26.07/7.51 YES 28.49/8.28 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 28.49/8.28 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.49/8.28 28.49/8.28 28.49/8.28 Termination w.r.t. Q of the given QTRS could be proven: 28.49/8.28 28.49/8.28 (0) QTRS 28.49/8.28 (1) QTRS Reverse [EQUIVALENT, 0 ms] 28.49/8.28 (2) QTRS 28.49/8.28 (3) DependencyPairsProof [EQUIVALENT, 10 ms] 28.49/8.28 (4) QDP 28.49/8.28 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 28.49/8.28 (6) QDP 28.49/8.28 (7) QDPOrderProof [EQUIVALENT, 35 ms] 28.49/8.28 (8) QDP 28.49/8.28 (9) QDPOrderProof [EQUIVALENT, 65 ms] 28.49/8.28 (10) QDP 28.49/8.28 (11) PisEmptyProof [EQUIVALENT, 0 ms] 28.49/8.28 (12) YES 28.49/8.28 28.49/8.28 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (0) 28.49/8.28 Obligation: 28.49/8.28 Q restricted rewrite system: 28.49/8.28 The TRS R consists of the following rules: 28.49/8.28 28.49/8.28 a(a(a(b(x1)))) -> b(a(a(a(x1)))) 28.49/8.28 b(a(b(a(x1)))) -> a(b(b(a(x1)))) 28.49/8.28 28.49/8.28 Q is empty. 28.49/8.28 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (1) QTRS Reverse (EQUIVALENT) 28.49/8.28 We applied the QTRS Reverse Processor [REVERSE]. 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (2) 28.49/8.28 Obligation: 28.49/8.28 Q restricted rewrite system: 28.49/8.28 The TRS R consists of the following rules: 28.49/8.28 28.49/8.28 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 28.49/8.28 a(b(a(b(x1)))) -> a(b(b(a(x1)))) 28.49/8.28 28.49/8.28 Q is empty. 28.49/8.28 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (3) DependencyPairsProof (EQUIVALENT) 28.49/8.28 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (4) 28.49/8.28 Obligation: 28.49/8.28 Q DP problem: 28.49/8.28 The TRS P consists of the following rules: 28.49/8.28 28.49/8.28 B(a(a(a(x1)))) -> A(a(a(b(x1)))) 28.49/8.28 B(a(a(a(x1)))) -> A(a(b(x1))) 28.49/8.28 B(a(a(a(x1)))) -> A(b(x1)) 28.49/8.28 B(a(a(a(x1)))) -> B(x1) 28.49/8.28 A(b(a(b(x1)))) -> A(b(b(a(x1)))) 28.49/8.28 A(b(a(b(x1)))) -> B(b(a(x1))) 28.49/8.28 A(b(a(b(x1)))) -> B(a(x1)) 28.49/8.28 A(b(a(b(x1)))) -> A(x1) 28.49/8.28 28.49/8.28 The TRS R consists of the following rules: 28.49/8.28 28.49/8.28 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 28.49/8.28 a(b(a(b(x1)))) -> a(b(b(a(x1)))) 28.49/8.28 28.49/8.28 Q is empty. 28.49/8.28 We have to consider all minimal (P,Q,R)-chains. 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (5) DependencyGraphProof (EQUIVALENT) 28.49/8.28 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (6) 28.49/8.28 Obligation: 28.49/8.28 Q DP problem: 28.49/8.28 The TRS P consists of the following rules: 28.49/8.28 28.49/8.28 B(a(a(a(x1)))) -> A(b(x1)) 28.49/8.28 A(b(a(b(x1)))) -> A(b(b(a(x1)))) 28.49/8.28 A(b(a(b(x1)))) -> B(b(a(x1))) 28.49/8.28 B(a(a(a(x1)))) -> B(x1) 28.49/8.28 A(b(a(b(x1)))) -> B(a(x1)) 28.49/8.28 A(b(a(b(x1)))) -> A(x1) 28.49/8.28 28.49/8.28 The TRS R consists of the following rules: 28.49/8.28 28.49/8.28 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 28.49/8.28 a(b(a(b(x1)))) -> a(b(b(a(x1)))) 28.49/8.28 28.49/8.28 Q is empty. 28.49/8.28 We have to consider all minimal (P,Q,R)-chains. 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (7) QDPOrderProof (EQUIVALENT) 28.49/8.28 We use the reduction pair processor [LPAR04,JAR06]. 28.49/8.28 28.49/8.28 28.49/8.28 The following pairs can be oriented strictly and are deleted. 28.49/8.28 28.49/8.28 B(a(a(a(x1)))) -> A(b(x1)) 28.49/8.28 A(b(a(b(x1)))) -> B(b(a(x1))) 28.49/8.28 B(a(a(a(x1)))) -> B(x1) 28.49/8.28 A(b(a(b(x1)))) -> B(a(x1)) 28.49/8.28 A(b(a(b(x1)))) -> A(x1) 28.49/8.28 The remaining pairs can at least be oriented weakly. 28.49/8.28 Used ordering: Polynomial interpretation [POLO]: 28.49/8.28 28.49/8.28 POL(A(x_1)) = 1 + x_1 28.49/8.28 POL(B(x_1)) = x_1 28.49/8.28 POL(a(x_1)) = 1 + x_1 28.49/8.28 POL(b(x_1)) = 1 + x_1 28.49/8.28 28.49/8.28 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.49/8.28 28.49/8.28 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 28.49/8.28 a(b(a(b(x1)))) -> a(b(b(a(x1)))) 28.49/8.28 28.49/8.28 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (8) 28.49/8.28 Obligation: 28.49/8.28 Q DP problem: 28.49/8.28 The TRS P consists of the following rules: 28.49/8.28 28.49/8.28 A(b(a(b(x1)))) -> A(b(b(a(x1)))) 28.49/8.28 28.49/8.28 The TRS R consists of the following rules: 28.49/8.28 28.49/8.28 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 28.49/8.28 a(b(a(b(x1)))) -> a(b(b(a(x1)))) 28.49/8.28 28.49/8.28 Q is empty. 28.49/8.28 We have to consider all minimal (P,Q,R)-chains. 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (9) QDPOrderProof (EQUIVALENT) 28.49/8.28 We use the reduction pair processor [LPAR04,JAR06]. 28.49/8.28 28.49/8.28 28.49/8.28 The following pairs can be oriented strictly and are deleted. 28.49/8.28 28.49/8.28 A(b(a(b(x1)))) -> A(b(b(a(x1)))) 28.49/8.28 The remaining pairs can at least be oriented weakly. 28.49/8.28 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 28.49/8.28 28.49/8.28 <<< 28.49/8.28 POL(A(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 28.49/8.28 >>> 28.49/8.28 28.49/8.28 <<< 28.49/8.28 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [0A, -I, 0A], [-I, -I, 0A]] * x_1 28.49/8.28 >>> 28.49/8.28 28.49/8.28 <<< 28.49/8.28 POL(a(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, -I, 1A], [-I, -I, -I]] * x_1 28.49/8.28 >>> 28.49/8.28 28.49/8.28 28.49/8.28 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.49/8.28 28.49/8.28 a(b(a(b(x1)))) -> a(b(b(a(x1)))) 28.49/8.28 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 28.49/8.28 28.49/8.28 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (10) 28.49/8.28 Obligation: 28.49/8.28 Q DP problem: 28.49/8.28 P is empty. 28.49/8.28 The TRS R consists of the following rules: 28.49/8.28 28.49/8.28 b(a(a(a(x1)))) -> a(a(a(b(x1)))) 28.49/8.28 a(b(a(b(x1)))) -> a(b(b(a(x1)))) 28.49/8.28 28.49/8.28 Q is empty. 28.49/8.28 We have to consider all minimal (P,Q,R)-chains. 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (11) PisEmptyProof (EQUIVALENT) 28.49/8.28 The TRS P is empty. Hence, there is no (P,Q,R) chain. 28.49/8.28 ---------------------------------------- 28.49/8.28 28.49/8.28 (12) 28.49/8.28 YES 28.68/8.37 EOF