38.09/10.73 YES 38.09/10.75 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 38.09/10.75 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 38.09/10.75 38.09/10.75 38.09/10.75 Termination w.r.t. Q of the given QTRS could be proven: 38.09/10.75 38.09/10.75 (0) QTRS 38.09/10.75 (1) QTRS Reverse [EQUIVALENT, 0 ms] 38.09/10.75 (2) QTRS 38.09/10.75 (3) DependencyPairsProof [EQUIVALENT, 5 ms] 38.09/10.75 (4) QDP 38.09/10.75 (5) QDPOrderProof [EQUIVALENT, 122 ms] 38.09/10.75 (6) QDP 38.09/10.75 (7) QDPOrderProof [EQUIVALENT, 0 ms] 38.09/10.75 (8) QDP 38.09/10.75 (9) QDPOrderProof [EQUIVALENT, 0 ms] 38.09/10.75 (10) QDP 38.09/10.75 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 38.09/10.75 (12) QDP 38.09/10.75 (13) QDPOrderProof [EQUIVALENT, 278 ms] 38.09/10.75 (14) QDP 38.09/10.75 (15) PisEmptyProof [EQUIVALENT, 0 ms] 38.09/10.75 (16) YES 38.09/10.75 38.09/10.75 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (0) 38.09/10.75 Obligation: 38.09/10.75 Q restricted rewrite system: 38.09/10.75 The TRS R consists of the following rules: 38.09/10.75 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 Q is empty. 38.09/10.75 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (1) QTRS Reverse (EQUIVALENT) 38.09/10.75 We applied the QTRS Reverse Processor [REVERSE]. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (2) 38.09/10.75 Obligation: 38.09/10.75 Q restricted rewrite system: 38.09/10.75 The TRS R consists of the following rules: 38.09/10.75 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 Q is empty. 38.09/10.75 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (3) DependencyPairsProof (EQUIVALENT) 38.09/10.75 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (4) 38.09/10.75 Obligation: 38.09/10.75 Q DP problem: 38.09/10.75 The TRS P consists of the following rules: 38.09/10.75 38.09/10.75 A(a(a(x1))) -> B(x1) 38.09/10.75 B(b(x1)) -> A(a(x1)) 38.09/10.75 B(b(x1)) -> A(x1) 38.09/10.75 A(a(x1)) -> A(b(a(x1))) 38.09/10.75 A(a(x1)) -> B(a(x1)) 38.09/10.75 38.09/10.75 The TRS R consists of the following rules: 38.09/10.75 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 Q is empty. 38.09/10.75 We have to consider all minimal (P,Q,R)-chains. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (5) QDPOrderProof (EQUIVALENT) 38.09/10.75 We use the reduction pair processor [LPAR04,JAR06]. 38.09/10.75 38.09/10.75 38.09/10.75 The following pairs can be oriented strictly and are deleted. 38.09/10.75 38.09/10.75 A(a(a(x1))) -> B(x1) 38.09/10.75 The remaining pairs can at least be oriented weakly. 38.09/10.75 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(A(x_1)) = [[0A]] + [[0A, -I, 0A]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(a(x_1)) = [[-I], [-I], [0A]] + [[1A, 0A, 0A], [0A, -I, -I], [1A, 0A, -I]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(B(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(b(x_1)) = [[-I], [0A], [0A]] + [[0A, 1A, -I], [1A, 0A, 0A], [0A, 1A, 0A]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 38.09/10.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 38.09/10.75 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (6) 38.09/10.75 Obligation: 38.09/10.75 Q DP problem: 38.09/10.75 The TRS P consists of the following rules: 38.09/10.75 38.09/10.75 B(b(x1)) -> A(a(x1)) 38.09/10.75 B(b(x1)) -> A(x1) 38.09/10.75 A(a(x1)) -> A(b(a(x1))) 38.09/10.75 A(a(x1)) -> B(a(x1)) 38.09/10.75 38.09/10.75 The TRS R consists of the following rules: 38.09/10.75 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 Q is empty. 38.09/10.75 We have to consider all minimal (P,Q,R)-chains. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (7) QDPOrderProof (EQUIVALENT) 38.09/10.75 We use the reduction pair processor [LPAR04,JAR06]. 38.09/10.75 38.09/10.75 38.09/10.75 The following pairs can be oriented strictly and are deleted. 38.09/10.75 38.09/10.75 B(b(x1)) -> A(x1) 38.09/10.75 The remaining pairs can at least be oriented weakly. 38.09/10.75 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(B(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(b(x_1)) = [[1A], [1A], [-I]] + [[0A, 1A, 1A], [1A, -I, 0A], [-I, -I, -I]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(A(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(a(x_1)) = [[0A], [1A], [-I]] + [[-I, 0A, 0A], [0A, 1A, 1A], [-I, -I, -I]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 38.09/10.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 38.09/10.75 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (8) 38.09/10.75 Obligation: 38.09/10.75 Q DP problem: 38.09/10.75 The TRS P consists of the following rules: 38.09/10.75 38.09/10.75 B(b(x1)) -> A(a(x1)) 38.09/10.75 A(a(x1)) -> A(b(a(x1))) 38.09/10.75 A(a(x1)) -> B(a(x1)) 38.09/10.75 38.09/10.75 The TRS R consists of the following rules: 38.09/10.75 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 Q is empty. 38.09/10.75 We have to consider all minimal (P,Q,R)-chains. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (9) QDPOrderProof (EQUIVALENT) 38.09/10.75 We use the reduction pair processor [LPAR04,JAR06]. 38.09/10.75 38.09/10.75 38.09/10.75 The following pairs can be oriented strictly and are deleted. 38.09/10.75 38.09/10.75 A(a(x1)) -> B(a(x1)) 38.09/10.75 The remaining pairs can at least be oriented weakly. 38.09/10.75 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(B(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(b(x_1)) = [[0A], [1A], [0A]] + [[0A, 1A, 0A], [1A, 0A, 0A], [-I, 0A, -I]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(A(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(a(x_1)) = [[1A], [0A], [0A]] + [[1A, 0A, 0A], [0A, -I, -I], [0A, -I, 0A]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 38.09/10.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 38.09/10.75 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (10) 38.09/10.75 Obligation: 38.09/10.75 Q DP problem: 38.09/10.75 The TRS P consists of the following rules: 38.09/10.75 38.09/10.75 B(b(x1)) -> A(a(x1)) 38.09/10.75 A(a(x1)) -> A(b(a(x1))) 38.09/10.75 38.09/10.75 The TRS R consists of the following rules: 38.09/10.75 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 Q is empty. 38.09/10.75 We have to consider all minimal (P,Q,R)-chains. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (11) DependencyGraphProof (EQUIVALENT) 38.09/10.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (12) 38.09/10.75 Obligation: 38.09/10.75 Q DP problem: 38.09/10.75 The TRS P consists of the following rules: 38.09/10.75 38.09/10.75 A(a(x1)) -> A(b(a(x1))) 38.09/10.75 38.09/10.75 The TRS R consists of the following rules: 38.09/10.75 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 Q is empty. 38.09/10.75 We have to consider all minimal (P,Q,R)-chains. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (13) QDPOrderProof (EQUIVALENT) 38.09/10.75 We use the reduction pair processor [LPAR04,JAR06]. 38.09/10.75 38.09/10.75 38.09/10.75 The following pairs can be oriented strictly and are deleted. 38.09/10.75 38.09/10.75 A(a(x1)) -> A(b(a(x1))) 38.09/10.75 The remaining pairs can at least be oriented weakly. 38.09/10.75 Used ordering: Matrix interpretation [MATRO] with arctic integers [ARCTIC,STERNAGEL_THIEMANN_RTA14]: 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(A(x_1)) = [[0A]] + [[-1A, -1A, 2A]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(a(x_1)) = [[0A], [-1A], [2A]] + [[-I, -1A, -1A], [-I, 0A, -I], [-1A, 2A, 1A]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 <<< 38.09/10.75 POL(b(x_1)) = [[2A], [-1A], [-I]] + [[-I, 2A, 1A], [-I, 0A, -I], [1A, -1A, -1A]] * x_1 38.09/10.75 >>> 38.09/10.75 38.09/10.75 38.09/10.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 38.09/10.75 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (14) 38.09/10.75 Obligation: 38.09/10.75 Q DP problem: 38.09/10.75 P is empty. 38.09/10.75 The TRS R consists of the following rules: 38.09/10.75 38.09/10.75 a(a(a(x1))) -> b(x1) 38.09/10.75 b(b(x1)) -> a(a(x1)) 38.09/10.75 a(a(x1)) -> a(b(a(x1))) 38.09/10.75 38.09/10.75 Q is empty. 38.09/10.75 We have to consider all minimal (P,Q,R)-chains. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (15) PisEmptyProof (EQUIVALENT) 38.09/10.75 The TRS P is empty. Hence, there is no (P,Q,R) chain. 38.09/10.75 ---------------------------------------- 38.09/10.75 38.09/10.75 (16) 38.09/10.75 YES 38.27/10.80 EOF