23.44/6.78 YES 25.24/7.26 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 25.24/7.26 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.24/7.26 25.24/7.26 25.24/7.26 Termination w.r.t. Q of the given QTRS could be proven: 25.24/7.26 25.24/7.26 (0) QTRS 25.24/7.26 (1) QTRS Reverse [EQUIVALENT, 0 ms] 25.24/7.26 (2) QTRS 25.24/7.26 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 25.24/7.26 (4) QDP 25.24/7.26 (5) DependencyGraphProof [EQUIVALENT, 2 ms] 25.24/7.26 (6) QDP 25.24/7.26 (7) QDPOrderProof [EQUIVALENT, 230 ms] 25.24/7.26 (8) QDP 25.24/7.26 (9) QDPOrderProof [EQUIVALENT, 98 ms] 25.24/7.26 (10) QDP 25.24/7.26 (11) PisEmptyProof [EQUIVALENT, 0 ms] 25.24/7.26 (12) YES 25.24/7.26 25.24/7.26 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (0) 25.24/7.26 Obligation: 25.24/7.26 Q restricted rewrite system: 25.24/7.26 The TRS R consists of the following rules: 25.24/7.26 25.24/7.26 a(a(a(b(x1)))) -> b(a(b(a(a(a(x1)))))) 25.24/7.26 b(a(x1)) -> x1 25.24/7.26 25.24/7.26 Q is empty. 25.24/7.26 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (1) QTRS Reverse (EQUIVALENT) 25.24/7.26 We applied the QTRS Reverse Processor [REVERSE]. 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (2) 25.24/7.26 Obligation: 25.24/7.26 Q restricted rewrite system: 25.24/7.26 The TRS R consists of the following rules: 25.24/7.26 25.24/7.26 b(a(a(a(x1)))) -> a(a(a(b(a(b(x1)))))) 25.24/7.26 a(b(x1)) -> x1 25.24/7.26 25.24/7.26 Q is empty. 25.24/7.26 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (3) DependencyPairsProof (EQUIVALENT) 25.24/7.26 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (4) 25.24/7.26 Obligation: 25.24/7.26 Q DP problem: 25.24/7.26 The TRS P consists of the following rules: 25.24/7.26 25.24/7.26 B(a(a(a(x1)))) -> A(a(a(b(a(b(x1)))))) 25.24/7.26 B(a(a(a(x1)))) -> A(a(b(a(b(x1))))) 25.24/7.26 B(a(a(a(x1)))) -> A(b(a(b(x1)))) 25.24/7.26 B(a(a(a(x1)))) -> B(a(b(x1))) 25.24/7.26 B(a(a(a(x1)))) -> A(b(x1)) 25.24/7.26 B(a(a(a(x1)))) -> B(x1) 25.24/7.26 25.24/7.26 The TRS R consists of the following rules: 25.24/7.26 25.24/7.26 b(a(a(a(x1)))) -> a(a(a(b(a(b(x1)))))) 25.24/7.26 a(b(x1)) -> x1 25.24/7.26 25.24/7.26 Q is empty. 25.24/7.26 We have to consider all minimal (P,Q,R)-chains. 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (5) DependencyGraphProof (EQUIVALENT) 25.24/7.26 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (6) 25.24/7.26 Obligation: 25.24/7.26 Q DP problem: 25.24/7.26 The TRS P consists of the following rules: 25.24/7.26 25.24/7.26 B(a(a(a(x1)))) -> B(x1) 25.24/7.26 B(a(a(a(x1)))) -> B(a(b(x1))) 25.24/7.26 25.24/7.26 The TRS R consists of the following rules: 25.24/7.26 25.24/7.26 b(a(a(a(x1)))) -> a(a(a(b(a(b(x1)))))) 25.24/7.26 a(b(x1)) -> x1 25.24/7.26 25.24/7.26 Q is empty. 25.24/7.26 We have to consider all minimal (P,Q,R)-chains. 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (7) QDPOrderProof (EQUIVALENT) 25.24/7.26 We use the reduction pair processor [LPAR04,JAR06]. 25.24/7.26 25.24/7.26 25.24/7.26 The following pairs can be oriented strictly and are deleted. 25.24/7.26 25.24/7.26 B(a(a(a(x1)))) -> B(x1) 25.24/7.26 The remaining pairs can at least be oriented weakly. 25.24/7.26 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 25.24/7.26 25.24/7.26 <<< 25.24/7.26 POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 25.24/7.26 >>> 25.24/7.26 25.24/7.26 <<< 25.24/7.26 POL(a(x_1)) = [[1A], [0A], [0A]] + [[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 25.24/7.26 >>> 25.24/7.26 25.24/7.26 <<< 25.24/7.26 POL(b(x_1)) = [[1A], [0A], [0A]] + [[0A, 1A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 25.24/7.26 >>> 25.24/7.26 25.24/7.26 25.24/7.26 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.24/7.26 25.24/7.26 b(a(a(a(x1)))) -> a(a(a(b(a(b(x1)))))) 25.24/7.26 a(b(x1)) -> x1 25.24/7.26 25.24/7.26 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (8) 25.24/7.26 Obligation: 25.24/7.26 Q DP problem: 25.24/7.26 The TRS P consists of the following rules: 25.24/7.26 25.24/7.26 B(a(a(a(x1)))) -> B(a(b(x1))) 25.24/7.26 25.24/7.26 The TRS R consists of the following rules: 25.24/7.26 25.24/7.26 b(a(a(a(x1)))) -> a(a(a(b(a(b(x1)))))) 25.24/7.26 a(b(x1)) -> x1 25.24/7.26 25.24/7.26 Q is empty. 25.24/7.26 We have to consider all minimal (P,Q,R)-chains. 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (9) QDPOrderProof (EQUIVALENT) 25.24/7.26 We use the reduction pair processor [LPAR04,JAR06]. 25.24/7.26 25.24/7.26 25.24/7.26 The following pairs can be oriented strictly and are deleted. 25.24/7.26 25.24/7.26 B(a(a(a(x1)))) -> B(a(b(x1))) 25.24/7.26 The remaining pairs can at least be oriented weakly. 25.24/7.26 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 25.24/7.26 25.24/7.26 <<< 25.24/7.26 POL(B(x_1)) = [[-I]] + [[0A, 1A, 0A]] * x_1 25.24/7.26 >>> 25.24/7.26 25.24/7.26 <<< 25.24/7.26 POL(a(x_1)) = [[0A], [1A], [1A]] + [[-I, 0A, 1A], [0A, -I, 0A], [-I, 0A, -I]] * x_1 25.24/7.26 >>> 25.24/7.26 25.24/7.26 <<< 25.24/7.26 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 25.24/7.26 >>> 25.24/7.26 25.24/7.26 25.24/7.26 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.24/7.26 25.24/7.26 b(a(a(a(x1)))) -> a(a(a(b(a(b(x1)))))) 25.24/7.26 a(b(x1)) -> x1 25.24/7.26 25.24/7.26 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (10) 25.24/7.26 Obligation: 25.24/7.26 Q DP problem: 25.24/7.26 P is empty. 25.24/7.26 The TRS R consists of the following rules: 25.24/7.26 25.24/7.26 b(a(a(a(x1)))) -> a(a(a(b(a(b(x1)))))) 25.24/7.26 a(b(x1)) -> x1 25.24/7.26 25.24/7.26 Q is empty. 25.24/7.26 We have to consider all minimal (P,Q,R)-chains. 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (11) PisEmptyProof (EQUIVALENT) 25.24/7.26 The TRS P is empty. Hence, there is no (P,Q,R) chain. 25.24/7.26 ---------------------------------------- 25.24/7.26 25.24/7.26 (12) 25.24/7.26 YES 25.57/7.39 EOF