28.80/8.47 YES 30.63/9.12 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 30.63/9.12 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.63/9.12 30.63/9.12 30.63/9.12 Termination w.r.t. Q of the given QTRS could be proven: 30.63/9.12 30.63/9.12 (0) QTRS 30.63/9.12 (1) QTRSRRRProof [EQUIVALENT, 39 ms] 30.63/9.12 (2) QTRS 30.63/9.12 (3) DependencyPairsProof [EQUIVALENT, 1 ms] 30.63/9.12 (4) QDP 30.63/9.12 (5) DependencyGraphProof [EQUIVALENT, 1 ms] 30.63/9.12 (6) QDP 30.63/9.12 (7) MRRProof [EQUIVALENT, 34 ms] 30.63/9.12 (8) QDP 30.63/9.12 (9) MRRProof [EQUIVALENT, 5 ms] 30.63/9.12 (10) QDP 30.63/9.12 (11) QDPOrderProof [EQUIVALENT, 51 ms] 30.63/9.12 (12) QDP 30.63/9.12 (13) PisEmptyProof [EQUIVALENT, 0 ms] 30.63/9.12 (14) YES 30.63/9.12 30.63/9.12 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (0) 30.63/9.12 Obligation: 30.63/9.12 Q restricted rewrite system: 30.63/9.12 The TRS R consists of the following rules: 30.63/9.12 30.63/9.12 b(b(a(b(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 30.63/9.12 b(a(a(a(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 30.63/9.12 Q is empty. 30.63/9.12 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (1) QTRSRRRProof (EQUIVALENT) 30.63/9.12 Used ordering: 30.63/9.12 Polynomial interpretation [POLO]: 30.63/9.12 30.63/9.12 POL(a(x_1)) = 1 + x_1 30.63/9.12 POL(b(x_1)) = x_1 30.63/9.12 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 30.63/9.12 30.63/9.12 b(a(a(a(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 30.63/9.12 30.63/9.12 30.63/9.12 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (2) 30.63/9.12 Obligation: 30.63/9.12 Q restricted rewrite system: 30.63/9.12 The TRS R consists of the following rules: 30.63/9.12 30.63/9.12 b(b(a(b(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 30.63/9.12 30.63/9.12 Q is empty. 30.63/9.12 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (3) DependencyPairsProof (EQUIVALENT) 30.63/9.12 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (4) 30.63/9.12 Obligation: 30.63/9.12 Q DP problem: 30.63/9.12 The TRS P consists of the following rules: 30.63/9.12 30.63/9.12 B(b(a(b(x1)))) -> B(b(b(a(x1)))) 30.63/9.12 B(b(a(b(x1)))) -> B(b(a(x1))) 30.63/9.12 B(b(a(b(x1)))) -> B(a(x1)) 30.63/9.12 B(b(a(a(x1)))) -> B(b(x1)) 30.63/9.12 B(b(a(a(x1)))) -> B(x1) 30.63/9.12 30.63/9.12 The TRS R consists of the following rules: 30.63/9.12 30.63/9.12 b(b(a(b(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 30.63/9.12 30.63/9.12 Q is empty. 30.63/9.12 We have to consider all minimal (P,Q,R)-chains. 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (5) DependencyGraphProof (EQUIVALENT) 30.63/9.12 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (6) 30.63/9.12 Obligation: 30.63/9.12 Q DP problem: 30.63/9.12 The TRS P consists of the following rules: 30.63/9.12 30.63/9.12 B(b(a(b(x1)))) -> B(b(a(x1))) 30.63/9.12 B(b(a(b(x1)))) -> B(b(b(a(x1)))) 30.63/9.12 B(b(a(a(x1)))) -> B(b(x1)) 30.63/9.12 B(b(a(a(x1)))) -> B(x1) 30.63/9.12 30.63/9.12 The TRS R consists of the following rules: 30.63/9.12 30.63/9.12 b(b(a(b(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 30.63/9.12 30.63/9.12 Q is empty. 30.63/9.12 We have to consider all minimal (P,Q,R)-chains. 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (7) MRRProof (EQUIVALENT) 30.63/9.12 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. 30.63/9.12 30.63/9.12 Strictly oriented dependency pairs: 30.63/9.12 30.63/9.12 B(b(a(a(x1)))) -> B(b(x1)) 30.63/9.12 B(b(a(a(x1)))) -> B(x1) 30.63/9.12 30.63/9.12 30.63/9.12 Used ordering: Polynomial interpretation [POLO]: 30.63/9.12 30.63/9.12 POL(B(x_1)) = x_1 30.63/9.12 POL(a(x_1)) = 1 + x_1 30.63/9.12 POL(b(x_1)) = x_1 30.63/9.12 30.63/9.12 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (8) 30.63/9.12 Obligation: 30.63/9.12 Q DP problem: 30.63/9.12 The TRS P consists of the following rules: 30.63/9.12 30.63/9.12 B(b(a(b(x1)))) -> B(b(a(x1))) 30.63/9.12 B(b(a(b(x1)))) -> B(b(b(a(x1)))) 30.63/9.12 30.63/9.12 The TRS R consists of the following rules: 30.63/9.12 30.63/9.12 b(b(a(b(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 30.63/9.12 30.63/9.12 Q is empty. 30.63/9.12 We have to consider all minimal (P,Q,R)-chains. 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (9) MRRProof (EQUIVALENT) 30.63/9.12 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. 30.63/9.12 30.63/9.12 Strictly oriented dependency pairs: 30.63/9.12 30.63/9.12 B(b(a(b(x1)))) -> B(b(a(x1))) 30.63/9.12 30.63/9.12 30.63/9.12 Used ordering: Polynomial interpretation [POLO]: 30.63/9.12 30.63/9.12 POL(B(x_1)) = 2*x_1 30.63/9.12 POL(a(x_1)) = 2 + 2*x_1 30.63/9.12 POL(b(x_1)) = 2 + 2*x_1 30.63/9.12 30.63/9.12 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (10) 30.63/9.12 Obligation: 30.63/9.12 Q DP problem: 30.63/9.12 The TRS P consists of the following rules: 30.63/9.12 30.63/9.12 B(b(a(b(x1)))) -> B(b(b(a(x1)))) 30.63/9.12 30.63/9.12 The TRS R consists of the following rules: 30.63/9.12 30.63/9.12 b(b(a(b(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 30.63/9.12 30.63/9.12 Q is empty. 30.63/9.12 We have to consider all minimal (P,Q,R)-chains. 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (11) QDPOrderProof (EQUIVALENT) 30.63/9.12 We use the reduction pair processor [LPAR04,JAR06]. 30.63/9.12 30.63/9.12 30.63/9.12 The following pairs can be oriented strictly and are deleted. 30.63/9.12 30.63/9.12 B(b(a(b(x1)))) -> B(b(b(a(x1)))) 30.63/9.12 The remaining pairs can at least be oriented weakly. 30.63/9.12 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 30.63/9.12 30.63/9.12 <<< 30.63/9.12 POL(B(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 30.63/9.12 >>> 30.63/9.12 30.63/9.12 <<< 30.63/9.12 POL(b(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, 0A, 1A]] * x_1 30.63/9.12 >>> 30.63/9.12 30.63/9.12 <<< 30.63/9.12 POL(a(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [0A, -I, 1A], [-I, -I, -I]] * x_1 30.63/9.12 >>> 30.63/9.12 30.63/9.12 30.63/9.12 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 30.63/9.12 30.63/9.12 b(b(a(b(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 30.63/9.12 30.63/9.12 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (12) 30.63/9.12 Obligation: 30.63/9.12 Q DP problem: 30.63/9.12 P is empty. 30.63/9.12 The TRS R consists of the following rules: 30.63/9.12 30.63/9.12 b(b(a(b(x1)))) -> b(b(b(a(x1)))) 30.63/9.12 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 30.63/9.12 30.63/9.12 Q is empty. 30.63/9.12 We have to consider all minimal (P,Q,R)-chains. 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (13) PisEmptyProof (EQUIVALENT) 30.63/9.12 The TRS P is empty. Hence, there is no (P,Q,R) chain. 30.63/9.12 ---------------------------------------- 30.63/9.12 30.63/9.12 (14) 30.63/9.12 YES 30.75/9.59 EOF