/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given RelTRS could be proven: (0) RelTRS (1) RelTRS Reverse [EQUIVALENT, 0 ms] (2) RelTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4) RelTRS (5) RelTRSRRRProof [EQUIVALENT, 1136 ms] (6) RelTRS (7) RelTRSRRRProof [EQUIVALENT, 301 ms] (8) RelTRS (9) RIsEmptyProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b(b(a(a(b(a(b(x1))))))) -> b(a(b(b(a(b(b(x1))))))) b(b(b(x1))) -> b(b(a(b(b(x1))))) The relative TRS consists of the following S rules: b(a(b(x1))) -> b(a(b(a(a(b(x1)))))) ---------------------------------------- (1) RelTRS Reverse (EQUIVALENT) We have reversed the following relative TRS [REVERSE]: The set of rules R is b(b(a(a(b(a(b(x1))))))) -> b(a(b(b(a(b(b(x1))))))) b(b(b(x1))) -> b(b(a(b(b(x1))))) The set of rules S is b(a(b(x1))) -> b(a(b(a(a(b(x1)))))) We have obtained the following relative TRS: The set of rules R is b(a(b(a(a(b(b(x1))))))) -> b(b(a(b(b(a(b(x1))))))) b(b(b(x1))) -> b(b(a(b(b(x1))))) The set of rules S is b(a(b(x1))) -> b(a(a(b(a(b(x1)))))) ---------------------------------------- (2) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b(a(b(a(a(b(b(x1))))))) -> b(b(a(b(b(a(b(x1))))))) b(b(b(x1))) -> b(b(a(b(b(x1))))) The relative TRS consists of the following S rules: b(a(b(x1))) -> b(a(a(b(a(b(x1)))))) ---------------------------------------- (3) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled ---------------------------------------- (4) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) The relative TRS consists of the following S rules: b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) ---------------------------------------- (5) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(b_{a_1}(x_1)) = [[0], [0]] + [[1, 2], [1, 2]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))) b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))))) Rules from S: none ---------------------------------------- (6) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) The relative TRS consists of the following S rules: b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) ---------------------------------------- (7) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(b_{b_1}(x_1)) = [[2], [2]] + [[2, 1], [1, 2]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) Rules from S: none ---------------------------------------- (8) Obligation: Relative term rewrite system: R is empty. The relative TRS consists of the following S rules: b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))) b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))) ---------------------------------------- (9) RIsEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (10) YES