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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given RelTRS could be proven: (0) RelTRS (1) RelTRS Reverse [EQUIVALENT, 0 ms] (2) RelTRS (3) FlatCCProof [EQUIVALENT, 0 ms] (4) RelTRS (5) RootLabelingProof [EQUIVALENT, 0 ms] (6) RelTRS (7) RelTRSRRRProof [EQUIVALENT, 37 ms] (8) RelTRS (9) RelTRSRRRProof [EQUIVALENT, 10 ms] (10) RelTRS (11) RelTRSRRRProof [EQUIVALENT, 123 ms] (12) RelTRS (13) RelTRSRRRProof [EQUIVALENT, 5 ms] (14) RelTRS (15) RIsEmptyProof [EQUIVALENT, 0 ms] (16) YES ---------------------------------------- (0) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b(c(a(x1))) -> a(b(a(b(x1)))) a(a(x1)) -> a(c(b(a(x1)))) The relative TRS consists of the following S rules: b(x1) -> c(b(x1)) ---------------------------------------- (1) RelTRS Reverse (EQUIVALENT) We have reversed the following relative TRS [REVERSE]: The set of rules R is b(c(a(x1))) -> a(b(a(b(x1)))) a(a(x1)) -> a(c(b(a(x1)))) The set of rules S is b(x1) -> c(b(x1)) We have obtained the following relative TRS: The set of rules R is a(c(b(x1))) -> b(a(b(a(x1)))) a(a(x1)) -> a(b(c(a(x1)))) The set of rules S is b(x1) -> b(c(x1)) ---------------------------------------- (2) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a(c(b(x1))) -> b(a(b(a(x1)))) a(a(x1)) -> a(b(c(a(x1)))) The relative TRS consists of the following S rules: b(x1) -> b(c(x1)) ---------------------------------------- (3) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] ---------------------------------------- (4) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a(a(x1)) -> a(b(c(a(x1)))) a(a(c(b(x1)))) -> a(b(a(b(a(x1))))) c(a(c(b(x1)))) -> c(b(a(b(a(x1))))) b(a(c(b(x1)))) -> b(b(a(b(a(x1))))) The relative TRS consists of the following S rules: b(x1) -> b(c(x1)) ---------------------------------------- (5) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled ---------------------------------------- (6) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) The relative TRS consists of the following S rules: b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1)) b_{b_1}(x1) -> b_{c_1}(c_{b_1}(x1)) b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1)) ---------------------------------------- (7) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = x_1 POL(a_{c_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = x_1 POL(b_{c_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = x_1 POL(c_{c_1}(x_1)) = x_1 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))) b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))) Rules from S: none ---------------------------------------- (8) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) The relative TRS consists of the following S rules: b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1)) b_{b_1}(x1) -> b_{c_1}(c_{b_1}(x1)) b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1)) ---------------------------------------- (9) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = x_1 POL(a_{c_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = x_1 POL(b_{c_1}(x_1)) = x_1 POL(c_{a_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = x_1 POL(c_{c_1}(x_1)) = x_1 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) Rules from S: none ---------------------------------------- (10) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) The relative TRS consists of the following S rules: b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1)) b_{b_1}(x1) -> b_{c_1}(c_{b_1}(x1)) b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1)) ---------------------------------------- (11) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(c_{a_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 2]] * x_1 >>> <<< POL(a_{c_1}(x_1)) = [[0], [2]] + [[2, 1], [0, 2]] * x_1 >>> <<< POL(c_{b_1}(x_1)) = [[0], [1]] + [[1, 1], [2, 2]] * x_1 >>> <<< POL(b_{c_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 1]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0], [0]] + [[2, 0], [0, 2]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[2], [2]] + [[2, 2], [2, 2]] * x_1 >>> <<< POL(c_{c_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: c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) Rules from S: b_{b_1}(x1) -> b_{c_1}(c_{b_1}(x1)) ---------------------------------------- (12) Obligation: Relative term rewrite system: The relative TRS consists of the following R rules: b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) The relative TRS consists of the following S rules: b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1)) b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1)) ---------------------------------------- (13) RelTRSRRRProof (EQUIVALENT) We used the following monotonic ordering for rule removal: Polynomial interpretation [POLO]: POL(a_{b_1}(x_1)) = x_1 POL(a_{c_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = 1 + x_1 POL(b_{b_1}(x_1)) = x_1 POL(b_{c_1}(x_1)) = 1 + x_1 POL(c_{a_1}(x_1)) = x_1 POL(c_{b_1}(x_1)) = 1 + x_1 POL(c_{c_1}(x_1)) = x_1 With this ordering the following rules can be removed [MATRO] because they are oriented strictly: Rules from R: b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))) Rules from S: none ---------------------------------------- (14) Obligation: Relative term rewrite system: R is empty. The relative TRS consists of the following S rules: b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1)) b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1)) ---------------------------------------- (15) RIsEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (16) YES