369.60/94.73 YES 369.95/94.79 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 369.95/94.79 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 369.95/94.79 369.95/94.79 369.95/94.79 Termination w.r.t. Q of the given QTRS could be proven: 369.95/94.79 369.95/94.79 (0) QTRS 369.95/94.79 (1) DependencyPairsProof [EQUIVALENT, 20 ms] 369.95/94.79 (2) QDP 369.95/94.79 (3) QDPOrderProof [EQUIVALENT, 24 ms] 369.95/94.79 (4) QDP 369.95/94.79 (5) QDPOrderProof [EQUIVALENT, 735 ms] 369.95/94.79 (6) QDP 369.95/94.79 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 369.95/94.79 (8) TRUE 369.95/94.79 369.95/94.79 369.95/94.79 ---------------------------------------- 369.95/94.79 369.95/94.79 (0) 369.95/94.79 Obligation: 369.95/94.79 Q restricted rewrite system: 369.95/94.79 The TRS R consists of the following rules: 369.95/94.79 369.95/94.79 b(b(b(b(x1)))) -> a(a(a(a(x1)))) 369.95/94.79 a(b(a(a(x1)))) -> b(b(b(b(x1)))) 369.95/94.79 b(a(b(a(x1)))) -> a(a(b(a(x1)))) 369.95/94.79 369.95/94.79 Q is empty. 369.95/94.79 369.95/94.79 ---------------------------------------- 369.95/94.79 369.95/94.79 (1) DependencyPairsProof (EQUIVALENT) 369.95/94.79 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 369.95/94.79 ---------------------------------------- 369.95/94.79 369.95/94.79 (2) 369.95/94.79 Obligation: 369.95/94.79 Q DP problem: 369.95/94.79 The TRS P consists of the following rules: 369.95/94.79 369.95/94.79 B(b(b(b(x1)))) -> A(a(a(a(x1)))) 369.95/94.79 B(b(b(b(x1)))) -> A(a(a(x1))) 369.95/94.79 B(b(b(b(x1)))) -> A(a(x1)) 369.95/94.79 B(b(b(b(x1)))) -> A(x1) 369.95/94.79 A(b(a(a(x1)))) -> B(b(b(b(x1)))) 369.95/94.79 A(b(a(a(x1)))) -> B(b(b(x1))) 369.95/94.79 A(b(a(a(x1)))) -> B(b(x1)) 369.95/94.79 A(b(a(a(x1)))) -> B(x1) 369.95/94.79 B(a(b(a(x1)))) -> A(a(b(a(x1)))) 369.95/94.79 369.95/94.79 The TRS R consists of the following rules: 369.95/94.79 369.95/94.79 b(b(b(b(x1)))) -> a(a(a(a(x1)))) 369.95/94.79 a(b(a(a(x1)))) -> b(b(b(b(x1)))) 369.95/94.79 b(a(b(a(x1)))) -> a(a(b(a(x1)))) 369.95/94.79 369.95/94.79 Q is empty. 369.95/94.79 We have to consider all minimal (P,Q,R)-chains. 369.95/94.79 ---------------------------------------- 369.95/94.79 369.95/94.79 (3) QDPOrderProof (EQUIVALENT) 369.95/94.79 We use the reduction pair processor [LPAR04,JAR06]. 369.95/94.79 369.95/94.79 369.95/94.79 The following pairs can be oriented strictly and are deleted. 369.95/94.79 369.95/94.79 B(b(b(b(x1)))) -> A(a(a(x1))) 369.95/94.79 B(b(b(b(x1)))) -> A(a(x1)) 369.95/94.79 B(b(b(b(x1)))) -> A(x1) 369.95/94.79 A(b(a(a(x1)))) -> B(b(b(x1))) 369.95/94.79 A(b(a(a(x1)))) -> B(b(x1)) 369.95/94.79 A(b(a(a(x1)))) -> B(x1) 369.95/94.79 The remaining pairs can at least be oriented weakly. 369.95/94.79 Used ordering: Polynomial interpretation [POLO]: 369.95/94.79 369.95/94.79 POL(A(x_1)) = x_1 369.95/94.79 POL(B(x_1)) = x_1 369.95/94.79 POL(a(x_1)) = 1 + x_1 369.95/94.79 POL(b(x_1)) = 1 + x_1 369.95/94.79 369.95/94.79 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 369.95/94.79 369.95/94.79 a(b(a(a(x1)))) -> b(b(b(b(x1)))) 369.95/94.79 b(b(b(b(x1)))) -> a(a(a(a(x1)))) 369.95/94.79 b(a(b(a(x1)))) -> a(a(b(a(x1)))) 369.95/94.79 369.95/94.79 369.95/94.79 ---------------------------------------- 369.95/94.79 369.95/94.79 (4) 369.95/94.79 Obligation: 369.95/94.79 Q DP problem: 369.95/94.79 The TRS P consists of the following rules: 369.95/94.79 369.95/94.79 B(b(b(b(x1)))) -> A(a(a(a(x1)))) 369.95/94.79 A(b(a(a(x1)))) -> B(b(b(b(x1)))) 369.95/94.79 B(a(b(a(x1)))) -> A(a(b(a(x1)))) 369.95/94.79 369.95/94.79 The TRS R consists of the following rules: 369.95/94.79 369.95/94.79 b(b(b(b(x1)))) -> a(a(a(a(x1)))) 369.95/94.79 a(b(a(a(x1)))) -> b(b(b(b(x1)))) 369.95/94.79 b(a(b(a(x1)))) -> a(a(b(a(x1)))) 369.95/94.79 369.95/94.79 Q is empty. 369.95/94.79 We have to consider all minimal (P,Q,R)-chains. 369.95/94.79 ---------------------------------------- 369.95/94.79 369.95/94.79 (5) QDPOrderProof (EQUIVALENT) 369.95/94.79 We use the reduction pair processor [LPAR04,JAR06]. 369.95/94.79 369.95/94.79 369.95/94.79 The following pairs can be oriented strictly and are deleted. 369.95/94.79 369.95/94.79 A(b(a(a(x1)))) -> B(b(b(b(x1)))) 369.95/94.79 The remaining pairs can at least be oriented weakly. 369.95/94.79 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 369.95/94.79 369.95/94.79 <<< 369.95/94.79 POL(B(x_1)) = [[0A]] + [[-I, -I, -I, -I, 0A]] * x_1 369.95/94.79 >>> 369.95/94.79 369.95/94.79 <<< 369.95/94.79 POL(b(x_1)) = [[0A], [0A], [0A], [0A], [-I]] + [[0A, 0A, 0A, 0A, 0A], [0A, -I, -I, -I, -I], [0A, 0A, 0A, -I, 0A], [-I, 0A, 0A, -I, 1A], [-I, 0A, -I, -I, 0A]] * x_1 369.95/94.79 >>> 369.95/94.79 369.95/94.79 <<< 369.95/94.79 POL(A(x_1)) = [[0A]] + [[-I, -I, 0A, 0A, 0A]] * x_1 369.95/94.79 >>> 369.95/94.79 369.95/94.79 <<< 369.95/94.79 POL(a(x_1)) = [[0A], [0A], [0A], [0A], [0A]] + [[0A, 0A, 0A, 0A, 0A], [0A, 0A, 0A, 0A, 0A], [0A, 0A, 0A, 0A, 0A], [0A, 0A, 0A, 0A, 0A], [0A, 0A, 0A, 0A, 0A]] * x_1 369.95/94.79 >>> 369.95/94.79 369.95/94.79 369.95/94.79 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 369.95/94.79 369.95/94.79 a(b(a(a(x1)))) -> b(b(b(b(x1)))) 369.95/94.79 b(b(b(b(x1)))) -> a(a(a(a(x1)))) 369.95/94.79 b(a(b(a(x1)))) -> a(a(b(a(x1)))) 369.95/94.79 369.95/94.79 369.95/94.79 ---------------------------------------- 369.95/94.79 369.95/94.79 (6) 369.95/94.79 Obligation: 369.95/94.79 Q DP problem: 369.95/94.79 The TRS P consists of the following rules: 369.95/94.79 369.95/94.79 B(b(b(b(x1)))) -> A(a(a(a(x1)))) 369.95/94.79 B(a(b(a(x1)))) -> A(a(b(a(x1)))) 369.95/94.79 369.95/94.79 The TRS R consists of the following rules: 369.95/94.79 369.95/94.79 b(b(b(b(x1)))) -> a(a(a(a(x1)))) 369.95/94.79 a(b(a(a(x1)))) -> b(b(b(b(x1)))) 369.95/94.79 b(a(b(a(x1)))) -> a(a(b(a(x1)))) 369.95/94.79 369.95/94.79 Q is empty. 369.95/94.79 We have to consider all minimal (P,Q,R)-chains. 369.95/94.79 ---------------------------------------- 369.95/94.79 369.95/94.79 (7) DependencyGraphProof (EQUIVALENT) 369.95/94.79 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 369.95/94.79 ---------------------------------------- 369.95/94.79 369.95/94.79 (8) 369.95/94.79 TRUE 370.30/94.93 EOF