37.64/10.31 YES 37.64/10.32 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 37.64/10.32 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 37.64/10.32 37.64/10.32 37.64/10.32 Termination of the given RelTRS could be proven: 37.64/10.32 37.64/10.32 (0) RelTRS 37.64/10.32 (1) FlatCCProof [EQUIVALENT, 0 ms] 37.64/10.32 (2) RelTRS 37.64/10.32 (3) RootLabelingProof [EQUIVALENT, 15 ms] 37.64/10.32 (4) RelTRS 37.64/10.32 (5) RelTRSRRRProof [EQUIVALENT, 2193 ms] 37.64/10.32 (6) RelTRS 37.64/10.32 (7) RelTRSRRRProof [EQUIVALENT, 5 ms] 37.64/10.32 (8) RelTRS 37.64/10.32 (9) RIsEmptyProof [EQUIVALENT, 0 ms] 37.64/10.32 (10) YES 37.64/10.32 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (0) 37.64/10.32 Obligation: 37.64/10.32 Relative term rewrite system: 37.64/10.32 The relative TRS consists of the following R rules: 37.64/10.32 37.64/10.32 a(b(c(x1))) -> c(b(c(x1))) 37.64/10.32 c(b(c(x1))) -> a(c(b(x1))) 37.64/10.32 b(c(c(x1))) -> b(b(b(x1))) 37.64/10.32 b(b(a(x1))) -> b(c(a(x1))) 37.64/10.32 37.64/10.32 The relative TRS consists of the following S rules: 37.64/10.32 37.64/10.32 a(c(a(x1))) -> a(b(a(x1))) 37.64/10.32 a(b(b(x1))) -> c(c(c(x1))) 37.64/10.32 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (1) FlatCCProof (EQUIVALENT) 37.64/10.32 We used flat context closure [ROOTLAB] 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (2) 37.64/10.32 Obligation: 37.64/10.32 Relative term rewrite system: 37.64/10.32 The relative TRS consists of the following R rules: 37.64/10.32 37.64/10.32 b(c(c(x1))) -> b(b(b(x1))) 37.64/10.32 b(b(a(x1))) -> b(c(a(x1))) 37.64/10.32 a(a(b(c(x1)))) -> a(c(b(c(x1)))) 37.64/10.32 b(a(b(c(x1)))) -> b(c(b(c(x1)))) 37.64/10.32 c(a(b(c(x1)))) -> c(c(b(c(x1)))) 37.64/10.32 a(c(b(c(x1)))) -> a(a(c(b(x1)))) 37.64/10.32 b(c(b(c(x1)))) -> b(a(c(b(x1)))) 37.64/10.32 c(c(b(c(x1)))) -> c(a(c(b(x1)))) 37.64/10.32 37.64/10.32 The relative TRS consists of the following S rules: 37.64/10.32 37.64/10.32 a(c(a(x1))) -> a(b(a(x1))) 37.64/10.32 a(a(b(b(x1)))) -> a(c(c(c(x1)))) 37.64/10.32 b(a(b(b(x1)))) -> b(c(c(c(x1)))) 37.64/10.32 c(a(b(b(x1)))) -> c(c(c(c(x1)))) 37.64/10.32 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (3) RootLabelingProof (EQUIVALENT) 37.64/10.32 We used plain root labeling [ROOTLAB] with the following heuristic: 37.64/10.32 LabelAll: All function symbols get labeled 37.64/10.32 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (4) 37.64/10.32 Obligation: 37.64/10.32 Relative term rewrite system: 37.64/10.32 The relative TRS consists of the following R rules: 37.64/10.32 37.64/10.32 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) 37.64/10.32 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1))) 37.64/10.32 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) 37.64/10.32 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 37.64/10.32 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 37.64/10.32 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 37.64/10.32 a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) 37.64/10.32 a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) 37.64/10.32 a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) 37.64/10.32 b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) 37.64/10.32 b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) 37.64/10.32 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) 37.64/10.32 c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) 37.64/10.32 c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) 37.64/10.32 c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) 37.64/10.32 37.64/10.32 The relative TRS consists of the following S rules: 37.64/10.32 37.64/10.32 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 37.64/10.32 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1))) 37.64/10.32 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 37.64/10.32 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (5) RelTRSRRRProof (EQUIVALENT) 37.64/10.32 We used the following monotonic ordering for rule removal: 37.64/10.32 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 37.64/10.32 37.64/10.32 <<< 37.64/10.32 POL(b_{c_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 37.64/10.32 >>> 37.64/10.32 37.64/10.32 <<< 37.64/10.32 POL(c_{c_1}(x_1)) = [[0], [1]] + [[2, 0], [1, 0]] * x_1 37.64/10.32 >>> 37.64/10.32 37.64/10.32 <<< 37.64/10.32 POL(c_{b_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 37.64/10.32 >>> 37.64/10.32 37.64/10.32 <<< 37.64/10.32 POL(b_{b_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 37.64/10.32 >>> 37.64/10.32 37.64/10.32 <<< 37.64/10.32 POL(c_{a_1}(x_1)) = [[0], [1]] + [[2, 0], [2, 2]] * x_1 37.64/10.32 >>> 37.64/10.32 37.64/10.32 <<< 37.64/10.32 POL(b_{a_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 0]] * x_1 37.64/10.32 >>> 37.64/10.32 37.64/10.32 <<< 37.64/10.32 POL(a_{b_1}(x_1)) = [[1], [1]] + [[2, 0], [2, 0]] * x_1 37.64/10.32 >>> 37.64/10.32 37.64/10.32 <<< 37.64/10.32 POL(a_{c_1}(x_1)) = [[0], [0]] + [[1, 2], [0, 2]] * x_1 37.64/10.32 >>> 37.64/10.32 37.64/10.32 <<< 37.64/10.32 POL(a_{a_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 2]] * x_1 37.64/10.32 >>> 37.64/10.32 37.64/10.32 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 37.64/10.32 Rules from R: 37.64/10.32 37.64/10.32 a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) 37.64/10.32 Rules from S: 37.64/10.32 37.64/10.32 a_{c_1}(c_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1))) 37.64/10.32 a_{c_1}(c_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1))) 37.64/10.32 a_{c_1}(c_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 37.64/10.32 b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 37.64/10.32 c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 37.64/10.32 37.64/10.32 37.64/10.32 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (6) 37.64/10.32 Obligation: 37.64/10.32 Relative term rewrite system: 37.64/10.32 The relative TRS consists of the following R rules: 37.64/10.32 37.64/10.32 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) 37.64/10.32 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1))) 37.64/10.32 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) 37.64/10.32 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 37.64/10.32 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 37.64/10.32 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 37.64/10.32 a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) 37.64/10.32 a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) 37.64/10.32 a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) 37.64/10.32 b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) 37.64/10.32 b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) 37.64/10.32 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) 37.64/10.32 c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) 37.64/10.32 c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) 37.64/10.32 c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) 37.64/10.32 37.64/10.32 The relative TRS consists of the following S rules: 37.64/10.32 37.64/10.32 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 37.64/10.32 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (7) RelTRSRRRProof (EQUIVALENT) 37.64/10.32 We used the following monotonic ordering for rule removal: 37.64/10.32 Knuth-Bendix order [KBO] with precedence:c_{a_1}_1 > a_{b_1}_1 > b_{a_1}_1 > b_{b_1}_1 > a_{a_1}_1 > c_{c_1}_1 > c_{b_1}_1 > a_{c_1}_1 > b_{c_1}_1 37.64/10.32 37.64/10.32 and weight map: 37.64/10.32 37.64/10.32 b_{c_1}_1=6 37.64/10.32 c_{c_1}_1=8 37.64/10.32 c_{b_1}_1=6 37.64/10.32 b_{b_1}_1=5 37.64/10.32 c_{a_1}_1=1 37.64/10.32 b_{a_1}_1=4 37.64/10.32 a_{b_1}_1=16 37.64/10.32 a_{c_1}_1=4 37.64/10.32 a_{a_1}_1=1 37.64/10.32 37.64/10.32 The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 37.64/10.32 Rules from R: 37.64/10.32 37.64/10.32 b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) 37.64/10.32 b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{b_1}(b_{c_1}(x1))) 37.64/10.32 b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) 37.64/10.32 b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1))) 37.64/10.32 b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1))) 37.64/10.32 b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1))) 37.64/10.32 a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) 37.64/10.32 a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) 37.64/10.32 a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) 37.64/10.32 b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) 37.64/10.32 b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) 37.64/10.32 b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) 37.64/10.32 c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) 37.64/10.32 c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) 37.64/10.32 c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) 37.64/10.32 Rules from S: 37.64/10.32 37.64/10.32 a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) 37.64/10.32 a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) 37.64/10.32 37.64/10.32 37.64/10.32 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (8) 37.64/10.32 Obligation: 37.64/10.32 Relative term rewrite system: 37.64/10.32 R is empty. 37.64/10.32 S is empty. 37.64/10.32 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (9) RIsEmptyProof (EQUIVALENT) 37.64/10.32 The TRS R is empty. Hence, termination is trivially proven. 37.64/10.32 ---------------------------------------- 37.64/10.32 37.64/10.32 (10) 37.64/10.32 YES 37.87/11.42 EOF