29.93/8.85 YES 30.69/8.99 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 30.69/8.99 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.69/8.99 30.69/8.99 30.69/8.99 Termination w.r.t. Q of the given QTRS could be proven: 30.69/8.99 30.69/8.99 (0) QTRS 30.69/8.99 (1) QTRS Reverse [EQUIVALENT, 0 ms] 30.69/8.99 (2) QTRS 30.69/8.99 (3) DependencyPairsProof [EQUIVALENT, 23 ms] 30.69/8.99 (4) QDP 30.69/8.99 (5) MRRProof [EQUIVALENT, 89 ms] 30.69/8.99 (6) QDP 30.69/8.99 (7) QDPOrderProof [EQUIVALENT, 546 ms] 30.69/8.99 (8) QDP 30.69/8.99 (9) QDPOrderProof [EQUIVALENT, 62 ms] 30.69/8.99 (10) QDP 30.69/8.99 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 30.69/8.99 (12) TRUE 30.69/8.99 30.69/8.99 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (0) 30.69/8.99 Obligation: 30.69/8.99 Q restricted rewrite system: 30.69/8.99 The TRS R consists of the following rules: 30.69/8.99 30.69/8.99 b(a(a(a(x1)))) -> b(b(a(b(x1)))) 30.69/8.99 b(b(b(a(x1)))) -> b(b(a(a(x1)))) 30.69/8.99 a(a(b(b(x1)))) -> a(a(a(a(x1)))) 30.69/8.99 30.69/8.99 Q is empty. 30.69/8.99 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (1) QTRS Reverse (EQUIVALENT) 30.69/8.99 We applied the QTRS Reverse Processor [REVERSE]. 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (2) 30.69/8.99 Obligation: 30.69/8.99 Q restricted rewrite system: 30.69/8.99 The TRS R consists of the following rules: 30.69/8.99 30.69/8.99 a(a(a(b(x1)))) -> b(a(b(b(x1)))) 30.69/8.99 a(b(b(b(x1)))) -> a(a(b(b(x1)))) 30.69/8.99 b(b(a(a(x1)))) -> a(a(a(a(x1)))) 30.69/8.99 30.69/8.99 Q is empty. 30.69/8.99 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (3) DependencyPairsProof (EQUIVALENT) 30.69/8.99 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (4) 30.69/8.99 Obligation: 30.69/8.99 Q DP problem: 30.69/8.99 The TRS P consists of the following rules: 30.69/8.99 30.69/8.99 A(a(a(b(x1)))) -> B(a(b(b(x1)))) 30.69/8.99 A(a(a(b(x1)))) -> A(b(b(x1))) 30.69/8.99 A(a(a(b(x1)))) -> B(b(x1)) 30.69/8.99 A(b(b(b(x1)))) -> A(a(b(b(x1)))) 30.69/8.99 A(b(b(b(x1)))) -> A(b(b(x1))) 30.69/8.99 B(b(a(a(x1)))) -> A(a(a(a(x1)))) 30.69/8.99 B(b(a(a(x1)))) -> A(a(a(x1))) 30.69/8.99 30.69/8.99 The TRS R consists of the following rules: 30.69/8.99 30.69/8.99 a(a(a(b(x1)))) -> b(a(b(b(x1)))) 30.69/8.99 a(b(b(b(x1)))) -> a(a(b(b(x1)))) 30.69/8.99 b(b(a(a(x1)))) -> a(a(a(a(x1)))) 30.69/8.99 30.69/8.99 Q is empty. 30.69/8.99 We have to consider all minimal (P,Q,R)-chains. 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (5) MRRProof (EQUIVALENT) 30.69/8.99 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 30.69/8.99 30.69/8.99 Strictly oriented dependency pairs: 30.69/8.99 30.69/8.99 A(a(a(b(x1)))) -> A(b(b(x1))) 30.69/8.99 A(a(a(b(x1)))) -> B(b(x1)) 30.69/8.99 A(b(b(b(x1)))) -> A(b(b(x1))) 30.69/8.99 B(b(a(a(x1)))) -> A(a(a(x1))) 30.69/8.99 30.69/8.99 30.69/8.99 Used ordering: Polynomial interpretation [POLO]: 30.69/8.99 30.69/8.99 POL(A(x_1)) = 2*x_1 30.69/8.99 POL(B(x_1)) = 2*x_1 30.69/8.99 POL(a(x_1)) = 2 + x_1 30.69/8.99 POL(b(x_1)) = 2 + x_1 30.69/8.99 30.69/8.99 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (6) 30.69/8.99 Obligation: 30.69/8.99 Q DP problem: 30.69/8.99 The TRS P consists of the following rules: 30.69/8.99 30.69/8.99 A(a(a(b(x1)))) -> B(a(b(b(x1)))) 30.69/8.99 A(b(b(b(x1)))) -> A(a(b(b(x1)))) 30.69/8.99 B(b(a(a(x1)))) -> A(a(a(a(x1)))) 30.69/8.99 30.69/8.99 The TRS R consists of the following rules: 30.69/8.99 30.69/8.99 a(a(a(b(x1)))) -> b(a(b(b(x1)))) 30.69/8.99 a(b(b(b(x1)))) -> a(a(b(b(x1)))) 30.69/8.99 b(b(a(a(x1)))) -> a(a(a(a(x1)))) 30.69/8.99 30.69/8.99 Q is empty. 30.69/8.99 We have to consider all minimal (P,Q,R)-chains. 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (7) QDPOrderProof (EQUIVALENT) 30.69/8.99 We use the reduction pair processor [LPAR04,JAR06]. 30.69/8.99 30.69/8.99 30.69/8.99 The following pairs can be oriented strictly and are deleted. 30.69/8.99 30.69/8.99 A(b(b(b(x1)))) -> A(a(b(b(x1)))) 30.69/8.99 The remaining pairs can at least be oriented weakly. 30.69/8.99 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 30.69/8.99 30.69/8.99 <<< 30.69/8.99 POL(A(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 30.69/8.99 >>> 30.69/8.99 30.69/8.99 <<< 30.69/8.99 POL(a(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, 1A], [1A, -I, -I]] * x_1 30.69/8.99 >>> 30.69/8.99 30.69/8.99 <<< 30.69/8.99 POL(b(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [0A, 1A, 1A], [-I, 0A, 0A]] * x_1 30.69/8.99 >>> 30.69/8.99 30.69/8.99 <<< 30.69/8.99 POL(B(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 30.69/8.99 >>> 30.69/8.99 30.69/8.99 30.69/8.99 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 30.69/8.99 30.69/8.99 b(b(a(a(x1)))) -> a(a(a(a(x1)))) 30.69/8.99 a(a(a(b(x1)))) -> b(a(b(b(x1)))) 30.69/8.99 a(b(b(b(x1)))) -> a(a(b(b(x1)))) 30.69/8.99 30.69/8.99 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (8) 30.69/8.99 Obligation: 30.69/8.99 Q DP problem: 30.69/8.99 The TRS P consists of the following rules: 30.69/8.99 30.69/8.99 A(a(a(b(x1)))) -> B(a(b(b(x1)))) 30.69/8.99 B(b(a(a(x1)))) -> A(a(a(a(x1)))) 30.69/8.99 30.69/8.99 The TRS R consists of the following rules: 30.69/8.99 30.69/8.99 a(a(a(b(x1)))) -> b(a(b(b(x1)))) 30.69/8.99 a(b(b(b(x1)))) -> a(a(b(b(x1)))) 30.69/8.99 b(b(a(a(x1)))) -> a(a(a(a(x1)))) 30.69/8.99 30.69/8.99 Q is empty. 30.69/8.99 We have to consider all minimal (P,Q,R)-chains. 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (9) QDPOrderProof (EQUIVALENT) 30.69/8.99 We use the reduction pair processor [LPAR04,JAR06]. 30.69/8.99 30.69/8.99 30.69/8.99 The following pairs can be oriented strictly and are deleted. 30.69/8.99 30.69/8.99 B(b(a(a(x1)))) -> A(a(a(a(x1)))) 30.69/8.99 The remaining pairs can at least be oriented weakly. 30.69/8.99 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 30.69/8.99 30.69/8.99 <<< 30.69/8.99 POL(A(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 30.69/8.99 >>> 30.69/8.99 30.69/8.99 <<< 30.69/8.99 POL(a(x_1)) = [[0A], [-I], [0A]] + [[0A, 0A, 1A], [0A, -I, -I], [-I, 0A, 0A]] * x_1 30.69/8.99 >>> 30.69/8.99 30.69/8.99 <<< 30.69/8.99 POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 1A, 0A], [-I, 0A, -I]] * x_1 30.69/8.99 >>> 30.69/8.99 30.69/8.99 <<< 30.69/8.99 POL(B(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 30.69/8.99 >>> 30.69/8.99 30.69/8.99 30.69/8.99 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 30.69/8.99 30.69/8.99 b(b(a(a(x1)))) -> a(a(a(a(x1)))) 30.69/8.99 a(a(a(b(x1)))) -> b(a(b(b(x1)))) 30.69/8.99 a(b(b(b(x1)))) -> a(a(b(b(x1)))) 30.69/8.99 30.69/8.99 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (10) 30.69/8.99 Obligation: 30.69/8.99 Q DP problem: 30.69/8.99 The TRS P consists of the following rules: 30.69/8.99 30.69/8.99 A(a(a(b(x1)))) -> B(a(b(b(x1)))) 30.69/8.99 30.69/8.99 The TRS R consists of the following rules: 30.69/8.99 30.69/8.99 a(a(a(b(x1)))) -> b(a(b(b(x1)))) 30.69/8.99 a(b(b(b(x1)))) -> a(a(b(b(x1)))) 30.69/8.99 b(b(a(a(x1)))) -> a(a(a(a(x1)))) 30.69/8.99 30.69/8.99 Q is empty. 30.69/8.99 We have to consider all minimal (P,Q,R)-chains. 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (11) DependencyGraphProof (EQUIVALENT) 30.69/8.99 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 30.69/8.99 ---------------------------------------- 30.69/8.99 30.69/8.99 (12) 30.69/8.99 TRUE 31.08/9.10 EOF