30.58/8.69 YES 38.30/10.68 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 38.30/10.68 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 38.30/10.68 38.30/10.68 38.30/10.68 Termination w.r.t. Q of the given QTRS could be proven: 38.30/10.68 38.30/10.68 (0) QTRS 38.30/10.68 (1) QTRSRRRProof [EQUIVALENT, 44 ms] 38.30/10.68 (2) QTRS 38.30/10.68 (3) DependencyPairsProof [EQUIVALENT, 18 ms] 38.30/10.68 (4) QDP 38.30/10.68 (5) QDPOrderProof [EQUIVALENT, 159 ms] 38.30/10.68 (6) QDP 38.30/10.68 (7) QDPOrderProof [EQUIVALENT, 267 ms] 38.30/10.68 (8) QDP 38.30/10.68 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 38.30/10.68 (10) TRUE 38.30/10.68 38.30/10.68 38.30/10.68 ---------------------------------------- 38.30/10.68 38.30/10.68 (0) 38.30/10.68 Obligation: 38.30/10.68 Q restricted rewrite system: 38.30/10.68 The TRS R consists of the following rules: 38.30/10.68 38.30/10.68 a(a(a(b(b(x1))))) -> b(b(b(x1))) 38.30/10.68 b(a(a(a(b(x1))))) -> a(a(a(b(a(a(a(x1))))))) 38.30/10.68 a(a(a(x1))) -> a(a(x1)) 38.30/10.68 b(b(x1)) -> a(b(a(x1))) 38.30/10.68 38.30/10.68 Q is empty. 38.30/10.68 38.30/10.68 ---------------------------------------- 38.30/10.68 38.30/10.68 (1) QTRSRRRProof (EQUIVALENT) 38.30/10.68 Used ordering: 38.30/10.68 Polynomial interpretation [POLO]: 38.30/10.68 38.30/10.68 POL(a(x_1)) = 1 + x_1 38.30/10.68 POL(b(x_1)) = 3 + x_1 38.30/10.68 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 38.30/10.68 38.30/10.68 a(a(a(x1))) -> a(a(x1)) 38.30/10.68 b(b(x1)) -> a(b(a(x1))) 38.30/10.68 38.30/10.68 38.30/10.68 38.30/10.68 38.30/10.68 ---------------------------------------- 38.30/10.68 38.30/10.68 (2) 38.30/10.68 Obligation: 38.30/10.68 Q restricted rewrite system: 38.30/10.68 The TRS R consists of the following rules: 38.30/10.68 38.30/10.68 a(a(a(b(b(x1))))) -> b(b(b(x1))) 38.30/10.68 b(a(a(a(b(x1))))) -> a(a(a(b(a(a(a(x1))))))) 38.30/10.68 38.30/10.68 Q is empty. 38.30/10.68 38.30/10.68 ---------------------------------------- 38.30/10.68 38.30/10.68 (3) DependencyPairsProof (EQUIVALENT) 38.30/10.68 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 38.30/10.68 ---------------------------------------- 38.30/10.68 38.30/10.68 (4) 38.30/10.68 Obligation: 38.30/10.68 Q DP problem: 38.30/10.68 The TRS P consists of the following rules: 38.30/10.68 38.30/10.68 A(a(a(b(b(x1))))) -> B(b(b(x1))) 38.30/10.68 B(a(a(a(b(x1))))) -> A(a(a(b(a(a(a(x1))))))) 38.30/10.68 B(a(a(a(b(x1))))) -> A(a(b(a(a(a(x1)))))) 38.30/10.68 B(a(a(a(b(x1))))) -> A(b(a(a(a(x1))))) 38.30/10.68 B(a(a(a(b(x1))))) -> B(a(a(a(x1)))) 38.30/10.68 B(a(a(a(b(x1))))) -> A(a(a(x1))) 38.30/10.68 B(a(a(a(b(x1))))) -> A(a(x1)) 38.30/10.68 B(a(a(a(b(x1))))) -> A(x1) 38.30/10.68 38.30/10.68 The TRS R consists of the following rules: 38.30/10.68 38.30/10.68 a(a(a(b(b(x1))))) -> b(b(b(x1))) 38.30/10.68 b(a(a(a(b(x1))))) -> a(a(a(b(a(a(a(x1))))))) 38.30/10.69 38.30/10.69 Q is empty. 38.30/10.69 We have to consider all minimal (P,Q,R)-chains. 38.30/10.69 ---------------------------------------- 38.30/10.69 38.30/10.69 (5) QDPOrderProof (EQUIVALENT) 38.30/10.69 We use the reduction pair processor [LPAR04,JAR06]. 38.30/10.69 38.30/10.69 38.30/10.69 The following pairs can be oriented strictly and are deleted. 38.30/10.69 38.30/10.69 B(a(a(a(b(x1))))) -> A(a(b(a(a(a(x1)))))) 38.30/10.69 B(a(a(a(b(x1))))) -> A(b(a(a(a(x1))))) 38.30/10.69 B(a(a(a(b(x1))))) -> B(a(a(a(x1)))) 38.30/10.69 B(a(a(a(b(x1))))) -> A(a(a(x1))) 38.30/10.69 B(a(a(a(b(x1))))) -> A(a(x1)) 38.30/10.69 B(a(a(a(b(x1))))) -> A(x1) 38.30/10.69 The remaining pairs can at least be oriented weakly. 38.30/10.69 Used ordering: Polynomial interpretation [POLO]: 38.30/10.69 38.30/10.69 POL(A(x_1)) = 1 + x_1 38.30/10.69 POL(B(x_1)) = 3 + x_1 38.30/10.69 POL(a(x_1)) = 1 + x_1 38.30/10.69 POL(b(x_1)) = 3 + x_1 38.30/10.69 38.30/10.69 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 38.30/10.69 38.30/10.69 b(a(a(a(b(x1))))) -> a(a(a(b(a(a(a(x1))))))) 38.30/10.69 a(a(a(b(b(x1))))) -> b(b(b(x1))) 38.30/10.69 38.30/10.69 38.30/10.69 ---------------------------------------- 38.30/10.69 38.30/10.69 (6) 38.30/10.69 Obligation: 38.30/10.69 Q DP problem: 38.30/10.69 The TRS P consists of the following rules: 38.30/10.69 38.30/10.69 A(a(a(b(b(x1))))) -> B(b(b(x1))) 38.30/10.69 B(a(a(a(b(x1))))) -> A(a(a(b(a(a(a(x1))))))) 38.30/10.69 38.30/10.69 The TRS R consists of the following rules: 38.30/10.69 38.30/10.69 a(a(a(b(b(x1))))) -> b(b(b(x1))) 38.30/10.69 b(a(a(a(b(x1))))) -> a(a(a(b(a(a(a(x1))))))) 38.30/10.69 38.30/10.69 Q is empty. 38.30/10.69 We have to consider all minimal (P,Q,R)-chains. 38.30/10.69 ---------------------------------------- 38.30/10.69 38.30/10.69 (7) QDPOrderProof (EQUIVALENT) 38.30/10.69 We use the reduction pair processor [LPAR04,JAR06]. 38.30/10.69 38.30/10.69 38.30/10.69 The following pairs can be oriented strictly and are deleted. 38.30/10.69 38.30/10.69 A(a(a(b(b(x1))))) -> B(b(b(x1))) 38.30/10.69 The remaining pairs can at least be oriented weakly. 38.30/10.69 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 38.30/10.69 38.30/10.69 <<< 38.30/10.69 POL(A(x_1)) = [[-I]] + [[0A, 0A, 0A]] * x_1 38.30/10.69 >>> 38.30/10.69 38.30/10.69 <<< 38.30/10.69 POL(a(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, -I], [0A, -I, -I]] * x_1 38.30/10.69 >>> 38.30/10.69 38.30/10.69 <<< 38.30/10.69 POL(b(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, -I], [1A, -I, -I], [0A, 0A, 0A]] * x_1 38.30/10.69 >>> 38.30/10.69 38.30/10.69 <<< 38.30/10.69 POL(B(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 38.30/10.69 >>> 38.30/10.69 38.30/10.69 38.30/10.69 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 38.30/10.69 38.30/10.69 b(a(a(a(b(x1))))) -> a(a(a(b(a(a(a(x1))))))) 38.30/10.69 a(a(a(b(b(x1))))) -> b(b(b(x1))) 38.30/10.69 38.30/10.69 38.30/10.69 ---------------------------------------- 38.30/10.69 38.30/10.69 (8) 38.30/10.69 Obligation: 38.30/10.69 Q DP problem: 38.30/10.69 The TRS P consists of the following rules: 38.30/10.69 38.30/10.69 B(a(a(a(b(x1))))) -> A(a(a(b(a(a(a(x1))))))) 38.30/10.69 38.30/10.69 The TRS R consists of the following rules: 38.30/10.69 38.30/10.69 a(a(a(b(b(x1))))) -> b(b(b(x1))) 38.30/10.69 b(a(a(a(b(x1))))) -> a(a(a(b(a(a(a(x1))))))) 38.30/10.69 38.30/10.69 Q is empty. 38.30/10.69 We have to consider all minimal (P,Q,R)-chains. 38.30/10.69 ---------------------------------------- 38.30/10.69 38.30/10.69 (9) DependencyGraphProof (EQUIVALENT) 38.30/10.69 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 38.30/10.69 ---------------------------------------- 38.30/10.69 38.30/10.69 (10) 38.30/10.69 TRUE 38.55/11.35 EOF