29.47/8.50 YES 29.75/8.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 29.75/8.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.75/8.51 29.75/8.51 29.75/8.51 Termination w.r.t. Q of the given QTRS could be proven: 29.75/8.51 29.75/8.51 (0) QTRS 29.75/8.51 (1) QTRS Reverse [EQUIVALENT, 0 ms] 29.75/8.51 (2) QTRS 29.75/8.51 (3) DependencyPairsProof [EQUIVALENT, 3 ms] 29.75/8.51 (4) QDP 29.75/8.51 (5) MRRProof [EQUIVALENT, 100 ms] 29.75/8.51 (6) QDP 29.75/8.51 (7) DependencyGraphProof [EQUIVALENT, 5 ms] 29.75/8.51 (8) QDP 29.75/8.51 (9) QDPOrderProof [EQUIVALENT, 89 ms] 29.75/8.51 (10) QDP 29.75/8.51 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 29.75/8.51 (12) TRUE 29.75/8.51 29.75/8.51 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (0) 29.75/8.51 Obligation: 29.75/8.51 Q restricted rewrite system: 29.75/8.51 The TRS R consists of the following rules: 29.75/8.51 29.75/8.51 a(b(a(b(x1)))) -> b(b(b(b(x1)))) 29.75/8.51 a(a(b(b(x1)))) -> a(a(a(b(x1)))) 29.75/8.51 b(a(b(a(x1)))) -> a(b(a(b(x1)))) 29.75/8.51 29.75/8.51 Q is empty. 29.75/8.51 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (1) QTRS Reverse (EQUIVALENT) 29.75/8.51 We applied the QTRS Reverse Processor [REVERSE]. 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (2) 29.75/8.51 Obligation: 29.75/8.51 Q restricted rewrite system: 29.75/8.51 The TRS R consists of the following rules: 29.75/8.51 29.75/8.51 b(a(b(a(x1)))) -> b(b(b(b(x1)))) 29.75/8.51 b(b(a(a(x1)))) -> b(a(a(a(x1)))) 29.75/8.51 a(b(a(b(x1)))) -> b(a(b(a(x1)))) 29.75/8.51 29.75/8.51 Q is empty. 29.75/8.51 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (3) DependencyPairsProof (EQUIVALENT) 29.75/8.51 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (4) 29.75/8.51 Obligation: 29.75/8.51 Q DP problem: 29.75/8.51 The TRS P consists of the following rules: 29.75/8.51 29.75/8.51 B(a(b(a(x1)))) -> B(b(b(b(x1)))) 29.75/8.51 B(a(b(a(x1)))) -> B(b(b(x1))) 29.75/8.51 B(a(b(a(x1)))) -> B(b(x1)) 29.75/8.51 B(a(b(a(x1)))) -> B(x1) 29.75/8.51 B(b(a(a(x1)))) -> B(a(a(a(x1)))) 29.75/8.51 B(b(a(a(x1)))) -> A(a(a(x1))) 29.75/8.51 A(b(a(b(x1)))) -> B(a(b(a(x1)))) 29.75/8.51 A(b(a(b(x1)))) -> A(b(a(x1))) 29.75/8.51 A(b(a(b(x1)))) -> B(a(x1)) 29.75/8.51 A(b(a(b(x1)))) -> A(x1) 29.75/8.51 29.75/8.51 The TRS R consists of the following rules: 29.75/8.51 29.75/8.51 b(a(b(a(x1)))) -> b(b(b(b(x1)))) 29.75/8.51 b(b(a(a(x1)))) -> b(a(a(a(x1)))) 29.75/8.51 a(b(a(b(x1)))) -> b(a(b(a(x1)))) 29.75/8.51 29.75/8.51 Q is empty. 29.75/8.51 We have to consider all minimal (P,Q,R)-chains. 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (5) MRRProof (EQUIVALENT) 29.75/8.51 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. 29.75/8.51 29.75/8.51 Strictly oriented dependency pairs: 29.75/8.51 29.75/8.51 B(a(b(a(x1)))) -> B(b(b(x1))) 29.75/8.51 B(a(b(a(x1)))) -> B(b(x1)) 29.75/8.51 B(a(b(a(x1)))) -> B(x1) 29.75/8.51 B(b(a(a(x1)))) -> A(a(a(x1))) 29.75/8.51 A(b(a(b(x1)))) -> A(b(a(x1))) 29.75/8.51 A(b(a(b(x1)))) -> B(a(x1)) 29.75/8.51 A(b(a(b(x1)))) -> A(x1) 29.75/8.51 29.75/8.51 29.75/8.51 Used ordering: Polynomial interpretation [POLO]: 29.75/8.51 29.75/8.51 POL(A(x_1)) = 2 + 3*x_1 29.75/8.51 POL(B(x_1)) = 2 + 3*x_1 29.75/8.51 POL(a(x_1)) = 2 + x_1 29.75/8.51 POL(b(x_1)) = 2 + x_1 29.75/8.51 29.75/8.51 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (6) 29.75/8.51 Obligation: 29.75/8.51 Q DP problem: 29.75/8.51 The TRS P consists of the following rules: 29.75/8.51 29.75/8.51 B(a(b(a(x1)))) -> B(b(b(b(x1)))) 29.75/8.51 B(b(a(a(x1)))) -> B(a(a(a(x1)))) 29.75/8.51 A(b(a(b(x1)))) -> B(a(b(a(x1)))) 29.75/8.51 29.75/8.51 The TRS R consists of the following rules: 29.75/8.51 29.75/8.51 b(a(b(a(x1)))) -> b(b(b(b(x1)))) 29.75/8.51 b(b(a(a(x1)))) -> b(a(a(a(x1)))) 29.75/8.51 a(b(a(b(x1)))) -> b(a(b(a(x1)))) 29.75/8.51 29.75/8.51 Q is empty. 29.75/8.51 We have to consider all minimal (P,Q,R)-chains. 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (7) DependencyGraphProof (EQUIVALENT) 29.75/8.51 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (8) 29.75/8.51 Obligation: 29.75/8.51 Q DP problem: 29.75/8.51 The TRS P consists of the following rules: 29.75/8.51 29.75/8.51 B(b(a(a(x1)))) -> B(a(a(a(x1)))) 29.75/8.51 B(a(b(a(x1)))) -> B(b(b(b(x1)))) 29.75/8.51 29.75/8.51 The TRS R consists of the following rules: 29.75/8.51 29.75/8.51 b(a(b(a(x1)))) -> b(b(b(b(x1)))) 29.75/8.51 b(b(a(a(x1)))) -> b(a(a(a(x1)))) 29.75/8.51 a(b(a(b(x1)))) -> b(a(b(a(x1)))) 29.75/8.51 29.75/8.51 Q is empty. 29.75/8.51 We have to consider all minimal (P,Q,R)-chains. 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (9) QDPOrderProof (EQUIVALENT) 29.75/8.51 We use the reduction pair processor [LPAR04,JAR06]. 29.75/8.51 29.75/8.51 29.75/8.51 The following pairs can be oriented strictly and are deleted. 29.75/8.51 29.75/8.51 B(b(a(a(x1)))) -> B(a(a(a(x1)))) 29.75/8.51 The remaining pairs can at least be oriented weakly. 29.75/8.51 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 29.75/8.51 29.75/8.51 <<< 29.75/8.51 POL(B(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 29.75/8.51 >>> 29.75/8.51 29.75/8.51 <<< 29.75/8.51 POL(b(x_1)) = [[-I], [-I], [0A]] + [[0A, 0A, -I], [-I, -I, -I], [1A, -I, 0A]] * x_1 29.75/8.51 >>> 29.75/8.51 29.75/8.51 <<< 29.75/8.51 POL(a(x_1)) = [[-I], [1A], [0A]] + [[0A, -I, -I], [-I, 0A, 0A], [1A, -I, 0A]] * x_1 29.75/8.51 >>> 29.75/8.51 29.75/8.51 29.75/8.51 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.75/8.51 29.75/8.51 a(b(a(b(x1)))) -> b(a(b(a(x1)))) 29.75/8.51 b(b(a(a(x1)))) -> b(a(a(a(x1)))) 29.75/8.51 b(a(b(a(x1)))) -> b(b(b(b(x1)))) 29.75/8.51 29.75/8.51 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (10) 29.75/8.51 Obligation: 29.75/8.51 Q DP problem: 29.75/8.51 The TRS P consists of the following rules: 29.75/8.51 29.75/8.51 B(a(b(a(x1)))) -> B(b(b(b(x1)))) 29.75/8.51 29.75/8.51 The TRS R consists of the following rules: 29.75/8.51 29.75/8.51 b(a(b(a(x1)))) -> b(b(b(b(x1)))) 29.75/8.51 b(b(a(a(x1)))) -> b(a(a(a(x1)))) 29.75/8.51 a(b(a(b(x1)))) -> b(a(b(a(x1)))) 29.75/8.51 29.75/8.51 Q is empty. 29.75/8.51 We have to consider all minimal (P,Q,R)-chains. 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (11) DependencyGraphProof (EQUIVALENT) 29.75/8.51 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 29.75/8.51 ---------------------------------------- 29.75/8.51 29.75/8.51 (12) 29.75/8.51 TRUE 30.03/8.64 EOF