YES 53.98/14.54 EOF