28.64/8.15 YES 28.64/8.16 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 28.64/8.16 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 28.64/8.16 28.64/8.16 28.64/8.16 Termination w.r.t. Q of the given QTRS could be proven: 28.64/8.16 28.64/8.16 (0) QTRS 28.64/8.16 (1) QTRS Reverse [EQUIVALENT, 0 ms] 28.64/8.16 (2) QTRS 28.64/8.16 (3) DependencyPairsProof [EQUIVALENT, 3 ms] 28.64/8.16 (4) QDP 28.64/8.16 (5) QDPOrderProof [EQUIVALENT, 133 ms] 28.64/8.16 (6) QDP 28.64/8.16 (7) QDPOrderProof [EQUIVALENT, 38 ms] 28.64/8.16 (8) QDP 28.64/8.16 (9) PisEmptyProof [EQUIVALENT, 0 ms] 28.64/8.16 (10) YES 28.64/8.16 28.64/8.16 28.64/8.16 ---------------------------------------- 28.64/8.16 28.64/8.16 (0) 28.64/8.16 Obligation: 28.64/8.16 Q restricted rewrite system: 28.64/8.16 The TRS R consists of the following rules: 28.64/8.16 28.64/8.16 a(a(b(x1))) -> b(b(a(a(x1)))) 28.64/8.16 b(a(b(x1))) -> a(a(a(a(x1)))) 28.64/8.16 28.64/8.16 Q is empty. 28.64/8.16 28.64/8.16 ---------------------------------------- 28.64/8.16 28.64/8.16 (1) QTRS Reverse (EQUIVALENT) 28.64/8.16 We applied the QTRS Reverse Processor [REVERSE]. 28.64/8.16 ---------------------------------------- 28.64/8.16 28.64/8.16 (2) 28.64/8.16 Obligation: 28.64/8.16 Q restricted rewrite system: 28.64/8.16 The TRS R consists of the following rules: 28.64/8.16 28.64/8.16 b(a(a(x1))) -> a(a(b(b(x1)))) 28.64/8.16 b(a(b(x1))) -> a(a(a(a(x1)))) 28.64/8.16 28.64/8.16 Q is empty. 28.64/8.16 28.64/8.16 ---------------------------------------- 28.64/8.21 28.64/8.21 (3) DependencyPairsProof (EQUIVALENT) 28.64/8.21 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 28.64/8.21 ---------------------------------------- 28.64/8.21 28.64/8.21 (4) 28.64/8.21 Obligation: 28.64/8.21 Q DP problem: 28.64/8.21 The TRS P consists of the following rules: 28.64/8.21 28.64/8.21 B(a(a(x1))) -> B(b(x1)) 28.64/8.21 B(a(a(x1))) -> B(x1) 28.64/8.21 28.64/8.21 The TRS R consists of the following rules: 28.64/8.21 28.64/8.21 b(a(a(x1))) -> a(a(b(b(x1)))) 28.64/8.21 b(a(b(x1))) -> a(a(a(a(x1)))) 28.64/8.21 28.64/8.21 Q is empty. 28.64/8.21 We have to consider all minimal (P,Q,R)-chains. 28.64/8.21 ---------------------------------------- 28.64/8.21 28.64/8.21 (5) QDPOrderProof (EQUIVALENT) 28.64/8.21 We use the reduction pair processor [LPAR04,JAR06]. 28.64/8.21 28.64/8.21 28.64/8.21 The following pairs can be oriented strictly and are deleted. 28.64/8.21 28.64/8.21 B(a(a(x1))) -> B(x1) 28.64/8.21 The remaining pairs can at least be oriented weakly. 28.64/8.21 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 28.64/8.21 28.64/8.21 <<< 28.64/8.21 POL(B(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 28.64/8.21 >>> 28.64/8.21 28.64/8.21 <<< 28.64/8.21 POL(a(x_1)) = [[1A], [-I], [-I]] + [[-I, 0A, 1A], [0A, 0A, 0A], [0A, -I, -I]] * x_1 28.64/8.21 >>> 28.64/8.21 28.64/8.21 <<< 28.64/8.21 POL(b(x_1)) = [[0A], [-I], [1A]] + [[0A, -I, -I], [1A, 0A, 0A], [1A, 0A, 0A]] * x_1 28.64/8.21 >>> 28.64/8.21 28.64/8.21 28.64/8.21 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.64/8.21 28.64/8.21 b(a(a(x1))) -> a(a(b(b(x1)))) 28.64/8.21 b(a(b(x1))) -> a(a(a(a(x1)))) 28.64/8.21 28.64/8.21 28.64/8.21 ---------------------------------------- 28.64/8.21 28.64/8.21 (6) 28.64/8.21 Obligation: 28.64/8.21 Q DP problem: 28.64/8.21 The TRS P consists of the following rules: 28.64/8.21 28.64/8.21 B(a(a(x1))) -> B(b(x1)) 28.64/8.21 28.64/8.21 The TRS R consists of the following rules: 28.64/8.21 28.64/8.21 b(a(a(x1))) -> a(a(b(b(x1)))) 28.64/8.21 b(a(b(x1))) -> a(a(a(a(x1)))) 28.64/8.21 28.64/8.21 Q is empty. 28.64/8.21 We have to consider all minimal (P,Q,R)-chains. 28.64/8.21 ---------------------------------------- 28.64/8.21 28.64/8.21 (7) QDPOrderProof (EQUIVALENT) 28.64/8.21 We use the reduction pair processor [LPAR04,JAR06]. 28.64/8.21 28.64/8.21 28.64/8.21 The following pairs can be oriented strictly and are deleted. 28.64/8.21 28.64/8.21 B(a(a(x1))) -> B(b(x1)) 28.64/8.21 The remaining pairs can at least be oriented weakly. 28.64/8.21 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 28.64/8.21 28.64/8.21 <<< 28.64/8.21 POL(B(x_1)) = [[-I]] + [[0A, 0A, 1A]] * x_1 28.64/8.21 >>> 28.64/8.21 28.64/8.21 <<< 28.64/8.21 POL(a(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [0A, 0A, -I], [1A, 0A, -I]] * x_1 28.64/8.21 >>> 28.64/8.21 28.64/8.21 <<< 28.64/8.21 POL(b(x_1)) = [[1A], [0A], [-I]] + [[0A, 1A, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 28.64/8.21 >>> 28.64/8.21 28.64/8.21 28.64/8.21 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 28.64/8.21 28.64/8.21 b(a(a(x1))) -> a(a(b(b(x1)))) 28.64/8.21 b(a(b(x1))) -> a(a(a(a(x1)))) 28.64/8.21 28.64/8.21 28.64/8.21 ---------------------------------------- 28.64/8.21 28.64/8.21 (8) 28.64/8.21 Obligation: 28.64/8.21 Q DP problem: 28.64/8.21 P is empty. 28.64/8.21 The TRS R consists of the following rules: 28.64/8.21 28.64/8.21 b(a(a(x1))) -> a(a(b(b(x1)))) 28.64/8.21 b(a(b(x1))) -> a(a(a(a(x1)))) 28.64/8.21 28.64/8.21 Q is empty. 28.64/8.21 We have to consider all minimal (P,Q,R)-chains. 28.64/8.21 ---------------------------------------- 28.64/8.21 28.64/8.21 (9) PisEmptyProof (EQUIVALENT) 28.64/8.21 The TRS P is empty. Hence, there is no (P,Q,R) chain. 28.64/8.21 ---------------------------------------- 28.64/8.21 28.64/8.21 (10) 28.64/8.21 YES 28.96/10.19 EOF