YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given RelTRS could be proven: (0) RelTRS (1) RelTRS Reverse [EQUIVALENT, 0 ms] (2) RelTRS (3) RootLabelingProof [EQUIVALENT, 0 ms] (4)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, 973 ms] (6) RelTRS (7) RelTRSRRRProof [EQUIVALENT, 449 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: a(a(b(b(a(a(x1)))))) -> a(a(b(b(b(b(a(a(x1)))))))) b(b(a(a(b(b(x1)))))) -> b(b(a(a(a(a(b(b(x1)))))))) The relative TRS consists of the following S rules: a(a(x1)) -> a(a(a(a(a(a(x1)))))) b(b(x1)) -> b(b(b(b(b(b(x1)))))) ---------------------------------------- (1) RelTRS Reverse (EQUIVALENT) We have reversed the following relative TRS [REVERSE]: The set of rules R is a(a(b(b(a(a(x1)))))) -> a(a(b(b(b(b(a(a(x1)))))))) b(b(a(a(b(b(x1)))))) -> b(b(a(a(a(a(b(b(x1)))))))) The set of rules S is a(a(x1)) -> a(a(a(a(a(a(x1)))))) b(b(x1)) -> b(b(b(b(b(b(x1)))))) We have obtained the following relative TRS: The set of rules R is a(a(b(b(a(a(x1)))))) -> a(a(b(b(b(b(a(a(x1)))))))) b(b(a(a(b(b(x1)))))) -> b(b(a(a(a(a(b(b(x1)))))))) The set of rules S is a(a(x1)) -> a(a(a(a(a(a(x1)))))) b(b(x1)) -> b(b(b(b(b(b(x1)))))) ---------------------------------------- (2) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a(a(b(b(a(a(x1)))))) -> a(a(b(b(b(b(a(a(x1)))))))) b(b(a(a(b(b(x1)))))) -> b(b(a(a(a(a(b(b(x1)))))))) The relative TRS consists of the following S rules: a(a(x1)) -> a(a(a(a(a(a(x1)))))) b(b(x1)) -> b(b(b(b(b(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: a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))) b_{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_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{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_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) The relative TRS consists of the following S rules: a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) b_{b_1}(b_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) ---------------------------------------- (5) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^3, +, *, >=, >) : <<< POL(a_{a_1}(x_1)) = [[0], [1], [1]] + [[1, 0, 0], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0], [0], [0]] + [[1, 0, 1], [0, 0, 0], [0, 0, 0]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 0], [0, 1, 0]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0], [0], [0]] + [[1, 0, 0], [0, 1, 1], [0, 0, 0]] * x_1 >>> With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{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_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) The relative TRS consists of the following S rules: a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) b_{b_1}(b_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) ---------------------------------------- (7) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^3, +, *, >=, >) : <<< POL(b_{b_1}(x_1)) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 1], [0, 0, 0]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0], [0], [1]] + [[1, 0, 1], [0, 0, 0], [0, 0, 1]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0], [0], [0]] + [[1, 0, 0], [0, 0, 0], [0, 1, 0]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0], [1], [0]] + [[1, 0, 0], [0, 1, 0], [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_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))) b_{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_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))) Rules from S: none ---------------------------------------- (8) Obligation: Relative term rewrite system: R is empty. The relative TRS consists of the following S rules: a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))) a_{a_1}(a_{b_1}(x1)) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))) b_{b_1}(b_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))) b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))) ---------------------------------------- (9) RIsEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (10) YES