YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 23 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 2 ms] (6) AND (7) QDP (8) QDPOrderProof [EQUIVALENT, 5558 ms] (9) QDP (10) PisEmptyProof [EQUIVALENT, 0 ms] (11) YES (12) QDP (13) QDPOrderProof [EQUIVALENT, 5917 ms] (14) QDP (15) PisEmptyProof [EQUIVALENT, 0 ms] (16) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: b(a(b(x1))) -> b(a(a(a(b(x1))))) b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(b(a(a(b(a(a(a(b(b(x1)))))))))))) b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) b(a(a(a(b(b(b(x1))))))) -> b(b(b(a(a(a(b(x1))))))) b(a(a(b(b(x1))))) -> b(x1) b(b(a(a(b(x1))))) -> b(x1) b(a(a(a(b(a(b(x1))))))) -> b(x1) b(a(b(a(a(a(b(x1))))))) -> b(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: b(a(b(x1))) -> b(a(a(a(b(x1))))) b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) b(b(a(a(b(x1))))) -> b(x1) b(a(a(b(b(x1))))) -> b(x1) b(a(b(a(a(a(b(x1))))))) -> b(x1) b(a(a(a(b(a(b(x1))))))) -> b(x1) Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: B(a(b(x1))) -> B(a(a(a(b(x1))))) B(a(a(b(a(a(a(b(x1)))))))) -> B(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(a(b(a(a(b(a(a(b(x1))))))))))) B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(a(a(b(x1))))))) B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(x1)))) B(b(b(a(a(a(b(x1))))))) -> B(a(a(a(b(b(b(x1))))))) B(b(b(a(a(a(b(x1))))))) -> B(b(b(x1))) B(b(b(a(a(a(b(x1))))))) -> B(b(x1)) The TRS R consists of the following rules: b(a(b(x1))) -> b(a(a(a(b(x1))))) b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) b(b(a(a(b(x1))))) -> b(x1) b(a(a(b(b(x1))))) -> b(x1) b(a(b(a(a(a(b(x1))))))) -> b(x1) b(a(a(a(b(a(b(x1))))))) -> b(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: B(b(b(a(a(a(b(x1))))))) -> B(b(x1)) B(b(b(a(a(a(b(x1))))))) -> B(b(b(x1))) The TRS R consists of the following rules: b(a(b(x1))) -> b(a(a(a(b(x1))))) b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) b(b(a(a(b(x1))))) -> b(x1) b(a(a(b(b(x1))))) -> b(x1) b(a(b(a(a(a(b(x1))))))) -> b(x1) b(a(a(a(b(a(b(x1))))))) -> b(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B(b(b(a(a(a(b(x1))))))) -> B(b(x1)) B(b(b(a(a(a(b(x1))))))) -> B(b(b(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(B(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 >>> <<< POL(b(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, -I, 0A]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [1A, -I, 0A], [0A, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) b(a(b(x1))) -> b(a(a(a(b(x1))))) b(a(a(a(b(a(b(x1))))))) -> b(x1) b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) b(b(a(a(b(x1))))) -> b(x1) b(a(a(b(b(x1))))) -> b(x1) b(a(b(a(a(a(b(x1))))))) -> b(x1) ---------------------------------------- (9) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b(a(b(x1))) -> b(a(a(a(b(x1))))) b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) b(b(a(a(b(x1))))) -> b(x1) b(a(a(b(b(x1))))) -> b(x1) b(a(b(a(a(a(b(x1))))))) -> b(x1) b(a(a(a(b(a(b(x1))))))) -> b(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (11) YES ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(x1)))) B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(a(a(b(x1))))))) The TRS R consists of the following rules: b(a(b(x1))) -> b(a(a(a(b(x1))))) b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) b(b(a(a(b(x1))))) -> b(x1) b(a(a(b(b(x1))))) -> b(x1) b(a(b(a(a(a(b(x1))))))) -> b(x1) b(a(a(a(b(a(b(x1))))))) -> b(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(x1)))) B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(a(a(b(x1))))))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 >>> <<< POL(b(x_1)) = [[0A], [-I], [0A]] + [[0A, -I, 0A], [-I, 0A, -I], [0A, 1A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) b(a(b(x1))) -> b(a(a(a(b(x1))))) b(a(a(a(b(a(b(x1))))))) -> b(x1) b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) b(b(a(a(b(x1))))) -> b(x1) b(a(a(b(b(x1))))) -> b(x1) b(a(b(a(a(a(b(x1))))))) -> b(x1) ---------------------------------------- (14) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b(a(b(x1))) -> b(a(a(a(b(x1))))) b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) b(b(a(a(b(x1))))) -> b(x1) b(a(a(b(b(x1))))) -> b(x1) b(a(b(a(a(a(b(x1))))))) -> b(x1) b(a(a(a(b(a(b(x1))))))) -> b(x1) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (16) YES