108.78/28.62 YES 115.77/30.41 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 115.77/30.41 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 115.77/30.41 115.77/30.41 115.77/30.41 Termination w.r.t. Q of the given QTRS could be proven: 115.77/30.41 115.77/30.41 (0) QTRS 115.77/30.41 (1) QTRS Reverse [EQUIVALENT, 0 ms] 115.77/30.41 (2) QTRS 115.77/30.41 (3) DependencyPairsProof [EQUIVALENT, 17 ms] 115.77/30.41 (4) QDP 115.77/30.41 (5) QDPOrderProof [EQUIVALENT, 631 ms] 115.77/30.41 (6) QDP 115.77/30.41 (7) QDPOrderProof [EQUIVALENT, 271 ms] 115.77/30.41 (8) QDP 115.77/30.41 (9) QDPOrderProof [EQUIVALENT, 320 ms] 115.77/30.41 (10) QDP 115.77/30.41 (11) QDPOrderProof [EQUIVALENT, 275 ms] 115.77/30.41 (12) QDP 115.77/30.41 (13) QDPOrderProof [EQUIVALENT, 310 ms] 115.77/30.41 (14) QDP 115.77/30.41 (15) QDPOrderProof [EQUIVALENT, 1486 ms] 115.77/30.41 (16) QDP 115.77/30.41 (17) DependencyGraphProof [EQUIVALENT, 0 ms] 115.77/30.41 (18) TRUE 115.77/30.41 115.77/30.41 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (0) 115.77/30.41 Obligation: 115.77/30.41 Q restricted rewrite system: 115.77/30.41 The TRS R consists of the following rules: 115.77/30.41 115.77/30.41 a(a(a(b(b(x1))))) -> b(b(a(b(b(a(a(a(x1)))))))) 115.77/30.41 b(b(a(x1))) -> x1 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 115.77/30.41 Q is empty. 115.77/30.41 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (1) QTRS Reverse (EQUIVALENT) 115.77/30.41 We applied the QTRS Reverse Processor [REVERSE]. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (2) 115.77/30.41 Obligation: 115.77/30.41 Q restricted rewrite system: 115.77/30.41 The TRS R consists of the following rules: 115.77/30.41 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 115.77/30.41 Q is empty. 115.77/30.41 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (3) DependencyPairsProof (EQUIVALENT) 115.77/30.41 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (4) 115.77/30.41 Obligation: 115.77/30.41 Q DP problem: 115.77/30.41 The TRS P consists of the following rules: 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1))))))) 115.77/30.41 B(b(a(a(a(x1))))) -> A(b(b(a(b(b(x1)))))) 115.77/30.41 B(b(a(a(a(x1))))) -> B(b(a(b(b(x1))))) 115.77/30.41 B(b(a(a(a(x1))))) -> B(a(b(b(x1)))) 115.77/30.41 B(b(a(a(a(x1))))) -> A(b(b(x1))) 115.77/30.41 B(b(a(a(a(x1))))) -> B(b(x1)) 115.77/30.41 B(b(a(a(a(x1))))) -> B(x1) 115.77/30.41 A(x1) -> B(b(b(x1))) 115.77/30.41 A(x1) -> B(b(x1)) 115.77/30.41 A(x1) -> B(x1) 115.77/30.41 115.77/30.41 The TRS R consists of the following rules: 115.77/30.41 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 115.77/30.41 Q is empty. 115.77/30.41 We have to consider all minimal (P,Q,R)-chains. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (5) QDPOrderProof (EQUIVALENT) 115.77/30.41 We use the reduction pair processor [LPAR04,JAR06]. 115.77/30.41 115.77/30.41 115.77/30.41 The following pairs can be oriented strictly and are deleted. 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> B(b(a(b(b(x1))))) 115.77/30.41 B(b(a(a(a(x1))))) -> B(b(x1)) 115.77/30.41 The remaining pairs can at least be oriented weakly. 115.77/30.41 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(B(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(a(x_1)) = [[1A], [0A], [0A]] + [[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 115.77/30.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 115.77/30.41 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 115.77/30.41 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (6) 115.77/30.41 Obligation: 115.77/30.41 Q DP problem: 115.77/30.41 The TRS P consists of the following rules: 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1))))))) 115.77/30.41 B(b(a(a(a(x1))))) -> A(b(b(a(b(b(x1)))))) 115.77/30.41 B(b(a(a(a(x1))))) -> B(a(b(b(x1)))) 115.77/30.41 B(b(a(a(a(x1))))) -> A(b(b(x1))) 115.77/30.41 B(b(a(a(a(x1))))) -> B(x1) 115.77/30.41 A(x1) -> B(b(b(x1))) 115.77/30.41 A(x1) -> B(b(x1)) 115.77/30.41 A(x1) -> B(x1) 115.77/30.41 115.77/30.41 The TRS R consists of the following rules: 115.77/30.41 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 115.77/30.41 Q is empty. 115.77/30.41 We have to consider all minimal (P,Q,R)-chains. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (7) QDPOrderProof (EQUIVALENT) 115.77/30.41 We use the reduction pair processor [LPAR04,JAR06]. 115.77/30.41 115.77/30.41 115.77/30.41 The following pairs can be oriented strictly and are deleted. 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> A(b(b(a(b(b(x1)))))) 115.77/30.41 B(b(a(a(a(x1))))) -> A(b(b(x1))) 115.77/30.41 The remaining pairs can at least be oriented weakly. 115.77/30.41 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(B(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [1A, 0A, 0A], [0A, -I, -I]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(a(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(A(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 115.77/30.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 115.77/30.41 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 115.77/30.41 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (8) 115.77/30.41 Obligation: 115.77/30.41 Q DP problem: 115.77/30.41 The TRS P consists of the following rules: 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1))))))) 115.77/30.41 B(b(a(a(a(x1))))) -> B(a(b(b(x1)))) 115.77/30.41 B(b(a(a(a(x1))))) -> B(x1) 115.77/30.41 A(x1) -> B(b(b(x1))) 115.77/30.41 A(x1) -> B(b(x1)) 115.77/30.41 A(x1) -> B(x1) 115.77/30.41 115.77/30.41 The TRS R consists of the following rules: 115.77/30.41 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 115.77/30.41 Q is empty. 115.77/30.41 We have to consider all minimal (P,Q,R)-chains. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (9) QDPOrderProof (EQUIVALENT) 115.77/30.41 We use the reduction pair processor [LPAR04,JAR06]. 115.77/30.41 115.77/30.41 115.77/30.41 The following pairs can be oriented strictly and are deleted. 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> B(a(b(b(x1)))) 115.77/30.41 B(b(a(a(a(x1))))) -> B(x1) 115.77/30.41 The remaining pairs can at least be oriented weakly. 115.77/30.41 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(B(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [1A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(a(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, -I], [0A, 0A, 0A], [1A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 115.77/30.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 115.77/30.41 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 115.77/30.41 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (10) 115.77/30.41 Obligation: 115.77/30.41 Q DP problem: 115.77/30.41 The TRS P consists of the following rules: 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1))))))) 115.77/30.41 A(x1) -> B(b(b(x1))) 115.77/30.41 A(x1) -> B(b(x1)) 115.77/30.41 A(x1) -> B(x1) 115.77/30.41 115.77/30.41 The TRS R consists of the following rules: 115.77/30.41 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 115.77/30.41 Q is empty. 115.77/30.41 We have to consider all minimal (P,Q,R)-chains. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (11) QDPOrderProof (EQUIVALENT) 115.77/30.41 We use the reduction pair processor [LPAR04,JAR06]. 115.77/30.41 115.77/30.41 115.77/30.41 The following pairs can be oriented strictly and are deleted. 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(b(b(a(b(b(x1))))))) 115.77/30.41 The remaining pairs can at least be oriented weakly. 115.77/30.41 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(B(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(b(x_1)) = [[0A], [-I], [-I]] + [[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(a(x_1)) = [[1A], [-I], [0A]] + [[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(A(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 115.77/30.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 115.77/30.41 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 115.77/30.41 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (12) 115.77/30.41 Obligation: 115.77/30.41 Q DP problem: 115.77/30.41 The TRS P consists of the following rules: 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 A(x1) -> B(b(b(x1))) 115.77/30.41 A(x1) -> B(b(x1)) 115.77/30.41 A(x1) -> B(x1) 115.77/30.41 115.77/30.41 The TRS R consists of the following rules: 115.77/30.41 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 115.77/30.41 Q is empty. 115.77/30.41 We have to consider all minimal (P,Q,R)-chains. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (13) QDPOrderProof (EQUIVALENT) 115.77/30.41 We use the reduction pair processor [LPAR04,JAR06]. 115.77/30.41 115.77/30.41 115.77/30.41 The following pairs can be oriented strictly and are deleted. 115.77/30.41 115.77/30.41 A(x1) -> B(x1) 115.77/30.41 The remaining pairs can at least be oriented weakly. 115.77/30.41 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(B(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(b(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, -I], [0A, -I, -I], [1A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(a(x_1)) = [[-I], [0A], [-I]] + [[0A, 0A, -I], [0A, 0A, 0A], [1A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(A(x_1)) = [[1A]] + [[1A, 1A, 1A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 115.77/30.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 115.77/30.41 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 115.77/30.41 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (14) 115.77/30.41 Obligation: 115.77/30.41 Q DP problem: 115.77/30.41 The TRS P consists of the following rules: 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 A(x1) -> B(b(b(x1))) 115.77/30.41 A(x1) -> B(b(x1)) 115.77/30.41 115.77/30.41 The TRS R consists of the following rules: 115.77/30.41 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 115.77/30.41 Q is empty. 115.77/30.41 We have to consider all minimal (P,Q,R)-chains. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (15) QDPOrderProof (EQUIVALENT) 115.77/30.41 We use the reduction pair processor [LPAR04,JAR06]. 115.77/30.41 115.77/30.41 115.77/30.41 The following pairs can be oriented strictly and are deleted. 115.77/30.41 115.77/30.41 A(x1) -> B(b(b(x1))) 115.77/30.41 A(x1) -> B(b(x1)) 115.77/30.41 The remaining pairs can at least be oriented weakly. 115.77/30.41 Used ordering: Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]: 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(B(x_1)) = [[0A]] + [[-1A, -I, -1A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(b(x_1)) = [[0A], [-1A], [-I]] + [[-I, -I, -1A], [0A, -I, 0A], [-I, -1A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(a(x_1)) = [[2A], [0A], [-1A]] + [[-1A, -1A, 1A], [2A, -1A, 0A], [-1A, 0A, 0A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 <<< 115.77/30.41 POL(A(x_1)) = [[1A]] + [[-1A, -1A, 1A]] * x_1 115.77/30.41 >>> 115.77/30.41 115.77/30.41 115.77/30.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 115.77/30.41 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 115.77/30.41 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (16) 115.77/30.41 Obligation: 115.77/30.41 Q DP problem: 115.77/30.41 The TRS P consists of the following rules: 115.77/30.41 115.77/30.41 B(b(a(a(a(x1))))) -> A(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 115.77/30.41 The TRS R consists of the following rules: 115.77/30.41 115.77/30.41 b(b(a(a(a(x1))))) -> a(a(a(b(b(a(b(b(x1)))))))) 115.77/30.41 a(b(b(x1))) -> x1 115.77/30.41 a(x1) -> b(b(b(x1))) 115.77/30.41 a(x1) -> b(x1) 115.77/30.41 115.77/30.41 Q is empty. 115.77/30.41 We have to consider all minimal (P,Q,R)-chains. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (17) DependencyGraphProof (EQUIVALENT) 115.77/30.41 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 115.77/30.41 ---------------------------------------- 115.77/30.41 115.77/30.41 (18) 115.77/30.41 TRUE 115.97/30.50 EOF