YES proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 27 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) QDP (7) QDPOrderProof [EQUIVALENT, 1030 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 388 ms] (10) QDP (11) QDPOrderProof [EQUIVALENT, 372 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 374 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(b(x1)))) -> b(b(b(a(b(x1))))) b(a(a(b(b(x1))))) -> b(a(b(b(a(a(b(x1))))))) b(a(a(a(b(b(x1)))))) -> b(a(a(b(b(a(a(a(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(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(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(b(a(b(x1)))) -> B(a(b(b(b(x1))))) B(b(a(b(x1)))) -> B(b(b(x1))) B(b(a(b(x1)))) -> B(b(x1)) B(b(a(a(b(x1))))) -> B(a(a(b(b(a(b(x1))))))) B(b(a(a(b(x1))))) -> B(b(a(b(x1)))) B(b(a(a(b(x1))))) -> B(a(b(x1))) B(b(a(a(a(b(x1)))))) -> B(a(a(a(b(b(a(a(b(x1))))))))) B(b(a(a(a(b(x1)))))) -> B(b(a(a(b(x1))))) B(b(a(a(a(b(x1)))))) -> B(a(a(b(x1)))) The TRS R consists of the following rules: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(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 1 SCC with 5 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: B(b(a(b(x1)))) -> B(b(x1)) B(b(a(b(x1)))) -> B(b(b(x1))) B(b(a(a(b(x1))))) -> B(b(a(b(x1)))) B(b(a(a(a(b(x1)))))) -> B(b(a(a(b(x1))))) The TRS R consists of the following rules: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B(b(a(a(a(b(x1)))))) -> B(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]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(b(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [0A], [-I]] + [[1A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: B(b(a(b(x1)))) -> B(b(x1)) B(b(a(b(x1)))) -> B(b(b(x1))) B(b(a(a(b(x1))))) -> B(b(a(b(x1)))) The TRS R consists of the following rules: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B(b(a(a(b(x1))))) -> B(b(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]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, -I], [0A, -I, -I]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [1A, -I, -I], [0A, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: B(b(a(b(x1)))) -> B(b(x1)) B(b(a(b(x1)))) -> B(b(b(x1))) The TRS R consists of the following rules: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. B(b(a(b(x1)))) -> 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, -I, 0A]] * x_1 >>> <<< POL(b(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [0A, -I, -I], [-I, -I, 0A]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [1A, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: B(b(a(b(x1)))) -> B(b(b(x1))) The TRS R consists of the following rules: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(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(b(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]] + [[-I, -I, 0A]] * x_1 >>> <<< POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(a(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 1A], [-I, -I, 0A], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1))))))))) ---------------------------------------- (14) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: b(b(a(b(x1)))) -> b(a(b(b(b(x1))))) b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1))))))) b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(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