53.46/14.37 YES 53.66/14.38 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 53.66/14.38 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 53.66/14.38 53.66/14.38 53.66/14.38 Termination of the given RelTRS could be proven: 53.66/14.38 53.66/14.38 (0) RelTRS 53.66/14.38 (1) RelTRS Reverse [EQUIVALENT, 0 ms] 53.66/14.38 (2) RelTRS 53.66/14.38 (3) FlatCCProof [EQUIVALENT, 0 ms] 53.66/14.38 (4) RelTRS 53.66/14.38 (5) RootLabelingProof [EQUIVALENT, 19 ms] 53.66/14.38 (6) RelTRS 53.66/14.38 (7) RelTRSRRRProof [EQUIVALENT, 3531 ms] 53.66/14.38 (8) RelTRS 53.66/14.38 (9) RelTRSRRRProof [EQUIVALENT, 0 ms] 53.66/14.38 (10) RelTRS 53.66/14.38 (11) RelTRSRRRProof [EQUIVALENT, 0 ms] 53.66/14.38 (12) RelTRS 53.66/14.38 (13) RelTRSRRRProof [EQUIVALENT, 0 ms] 53.66/14.38 (14) RelTRS 53.66/14.38 (15) RelTRSRRRProof [EQUIVALENT, 183 ms] 53.66/14.38 (16) RelTRS 53.66/14.38 (17) RelTRSRRRProof [EQUIVALENT, 3 ms] 53.66/14.38 (18) RelTRS 53.66/14.38 (19) RIsEmptyProof [EQUIVALENT, 1 ms] 53.66/14.38 (20) YES 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (0) 53.66/14.38 Obligation: 53.66/14.38 Relative term rewrite system: 53.66/14.38 The relative TRS consists of the following R rules: 53.66/14.38 53.66/14.38 b(b(c(x1))) -> b(c(c(x1))) 53.66/14.38 a(a(a(x1))) -> c(b(a(x1))) 53.66/14.38 b(b(c(x1))) -> b(a(a(x1))) 53.66/14.38 c(b(c(x1))) -> a(a(a(x1))) 53.66/14.38 53.66/14.38 The relative TRS consists of the following S rules: 53.66/14.38 53.66/14.38 a(b(b(x1))) -> a(c(a(x1))) 53.66/14.38 a(c(b(x1))) -> b(a(c(x1))) 53.66/14.38 b(a(a(x1))) -> b(a(c(x1))) 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (1) RelTRS Reverse (EQUIVALENT) 53.66/14.38 We have reversed the following relative TRS [REVERSE]: 53.66/14.38 The set of rules R is 53.66/14.38 b(b(c(x1))) -> b(c(c(x1))) 53.66/14.38 a(a(a(x1))) -> c(b(a(x1))) 53.66/14.38 b(b(c(x1))) -> b(a(a(x1))) 53.66/14.38 c(b(c(x1))) -> a(a(a(x1))) 53.66/14.38 53.66/14.38 The set of rules S is 53.66/14.38 a(b(b(x1))) -> a(c(a(x1))) 53.66/14.38 a(c(b(x1))) -> b(a(c(x1))) 53.66/14.38 b(a(a(x1))) -> b(a(c(x1))) 53.66/14.38 53.66/14.38 We have obtained the following relative TRS: 53.66/14.38 The set of rules R is 53.66/14.38 c(b(b(x1))) -> c(c(b(x1))) 53.66/14.38 a(a(a(x1))) -> a(b(c(x1))) 53.66/14.38 c(b(b(x1))) -> a(a(b(x1))) 53.66/14.38 c(b(c(x1))) -> a(a(a(x1))) 53.66/14.38 53.66/14.38 The set of rules S is 53.66/14.38 b(b(a(x1))) -> a(c(a(x1))) 53.66/14.38 b(c(a(x1))) -> c(a(b(x1))) 53.66/14.38 a(a(b(x1))) -> c(a(b(x1))) 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (2) 53.66/14.38 Obligation: 53.66/14.38 Relative term rewrite system: 53.66/14.38 The relative TRS consists of the following R rules: 53.66/14.38 53.66/14.38 c(b(b(x1))) -> c(c(b(x1))) 53.66/14.38 a(a(a(x1))) -> a(b(c(x1))) 53.66/14.38 c(b(b(x1))) -> a(a(b(x1))) 53.66/14.38 c(b(c(x1))) -> a(a(a(x1))) 53.66/14.38 53.66/14.38 The relative TRS consists of the following S rules: 53.66/14.38 53.66/14.38 b(b(a(x1))) -> a(c(a(x1))) 53.66/14.38 b(c(a(x1))) -> c(a(b(x1))) 53.66/14.38 a(a(b(x1))) -> c(a(b(x1))) 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (3) FlatCCProof (EQUIVALENT) 53.66/14.38 We used flat context closure [ROOTLAB] 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (4) 53.66/14.38 Obligation: 53.66/14.38 Relative term rewrite system: 53.66/14.38 The relative TRS consists of the following R rules: 53.66/14.38 53.66/14.38 c(b(b(x1))) -> c(c(b(x1))) 53.66/14.38 a(a(a(x1))) -> a(b(c(x1))) 53.66/14.38 c(c(b(b(x1)))) -> c(a(a(b(x1)))) 53.66/14.38 b(c(b(b(x1)))) -> b(a(a(b(x1)))) 53.66/14.38 a(c(b(b(x1)))) -> a(a(a(b(x1)))) 53.66/14.38 c(c(b(c(x1)))) -> c(a(a(a(x1)))) 53.66/14.38 b(c(b(c(x1)))) -> b(a(a(a(x1)))) 53.66/14.38 a(c(b(c(x1)))) -> a(a(a(a(x1)))) 53.66/14.38 53.66/14.38 The relative TRS consists of the following S rules: 53.66/14.38 53.66/14.38 c(b(b(a(x1)))) -> c(a(c(a(x1)))) 53.66/14.38 b(b(b(a(x1)))) -> b(a(c(a(x1)))) 53.66/14.38 a(b(b(a(x1)))) -> a(a(c(a(x1)))) 53.66/14.38 c(b(c(a(x1)))) -> c(c(a(b(x1)))) 53.66/14.38 b(b(c(a(x1)))) -> b(c(a(b(x1)))) 53.66/14.38 a(b(c(a(x1)))) -> a(c(a(b(x1)))) 53.66/14.38 c(a(a(b(x1)))) -> c(c(a(b(x1)))) 53.66/14.38 b(a(a(b(x1)))) -> b(c(a(b(x1)))) 53.66/14.38 a(a(a(b(x1)))) -> a(c(a(b(x1)))) 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (5) RootLabelingProof (EQUIVALENT) 53.66/14.38 We used plain root labeling [ROOTLAB] with the following heuristic: 53.66/14.38 LabelAll: All function symbols get labeled 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (6) 53.66/14.38 Obligation: 53.66/14.38 Relative term rewrite system: 53.66/14.38 The relative TRS consists of the following R rules: 53.66/14.38 53.66/14.38 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 53.66/14.38 The relative TRS consists of the following S rules: 53.66/14.38 53.66/14.38 c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.66/14.38 b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.66/14.38 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.66/14.38 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.66/14.38 a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.66/14.38 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.66/14.38 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.66/14.38 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (7) RelTRSRRRProof (EQUIVALENT) 53.66/14.38 We used the following monotonic ordering for rule removal: 53.66/14.38 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 53.66/14.38 53.66/14.38 <<< 53.66/14.38 POL(c_{b_1}(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 53.66/14.38 >>> 53.66/14.38 53.66/14.38 <<< 53.66/14.38 POL(b_{b_1}(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 53.66/14.38 >>> 53.66/14.38 53.66/14.38 <<< 53.66/14.38 POL(b_{c_1}(x_1)) = [[2], [0]] + [[1, 0], [0, 0]] * x_1 53.66/14.38 >>> 53.66/14.38 53.66/14.38 <<< 53.66/14.38 POL(c_{c_1}(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 53.66/14.38 >>> 53.66/14.38 53.66/14.38 <<< 53.66/14.38 POL(b_{a_1}(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 53.66/14.38 >>> 53.66/14.38 53.66/14.38 <<< 53.66/14.38 POL(a_{a_1}(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 53.66/14.38 >>> 53.66/14.38 53.66/14.38 <<< 53.66/14.38 POL(a_{c_1}(x_1)) = [[1], [0]] + [[1, 1], [0, 0]] * x_1 53.66/14.38 >>> 53.66/14.38 53.66/14.38 <<< 53.66/14.38 POL(a_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 53.66/14.38 >>> 53.66/14.38 53.66/14.38 <<< 53.66/14.38 POL(c_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 53.66/14.38 >>> 53.66/14.38 53.66/14.38 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 53.66/14.38 Rules from R: 53.66/14.38 53.66/14.38 a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) 53.66/14.38 Rules from S: 53.66/14.38 53.66/14.38 c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.66/14.38 b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.66/14.38 b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.66/14.38 b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.66/14.38 c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 53.66/14.38 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (8) 53.66/14.38 Obligation: 53.66/14.38 Relative term rewrite system: 53.66/14.38 The relative TRS consists of the following R rules: 53.66/14.38 53.66/14.38 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) 53.66/14.38 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 53.66/14.38 The relative TRS consists of the following S rules: 53.66/14.38 53.66/14.38 a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.66/14.38 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.66/14.38 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.66/14.38 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (9) RelTRSRRRProof (EQUIVALENT) 53.66/14.38 We used the following monotonic ordering for rule removal: 53.66/14.38 Polynomial interpretation [POLO]: 53.66/14.38 53.66/14.38 POL(a_{a_1}(x_1)) = x_1 53.66/14.38 POL(a_{b_1}(x_1)) = 1 + x_1 53.66/14.38 POL(a_{c_1}(x_1)) = x_1 53.66/14.38 POL(b_{a_1}(x_1)) = x_1 53.66/14.38 POL(b_{b_1}(x_1)) = 1 + x_1 53.66/14.38 POL(b_{c_1}(x_1)) = x_1 53.66/14.38 POL(c_{a_1}(x_1)) = x_1 53.66/14.38 POL(c_{b_1}(x_1)) = x_1 53.66/14.38 POL(c_{c_1}(x_1)) = x_1 53.66/14.38 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 53.66/14.38 Rules from R: 53.66/14.38 53.66/14.38 c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1))) 53.66/14.38 c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1))) 53.66/14.38 Rules from S: 53.66/14.38 53.66/14.38 a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) 53.66/14.38 a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) 53.66/14.38 a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) 53.66/14.38 53.66/14.38 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (10) 53.66/14.38 Obligation: 53.66/14.38 Relative term rewrite system: 53.66/14.38 The relative TRS consists of the following R rules: 53.66/14.38 53.66/14.38 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 53.66/14.38 The relative TRS consists of the following S rules: 53.66/14.38 53.66/14.38 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.38 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.38 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (11) RelTRSRRRProof (EQUIVALENT) 53.66/14.38 We used the following monotonic ordering for rule removal: 53.66/14.38 Polynomial interpretation [POLO]: 53.66/14.38 53.66/14.38 POL(a_{a_1}(x_1)) = x_1 53.66/14.38 POL(a_{b_1}(x_1)) = x_1 53.66/14.38 POL(a_{c_1}(x_1)) = x_1 53.66/14.38 POL(b_{a_1}(x_1)) = x_1 53.66/14.38 POL(b_{b_1}(x_1)) = x_1 53.66/14.38 POL(b_{c_1}(x_1)) = x_1 53.66/14.38 POL(c_{a_1}(x_1)) = x_1 53.66/14.38 POL(c_{b_1}(x_1)) = x_1 53.66/14.38 POL(c_{c_1}(x_1)) = 1 + x_1 53.66/14.38 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 53.66/14.38 Rules from R: 53.66/14.38 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.38 b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) 53.66/14.38 Rules from S: 53.66/14.38 none 53.66/14.38 53.66/14.38 53.66/14.38 53.66/14.38 53.66/14.38 ---------------------------------------- 53.66/14.38 53.66/14.38 (12) 53.66/14.38 Obligation: 53.66/14.39 Relative term rewrite system: 53.66/14.39 The relative TRS consists of the following R rules: 53.66/14.39 53.66/14.39 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 53.66/14.39 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.39 a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.39 53.66/14.39 The relative TRS consists of the following S rules: 53.66/14.39 53.66/14.39 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.39 53.66/14.39 53.66/14.39 ---------------------------------------- 53.66/14.39 53.66/14.39 (13) RelTRSRRRProof (EQUIVALENT) 53.66/14.39 We used the following monotonic ordering for rule removal: 53.66/14.39 Polynomial interpretation [POLO]: 53.66/14.39 53.66/14.39 POL(a_{a_1}(x_1)) = x_1 53.66/14.39 POL(a_{b_1}(x_1)) = 1 + x_1 53.66/14.39 POL(a_{c_1}(x_1)) = 1 + x_1 53.66/14.39 POL(b_{a_1}(x_1)) = x_1 53.66/14.39 POL(b_{b_1}(x_1)) = x_1 53.66/14.39 POL(b_{c_1}(x_1)) = x_1 53.66/14.39 POL(c_{a_1}(x_1)) = x_1 53.66/14.39 POL(c_{b_1}(x_1)) = x_1 53.66/14.39 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 53.66/14.39 Rules from R: 53.66/14.39 53.66/14.39 a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.39 Rules from S: 53.66/14.39 none 53.66/14.39 53.66/14.39 53.66/14.39 53.66/14.39 53.66/14.39 ---------------------------------------- 53.66/14.39 53.66/14.39 (14) 53.66/14.39 Obligation: 53.66/14.39 Relative term rewrite system: 53.66/14.39 The relative TRS consists of the following R rules: 53.66/14.39 53.66/14.39 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 53.66/14.39 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.39 53.66/14.39 The relative TRS consists of the following S rules: 53.66/14.39 53.66/14.39 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.39 53.66/14.39 53.66/14.39 ---------------------------------------- 53.66/14.39 53.66/14.39 (15) RelTRSRRRProof (EQUIVALENT) 53.66/14.39 We used the following monotonic ordering for rule removal: 53.66/14.39 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 53.66/14.39 53.66/14.39 <<< 53.66/14.39 POL(a_{a_1}(x_1)) = [[0], [2]] + [[2, 0], [0, 0]] * x_1 53.66/14.39 >>> 53.66/14.39 53.66/14.39 <<< 53.66/14.39 POL(a_{b_1}(x_1)) = [[1], [2]] + [[2, 0], [0, 0]] * x_1 53.66/14.39 >>> 53.66/14.39 53.66/14.39 <<< 53.66/14.39 POL(b_{c_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 53.66/14.39 >>> 53.66/14.39 53.66/14.39 <<< 53.66/14.39 POL(c_{b_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 53.66/14.39 >>> 53.66/14.39 53.66/14.39 <<< 53.66/14.39 POL(c_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 53.66/14.39 >>> 53.66/14.39 53.66/14.39 <<< 53.66/14.39 POL(b_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 53.66/14.39 >>> 53.66/14.39 53.66/14.39 <<< 53.66/14.39 POL(b_{b_1}(x_1)) = [[0], [2]] + [[2, 0], [0, 0]] * x_1 53.66/14.39 >>> 53.66/14.39 53.66/14.39 <<< 53.66/14.39 POL(a_{c_1}(x_1)) = [[2], [0]] + [[2, 0], [0, 0]] * x_1 53.66/14.39 >>> 53.66/14.39 53.66/14.39 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 53.66/14.39 Rules from R: 53.66/14.39 53.66/14.39 a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1))) 53.66/14.39 Rules from S: 53.66/14.39 53.66/14.39 b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 53.66/14.39 53.66/14.39 53.66/14.39 53.66/14.39 ---------------------------------------- 53.66/14.39 53.66/14.39 (16) 53.66/14.39 Obligation: 53.66/14.39 Relative term rewrite system: 53.66/14.39 The relative TRS consists of the following R rules: 53.66/14.39 53.66/14.39 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.39 53.66/14.39 The relative TRS consists of the following S rules: 53.66/14.39 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.39 53.66/14.39 53.66/14.39 ---------------------------------------- 53.66/14.39 53.66/14.39 (17) RelTRSRRRProof (EQUIVALENT) 53.66/14.39 We used the following monotonic ordering for rule removal: 53.66/14.39 Knuth-Bendix order [KBO] with precedence:a_{b_1}_1 > b_{a_1}_1 > c_{a_1}_1 > b_{c_1}_1 > a_{a_1}_1 > c_{b_1}_1 > b_{b_1}_1 53.66/14.39 53.66/14.39 and weight map: 53.66/14.39 53.66/14.39 b_{c_1}_1=2 53.66/14.39 c_{b_1}_1=5 53.66/14.39 c_{a_1}_1=1 53.66/14.39 b_{a_1}_1=1 53.66/14.39 a_{a_1}_1=2 53.66/14.39 a_{b_1}_1=1 53.66/14.39 b_{b_1}_1=1 53.66/14.39 53.66/14.39 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 53.66/14.39 Rules from R: 53.66/14.39 53.66/14.39 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) 53.66/14.39 Rules from S: 53.66/14.39 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) 53.66/14.39 b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) 53.66/14.39 53.66/14.39 53.66/14.39 53.66/14.39 53.66/14.39 ---------------------------------------- 53.66/14.39 53.66/14.39 (18) 53.66/14.39 Obligation: 53.66/14.39 Relative term rewrite system: 53.66/14.39 R is empty. 53.66/14.39 S is empty. 53.66/14.39 53.66/14.39 ---------------------------------------- 53.66/14.39 53.66/14.39 (19) RIsEmptyProof (EQUIVALENT) 53.66/14.39 The TRS R is empty. Hence, termination is trivially proven. 53.66/14.39 ---------------------------------------- 53.66/14.39 53.66/14.39 (20) 53.66/14.39 YES 53.74/14.45 EOF