12.29/4.02 YES 12.29/4.04 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 12.29/4.04 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 12.29/4.04 12.29/4.04 12.29/4.04 Termination of the given RelTRS could be proven: 12.29/4.04 12.29/4.04 (0) RelTRS 12.29/4.04 (1) RootLabelingProof [EQUIVALENT, 0 ms] 12.29/4.04 (2) RelTRS 12.29/4.04 (3) RelTRSRRRProof [EQUIVALENT, 133 ms] 12.29/4.04 (4) RelTRS 12.29/4.04 (5) RelTRSRRRProof [EQUIVALENT, 101 ms] 12.29/4.04 (6) RelTRS 12.29/4.04 (7) RelTRSRRRProof [EQUIVALENT, 10 ms] 12.29/4.04 (8) RelTRS 12.29/4.04 (9) RelTRSRRRProof [EQUIVALENT, 0 ms] 12.29/4.04 (10) RelTRS 12.29/4.04 (11) RIsEmptyProof [EQUIVALENT, 0 ms] 12.29/4.04 (12) YES 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (0) 12.29/4.04 Obligation: 12.29/4.04 Relative term rewrite system: 12.29/4.04 The relative TRS consists of the following R rules: 12.29/4.04 12.29/4.04 a(b(a(x1))) -> a(b(b(a(x1)))) 12.29/4.04 b(a(b(x1))) -> b(a(a(b(x1)))) 12.29/4.04 12.29/4.04 The relative TRS consists of the following S rules: 12.29/4.04 12.29/4.04 a(x1) -> a(a(a(x1))) 12.29/4.04 b(x1) -> b(b(b(x1))) 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (1) RootLabelingProof (EQUIVALENT) 12.29/4.04 We used plain root labeling [ROOTLAB] with the following heuristic: 12.29/4.04 LabelAll: All function symbols get labeled 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (2) 12.29/4.04 Obligation: 12.29/4.04 Relative term rewrite system: 12.29/4.04 The relative TRS consists of the following R rules: 12.29/4.04 12.29/4.04 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 12.29/4.04 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 12.29/4.04 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 12.29/4.04 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 12.29/4.04 12.29/4.04 The relative TRS consists of the following S rules: 12.29/4.04 12.29/4.04 a_{a_1}(x1) -> a_{a_1}(a_{a_1}(a_{a_1}(x1))) 12.29/4.04 a_{b_1}(x1) -> a_{a_1}(a_{a_1}(a_{b_1}(x1))) 12.29/4.04 b_{a_1}(x1) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) 12.29/4.04 b_{b_1}(x1) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (3) RelTRSRRRProof (EQUIVALENT) 12.29/4.04 We used the following monotonic ordering for rule removal: 12.29/4.04 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(a_{b_1}(x_1)) = [[0], [1]] + [[1, 2], [2, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(b_{a_1}(x_1)) = [[0], [0]] + [[2, 0], [1, 1]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(a_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(b_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 12.29/4.04 Rules from R: 12.29/4.04 12.29/4.04 a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) 12.29/4.04 Rules from S: 12.29/4.04 none 12.29/4.04 12.29/4.04 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (4) 12.29/4.04 Obligation: 12.29/4.04 Relative term rewrite system: 12.29/4.04 The relative TRS consists of the following R rules: 12.29/4.04 12.29/4.04 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 12.29/4.04 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 12.29/4.04 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 12.29/4.04 12.29/4.04 The relative TRS consists of the following S rules: 12.29/4.04 12.29/4.04 a_{a_1}(x1) -> a_{a_1}(a_{a_1}(a_{a_1}(x1))) 12.29/4.04 a_{b_1}(x1) -> a_{a_1}(a_{a_1}(a_{b_1}(x1))) 12.29/4.04 b_{a_1}(x1) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) 12.29/4.04 b_{b_1}(x1) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (5) RelTRSRRRProof (EQUIVALENT) 12.29/4.04 We used the following monotonic ordering for rule removal: 12.29/4.04 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(a_{b_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 2]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(b_{a_1}(x_1)) = [[0], [1]] + [[2, 1], [0, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(a_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(b_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 12.29/4.04 Rules from R: 12.29/4.04 12.29/4.04 b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) 12.29/4.04 Rules from S: 12.29/4.04 none 12.29/4.04 12.29/4.04 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (6) 12.29/4.04 Obligation: 12.29/4.04 Relative term rewrite system: 12.29/4.04 The relative TRS consists of the following R rules: 12.29/4.04 12.29/4.04 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 12.29/4.04 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 12.29/4.04 12.29/4.04 The relative TRS consists of the following S rules: 12.29/4.04 12.29/4.04 a_{a_1}(x1) -> a_{a_1}(a_{a_1}(a_{a_1}(x1))) 12.29/4.04 a_{b_1}(x1) -> a_{a_1}(a_{a_1}(a_{b_1}(x1))) 12.29/4.04 b_{a_1}(x1) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) 12.29/4.04 b_{b_1}(x1) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (7) RelTRSRRRProof (EQUIVALENT) 12.29/4.04 We used the following monotonic ordering for rule removal: 12.29/4.04 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(a_{b_1}(x_1)) = [[0], [0]] + [[2, 1], [0, 2]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(b_{a_1}(x_1)) = [[0], [2]] + [[2, 0], [0, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(a_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(b_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 12.29/4.04 Rules from R: 12.29/4.04 12.29/4.04 a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) 12.29/4.04 Rules from S: 12.29/4.04 none 12.29/4.04 12.29/4.04 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (8) 12.29/4.04 Obligation: 12.29/4.04 Relative term rewrite system: 12.29/4.04 The relative TRS consists of the following R rules: 12.29/4.04 12.29/4.04 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 12.29/4.04 12.29/4.04 The relative TRS consists of the following S rules: 12.29/4.04 12.29/4.04 a_{a_1}(x1) -> a_{a_1}(a_{a_1}(a_{a_1}(x1))) 12.29/4.04 a_{b_1}(x1) -> a_{a_1}(a_{a_1}(a_{b_1}(x1))) 12.29/4.04 b_{a_1}(x1) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) 12.29/4.04 b_{b_1}(x1) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (9) RelTRSRRRProof (EQUIVALENT) 12.29/4.04 We used the following monotonic ordering for rule removal: 12.29/4.04 Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(b_{a_1}(x_1)) = [[0], [2]] + [[2, 1], [0, 2]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(a_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [1, 1]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(b_{b_1}(x_1)) = [[0], [2]] + [[1, 0], [0, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 <<< 12.29/4.04 POL(a_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 12.29/4.04 >>> 12.29/4.04 12.29/4.04 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: 12.29/4.04 Rules from R: 12.29/4.04 12.29/4.04 b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) 12.29/4.04 Rules from S: 12.29/4.04 none 12.29/4.04 12.29/4.04 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (10) 12.29/4.04 Obligation: 12.29/4.04 Relative term rewrite system: 12.29/4.04 R is empty. 12.29/4.04 The relative TRS consists of the following S rules: 12.29/4.04 12.29/4.04 a_{a_1}(x1) -> a_{a_1}(a_{a_1}(a_{a_1}(x1))) 12.29/4.04 a_{b_1}(x1) -> a_{a_1}(a_{a_1}(a_{b_1}(x1))) 12.29/4.04 b_{a_1}(x1) -> b_{b_1}(b_{b_1}(b_{a_1}(x1))) 12.29/4.04 b_{b_1}(x1) -> b_{b_1}(b_{b_1}(b_{b_1}(x1))) 12.29/4.04 12.29/4.04 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (11) RIsEmptyProof (EQUIVALENT) 12.29/4.04 The TRS R is empty. Hence, termination is trivially proven. 12.29/4.04 ---------------------------------------- 12.29/4.04 12.29/4.04 (12) 12.29/4.04 YES 12.71/4.11 EOF