/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) FlatCCProof [EQUIVALENT, 0 ms] (4) QTRS (5) RootLabelingProof [EQUIVALENT, 0 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 36 ms] (8) QTRS (9) DependencyPairsProof [EQUIVALENT, 0 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) UsableRulesProof [EQUIVALENT, 0 ms] (15) QDP (16) QDPOrderProof [EQUIVALENT, 20 ms] (17) QDP (18) PisEmptyProof [EQUIVALENT, 0 ms] (19) YES (20) QDP (21) QDPOrderProof [EQUIVALENT, 17 ms] (22) QDP (23) QDPOrderProof [EQUIVALENT, 193 ms] (24) QDP (25) QDPOrderProof [EQUIVALENT, 417 ms] (26) QDP (27) PisEmptyProof [EQUIVALENT, 0 ms] (28) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(a(a(x1)))) -> b(a(a(a(x1)))) a(b(b(b(x1)))) -> b(a(b(a(x1)))) b(b(a(a(x1)))) -> a(b(b(a(x1)))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(a(a(x1)))) -> a(a(a(b(x1)))) b(b(b(a(x1)))) -> a(b(a(b(x1)))) a(a(b(b(x1)))) -> a(b(b(a(x1)))) Q is empty. ---------------------------------------- (3) FlatCCProof (EQUIVALENT) We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete. ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(a(a(x1)))) -> a(a(a(b(x1)))) a(a(b(b(x1)))) -> a(b(b(a(x1)))) a(b(b(b(a(x1))))) -> a(a(b(a(b(x1))))) b(b(b(b(a(x1))))) -> b(a(b(a(b(x1))))) Q is empty. ---------------------------------------- (5) RootLabelingProof (EQUIVALENT) We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled As Q is empty the root labeling was sound AND complete. ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a_{a_1}(x_1)) = 1 + x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 1 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. ---------------------------------------- (9) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1)) A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{B_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = 1 + x_1 POL(b_{a_1}(x_1)) = x_1 POL(b_{b_1}(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (17) Obligation: Q DP problem: P is empty. R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(A_{A_1}(x_1)) = x_1 POL(a_{a_1}(x_1)) = x_1 POL(a_{b_1}(x_1)) = x_1 POL(b_{a_1}(x_1)) = 1 + x_1 POL(b_{b_1}(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A_{A_1}(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 1A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [0A, -I, -I], [1A, 0A, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [-I], [0A]] + [[-I, 0A, -I], [0A, -I, -I], [1A, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A_{A_1}(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 >>> <<< POL(a_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, 1A], [0A, 0A, 1A]] * x_1 >>> <<< POL(a_{b_1}(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, -I, 0A]] * x_1 >>> <<< POL(b_{b_1}(x_1)) = [[0A], [0A], [1A]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(b_{a_1}(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, 0A, 0A], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) ---------------------------------------- (26) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (28) YES