37.33/10.45 YES 38.14/10.64 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 38.14/10.64 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 38.14/10.64 38.14/10.64 38.14/10.64 Termination w.r.t. Q of the given QTRS could be proven: 38.14/10.64 38.14/10.64 (0) QTRS 38.14/10.64 (1) QTRS Reverse [EQUIVALENT, 0 ms] 38.14/10.64 (2) QTRS 38.14/10.64 (3) DependencyPairsProof [EQUIVALENT, 16 ms] 38.14/10.64 (4) QDP 38.14/10.64 (5) MRRProof [EQUIVALENT, 71 ms] 38.14/10.64 (6) QDP 38.14/10.64 (7) QDPOrderProof [EQUIVALENT, 43 ms] 38.14/10.64 (8) QDP 38.14/10.64 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 38.14/10.64 (10) TRUE 38.14/10.64 38.14/10.64 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (0) 38.14/10.64 Obligation: 38.14/10.64 Q restricted rewrite system: 38.14/10.64 The TRS R consists of the following rules: 38.14/10.64 38.14/10.64 a(a(a(a(x1)))) -> a(b(a(b(x1)))) 38.14/10.64 b(a(b(x1))) -> a(b(a(x1))) 38.14/10.64 38.14/10.64 Q is empty. 38.14/10.64 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (1) QTRS Reverse (EQUIVALENT) 38.14/10.64 We applied the QTRS Reverse Processor [REVERSE]. 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (2) 38.14/10.64 Obligation: 38.14/10.64 Q restricted rewrite system: 38.14/10.64 The TRS R consists of the following rules: 38.14/10.64 38.14/10.64 a(a(a(a(x1)))) -> b(a(b(a(x1)))) 38.14/10.64 b(a(b(x1))) -> a(b(a(x1))) 38.14/10.64 38.14/10.64 Q is empty. 38.14/10.64 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (3) DependencyPairsProof (EQUIVALENT) 38.14/10.64 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (4) 38.14/10.64 Obligation: 38.14/10.64 Q DP problem: 38.14/10.64 The TRS P consists of the following rules: 38.14/10.64 38.14/10.64 A(a(a(a(x1)))) -> B(a(b(a(x1)))) 38.14/10.64 A(a(a(a(x1)))) -> A(b(a(x1))) 38.14/10.64 A(a(a(a(x1)))) -> B(a(x1)) 38.14/10.64 B(a(b(x1))) -> A(b(a(x1))) 38.14/10.64 B(a(b(x1))) -> B(a(x1)) 38.14/10.64 B(a(b(x1))) -> A(x1) 38.14/10.64 38.14/10.64 The TRS R consists of the following rules: 38.14/10.64 38.14/10.64 a(a(a(a(x1)))) -> b(a(b(a(x1)))) 38.14/10.64 b(a(b(x1))) -> a(b(a(x1))) 38.14/10.64 38.14/10.64 Q is empty. 38.14/10.64 We have to consider all minimal (P,Q,R)-chains. 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (5) MRRProof (EQUIVALENT) 38.14/10.64 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. 38.14/10.64 38.14/10.64 Strictly oriented dependency pairs: 38.14/10.64 38.14/10.64 A(a(a(a(x1)))) -> A(b(a(x1))) 38.14/10.64 A(a(a(a(x1)))) -> B(a(x1)) 38.14/10.64 B(a(b(x1))) -> B(a(x1)) 38.14/10.64 B(a(b(x1))) -> A(x1) 38.14/10.64 38.14/10.64 38.14/10.64 Used ordering: Polynomial interpretation [POLO]: 38.14/10.64 38.14/10.64 POL(A(x_1)) = 2 + 2*x_1 38.14/10.64 POL(B(x_1)) = 2 + 2*x_1 38.14/10.64 POL(a(x_1)) = 1 + x_1 38.14/10.64 POL(b(x_1)) = 1 + x_1 38.14/10.64 38.14/10.64 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (6) 38.14/10.64 Obligation: 38.14/10.64 Q DP problem: 38.14/10.64 The TRS P consists of the following rules: 38.14/10.64 38.14/10.64 A(a(a(a(x1)))) -> B(a(b(a(x1)))) 38.14/10.64 B(a(b(x1))) -> A(b(a(x1))) 38.14/10.64 38.14/10.64 The TRS R consists of the following rules: 38.14/10.64 38.14/10.64 a(a(a(a(x1)))) -> b(a(b(a(x1)))) 38.14/10.64 b(a(b(x1))) -> a(b(a(x1))) 38.14/10.64 38.14/10.64 Q is empty. 38.14/10.64 We have to consider all minimal (P,Q,R)-chains. 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (7) QDPOrderProof (EQUIVALENT) 38.14/10.64 We use the reduction pair processor [LPAR04,JAR06]. 38.14/10.64 38.14/10.64 38.14/10.64 The following pairs can be oriented strictly and are deleted. 38.14/10.64 38.14/10.64 A(a(a(a(x1)))) -> B(a(b(a(x1)))) 38.14/10.64 The remaining pairs can at least be oriented weakly. 38.14/10.64 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 38.14/10.64 38.14/10.64 <<< 38.14/10.64 POL(A(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 38.14/10.64 >>> 38.14/10.64 38.14/10.64 <<< 38.14/10.64 POL(a(x_1)) = [[0A], [1A], [-I]] + [[-I, 0A, -I], [0A, 1A, 0A], [0A, -I, -I]] * x_1 38.14/10.64 >>> 38.14/10.64 38.14/10.64 <<< 38.14/10.64 POL(B(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 38.14/10.64 >>> 38.14/10.64 38.14/10.64 <<< 38.14/10.64 POL(b(x_1)) = [[0A], [-I], [0A]] + [[1A, 0A, 0A], [0A, -I, 1A], [0A, -I, 1A]] * x_1 38.14/10.64 >>> 38.14/10.64 38.14/10.64 38.14/10.64 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 38.14/10.64 38.14/10.64 b(a(b(x1))) -> a(b(a(x1))) 38.14/10.64 a(a(a(a(x1)))) -> b(a(b(a(x1)))) 38.14/10.64 38.14/10.64 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (8) 38.14/10.64 Obligation: 38.14/10.64 Q DP problem: 38.14/10.64 The TRS P consists of the following rules: 38.14/10.64 38.14/10.64 B(a(b(x1))) -> A(b(a(x1))) 38.14/10.64 38.14/10.64 The TRS R consists of the following rules: 38.14/10.64 38.14/10.64 a(a(a(a(x1)))) -> b(a(b(a(x1)))) 38.14/10.64 b(a(b(x1))) -> a(b(a(x1))) 38.14/10.64 38.14/10.64 Q is empty. 38.14/10.64 We have to consider all minimal (P,Q,R)-chains. 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (9) DependencyGraphProof (EQUIVALENT) 38.14/10.64 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 38.14/10.64 ---------------------------------------- 38.14/10.64 38.14/10.64 (10) 38.14/10.64 TRUE 38.29/12.21 EOF