157.50/40.91 YES 157.70/41.01 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 157.70/41.01 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 157.70/41.01 157.70/41.01 157.70/41.01 Termination w.r.t. Q of the given QTRS could be proven: 157.70/41.01 157.70/41.01 (0) QTRS 157.70/41.01 (1) QTRS Reverse [EQUIVALENT, 0 ms] 157.70/41.01 (2) QTRS 157.70/41.01 (3) DependencyPairsProof [EQUIVALENT, 40 ms] 157.70/41.01 (4) QDP 157.70/41.01 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 157.70/41.01 (6) AND 157.70/41.01 (7) QDP 157.70/41.01 (8) QDPOrderProof [EQUIVALENT, 6792 ms] 157.70/41.01 (9) QDP 157.70/41.01 (10) PisEmptyProof [EQUIVALENT, 0 ms] 157.70/41.01 (11) YES 157.70/41.01 (12) QDP 157.70/41.01 (13) QDPOrderProof [EQUIVALENT, 6895 ms] 157.70/41.01 (14) QDP 157.70/41.01 (15) PisEmptyProof [EQUIVALENT, 0 ms] 157.70/41.01 (16) YES 157.70/41.01 157.70/41.01 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (0) 157.70/41.01 Obligation: 157.70/41.01 Q restricted rewrite system: 157.70/41.01 The TRS R consists of the following rules: 157.70/41.01 157.70/41.01 b(a(b(x1))) -> b(a(a(a(b(x1))))) 157.70/41.01 b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(b(a(a(b(a(a(a(b(b(x1)))))))))))) 157.70/41.01 b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) 157.70/41.01 b(a(a(a(b(b(b(x1))))))) -> b(b(b(a(a(a(b(x1))))))) 157.70/41.01 b(a(a(b(b(x1))))) -> b(x1) 157.70/41.01 b(b(a(a(b(x1))))) -> b(x1) 157.70/41.01 b(a(a(a(b(a(b(x1))))))) -> b(x1) 157.70/41.01 b(a(b(a(a(a(b(x1))))))) -> b(x1) 157.70/41.01 157.70/41.01 Q is empty. 157.70/41.01 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (1) QTRS Reverse (EQUIVALENT) 157.70/41.01 We applied the QTRS Reverse Processor [REVERSE]. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (2) 157.70/41.01 Obligation: 157.70/41.01 Q restricted rewrite system: 157.70/41.01 The TRS R consists of the following rules: 157.70/41.01 157.70/41.01 b(a(b(x1))) -> b(a(a(a(b(x1))))) 157.70/41.01 b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) 157.70/41.01 b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) 157.70/41.01 b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) 157.70/41.01 b(b(a(a(b(x1))))) -> b(x1) 157.70/41.01 b(a(a(b(b(x1))))) -> b(x1) 157.70/41.01 b(a(b(a(a(a(b(x1))))))) -> b(x1) 157.70/41.01 b(a(a(a(b(a(b(x1))))))) -> b(x1) 157.70/41.01 157.70/41.01 Q is empty. 157.70/41.01 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (3) DependencyPairsProof (EQUIVALENT) 157.70/41.01 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (4) 157.70/41.01 Obligation: 157.70/41.01 Q DP problem: 157.70/41.01 The TRS P consists of the following rules: 157.70/41.01 157.70/41.01 B(a(b(x1))) -> B(a(a(a(b(x1))))) 157.70/41.01 B(a(a(b(a(a(a(b(x1)))))))) -> B(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) 157.70/41.01 B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(a(b(a(a(b(a(a(b(x1))))))))))) 157.70/41.01 B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(a(a(b(x1))))))) 157.70/41.01 B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(x1)))) 157.70/41.01 B(b(b(a(a(a(b(x1))))))) -> B(a(a(a(b(b(b(x1))))))) 157.70/41.01 B(b(b(a(a(a(b(x1))))))) -> B(b(b(x1))) 157.70/41.01 B(b(b(a(a(a(b(x1))))))) -> B(b(x1)) 157.70/41.01 157.70/41.01 The TRS R consists of the following rules: 157.70/41.01 157.70/41.01 b(a(b(x1))) -> b(a(a(a(b(x1))))) 157.70/41.01 b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) 157.70/41.01 b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) 157.70/41.01 b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) 157.70/41.01 b(b(a(a(b(x1))))) -> b(x1) 157.70/41.01 b(a(a(b(b(x1))))) -> b(x1) 157.70/41.01 b(a(b(a(a(a(b(x1))))))) -> b(x1) 157.70/41.01 b(a(a(a(b(a(b(x1))))))) -> b(x1) 157.70/41.01 157.70/41.01 Q is empty. 157.70/41.01 We have to consider all minimal (P,Q,R)-chains. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (5) DependencyGraphProof (EQUIVALENT) 157.70/41.01 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (6) 157.70/41.01 Complex Obligation (AND) 157.70/41.01 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (7) 157.70/41.01 Obligation: 157.70/41.01 Q DP problem: 157.70/41.01 The TRS P consists of the following rules: 157.70/41.01 157.70/41.01 B(b(b(a(a(a(b(x1))))))) -> B(b(x1)) 157.70/41.01 B(b(b(a(a(a(b(x1))))))) -> B(b(b(x1))) 157.70/41.01 157.70/41.01 The TRS R consists of the following rules: 157.70/41.01 157.70/41.01 b(a(b(x1))) -> b(a(a(a(b(x1))))) 157.70/41.01 b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) 157.70/41.01 b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) 157.70/41.01 b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) 157.70/41.01 b(b(a(a(b(x1))))) -> b(x1) 157.70/41.01 b(a(a(b(b(x1))))) -> b(x1) 157.70/41.01 b(a(b(a(a(a(b(x1))))))) -> b(x1) 157.70/41.01 b(a(a(a(b(a(b(x1))))))) -> b(x1) 157.70/41.01 157.70/41.01 Q is empty. 157.70/41.01 We have to consider all minimal (P,Q,R)-chains. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (8) QDPOrderProof (EQUIVALENT) 157.70/41.01 We use the reduction pair processor [LPAR04,JAR06]. 157.70/41.01 157.70/41.01 157.70/41.01 The following pairs can be oriented strictly and are deleted. 157.70/41.01 157.70/41.01 B(b(b(a(a(a(b(x1))))))) -> B(b(x1)) 157.70/41.01 B(b(b(a(a(a(b(x1))))))) -> B(b(b(x1))) 157.70/41.01 The remaining pairs can at least be oriented weakly. 157.70/41.01 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 157.70/41.01 157.70/41.01 <<< 157.70/41.01 POL(B(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 157.70/41.01 >>> 157.70/41.01 157.70/41.01 <<< 157.70/41.01 POL(b(x_1)) = [[1A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, -I], [0A, -I, 0A]] * x_1 157.70/41.01 >>> 157.70/41.01 157.70/41.01 <<< 157.70/41.01 POL(a(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [1A, -I, 0A], [0A, 0A, -I]] * x_1 157.70/41.01 >>> 157.70/41.01 157.70/41.01 157.70/41.01 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 157.70/41.01 157.70/41.01 b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) 157.70/41.01 b(a(b(x1))) -> b(a(a(a(b(x1))))) 157.70/41.01 b(a(a(a(b(a(b(x1))))))) -> b(x1) 157.70/41.01 b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) 157.70/41.01 b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) 157.70/41.01 b(b(a(a(b(x1))))) -> b(x1) 157.70/41.01 b(a(a(b(b(x1))))) -> b(x1) 157.70/41.01 b(a(b(a(a(a(b(x1))))))) -> b(x1) 157.70/41.01 157.70/41.01 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (9) 157.70/41.01 Obligation: 157.70/41.01 Q DP problem: 157.70/41.01 P is empty. 157.70/41.01 The TRS R consists of the following rules: 157.70/41.01 157.70/41.01 b(a(b(x1))) -> b(a(a(a(b(x1))))) 157.70/41.01 b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) 157.70/41.01 b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) 157.70/41.01 b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) 157.70/41.01 b(b(a(a(b(x1))))) -> b(x1) 157.70/41.01 b(a(a(b(b(x1))))) -> b(x1) 157.70/41.01 b(a(b(a(a(a(b(x1))))))) -> b(x1) 157.70/41.01 b(a(a(a(b(a(b(x1))))))) -> b(x1) 157.70/41.01 157.70/41.01 Q is empty. 157.70/41.01 We have to consider all minimal (P,Q,R)-chains. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (10) PisEmptyProof (EQUIVALENT) 157.70/41.01 The TRS P is empty. Hence, there is no (P,Q,R) chain. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (11) 157.70/41.01 YES 157.70/41.01 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (12) 157.70/41.01 Obligation: 157.70/41.01 Q DP problem: 157.70/41.01 The TRS P consists of the following rules: 157.70/41.01 157.70/41.01 B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(x1)))) 157.70/41.01 B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(a(a(b(x1))))))) 157.70/41.01 157.70/41.01 The TRS R consists of the following rules: 157.70/41.01 157.70/41.01 b(a(b(x1))) -> b(a(a(a(b(x1))))) 157.70/41.01 b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) 157.70/41.01 b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) 157.70/41.01 b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) 157.70/41.01 b(b(a(a(b(x1))))) -> b(x1) 157.70/41.01 b(a(a(b(b(x1))))) -> b(x1) 157.70/41.01 b(a(b(a(a(a(b(x1))))))) -> b(x1) 157.70/41.01 b(a(a(a(b(a(b(x1))))))) -> b(x1) 157.70/41.01 157.70/41.01 Q is empty. 157.70/41.01 We have to consider all minimal (P,Q,R)-chains. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (13) QDPOrderProof (EQUIVALENT) 157.70/41.01 We use the reduction pair processor [LPAR04,JAR06]. 157.70/41.01 157.70/41.01 157.70/41.01 The following pairs can be oriented strictly and are deleted. 157.70/41.01 157.70/41.01 B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(x1)))) 157.70/41.01 B(a(a(b(a(a(a(b(x1)))))))) -> B(a(a(b(a(a(b(x1))))))) 157.70/41.01 The remaining pairs can at least be oriented weakly. 157.70/41.01 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 157.70/41.01 157.70/41.01 <<< 157.70/41.01 POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 157.70/41.01 >>> 157.70/41.01 157.70/41.01 <<< 157.70/41.01 POL(a(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, -I]] * x_1 157.70/41.01 >>> 157.70/41.01 157.70/41.01 <<< 157.70/41.01 POL(b(x_1)) = [[0A], [-I], [0A]] + [[0A, -I, 0A], [-I, 0A, -I], [0A, 1A, 0A]] * x_1 157.70/41.01 >>> 157.70/41.01 157.70/41.01 157.70/41.01 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 157.70/41.01 157.70/41.01 b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) 157.70/41.01 b(a(b(x1))) -> b(a(a(a(b(x1))))) 157.70/41.01 b(a(a(a(b(a(b(x1))))))) -> b(x1) 157.70/41.01 b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) 157.70/41.01 b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) 157.70/41.01 b(b(a(a(b(x1))))) -> b(x1) 157.70/41.01 b(a(a(b(b(x1))))) -> b(x1) 157.70/41.01 b(a(b(a(a(a(b(x1))))))) -> b(x1) 157.70/41.01 157.70/41.01 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (14) 157.70/41.01 Obligation: 157.70/41.01 Q DP problem: 157.70/41.01 P is empty. 157.70/41.01 The TRS R consists of the following rules: 157.70/41.01 157.70/41.01 b(a(b(x1))) -> b(a(a(a(b(x1))))) 157.70/41.01 b(a(a(b(a(a(a(b(x1)))))))) -> b(b(a(a(a(b(a(a(b(a(a(b(x1)))))))))))) 157.70/41.01 b(a(a(a(b(a(a(a(b(x1))))))))) -> b(x1) 157.70/41.01 b(b(b(a(a(a(b(x1))))))) -> b(a(a(a(b(b(b(x1))))))) 157.70/41.01 b(b(a(a(b(x1))))) -> b(x1) 157.70/41.01 b(a(a(b(b(x1))))) -> b(x1) 157.70/41.01 b(a(b(a(a(a(b(x1))))))) -> b(x1) 157.70/41.01 b(a(a(a(b(a(b(x1))))))) -> b(x1) 157.70/41.01 157.70/41.01 Q is empty. 157.70/41.01 We have to consider all minimal (P,Q,R)-chains. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (15) PisEmptyProof (EQUIVALENT) 157.70/41.01 The TRS P is empty. Hence, there is no (P,Q,R) chain. 157.70/41.01 ---------------------------------------- 157.70/41.01 157.70/41.01 (16) 157.70/41.01 YES 158.01/41.06 EOF