25.59/7.34 YES 25.59/7.36 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 25.59/7.36 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.59/7.36 25.59/7.36 25.59/7.36 Termination w.r.t. Q of the given QTRS could be proven: 25.59/7.36 25.59/7.36 (0) QTRS 25.59/7.36 (1) DependencyPairsProof [EQUIVALENT, 21 ms] 25.59/7.36 (2) QDP 25.59/7.36 (3) DependencyGraphProof [EQUIVALENT, 9 ms] 25.59/7.36 (4) QDP 25.59/7.36 (5) QDPOrderProof [EQUIVALENT, 101 ms] 25.59/7.36 (6) QDP 25.59/7.36 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 25.59/7.36 (8) QDP 25.59/7.36 (9) UsableRulesProof [EQUIVALENT, 0 ms] 25.59/7.36 (10) QDP 25.59/7.36 (11) QDPOrderProof [EQUIVALENT, 0 ms] 25.59/7.36 (12) QDP 25.59/7.36 (13) PisEmptyProof [EQUIVALENT, 0 ms] 25.59/7.36 (14) YES 25.59/7.36 25.59/7.36 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (0) 25.59/7.36 Obligation: 25.59/7.36 Q restricted rewrite system: 25.59/7.36 The TRS R consists of the following rules: 25.59/7.36 25.59/7.36 a(a(x1)) -> b(b(b(c(x1)))) 25.59/7.36 b(c(b(x1))) -> a(b(c(x1))) 25.59/7.36 25.59/7.36 Q is empty. 25.59/7.36 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (1) DependencyPairsProof (EQUIVALENT) 25.59/7.36 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (2) 25.59/7.36 Obligation: 25.59/7.36 Q DP problem: 25.59/7.36 The TRS P consists of the following rules: 25.59/7.36 25.59/7.36 A(a(x1)) -> B(b(b(c(x1)))) 25.59/7.36 A(a(x1)) -> B(b(c(x1))) 25.59/7.36 A(a(x1)) -> B(c(x1)) 25.59/7.36 B(c(b(x1))) -> A(b(c(x1))) 25.59/7.36 B(c(b(x1))) -> B(c(x1)) 25.59/7.36 25.59/7.36 The TRS R consists of the following rules: 25.59/7.36 25.59/7.36 a(a(x1)) -> b(b(b(c(x1)))) 25.59/7.36 b(c(b(x1))) -> a(b(c(x1))) 25.59/7.36 25.59/7.36 Q is empty. 25.59/7.36 We have to consider all minimal (P,Q,R)-chains. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (3) DependencyGraphProof (EQUIVALENT) 25.59/7.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (4) 25.59/7.36 Obligation: 25.59/7.36 Q DP problem: 25.59/7.36 The TRS P consists of the following rules: 25.59/7.36 25.59/7.36 A(a(x1)) -> B(c(x1)) 25.59/7.36 B(c(b(x1))) -> A(b(c(x1))) 25.59/7.36 B(c(b(x1))) -> B(c(x1)) 25.59/7.36 25.59/7.36 The TRS R consists of the following rules: 25.59/7.36 25.59/7.36 a(a(x1)) -> b(b(b(c(x1)))) 25.59/7.36 b(c(b(x1))) -> a(b(c(x1))) 25.59/7.36 25.59/7.36 Q is empty. 25.59/7.36 We have to consider all minimal (P,Q,R)-chains. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (5) QDPOrderProof (EQUIVALENT) 25.59/7.36 We use the reduction pair processor [LPAR04,JAR06]. 25.59/7.36 25.59/7.36 25.59/7.36 The following pairs can be oriented strictly and are deleted. 25.59/7.36 25.59/7.36 A(a(x1)) -> B(c(x1)) 25.59/7.36 The remaining pairs can at least be oriented weakly. 25.59/7.36 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 25.59/7.36 25.59/7.36 <<< 25.59/7.36 POL(A(x_1)) = [[-I]] + [[0A, -I, 0A]] * x_1 25.59/7.36 >>> 25.59/7.36 25.59/7.36 <<< 25.59/7.36 POL(a(x_1)) = [[0A], [0A], [1A]] + [[0A, -I, 0A], [0A, 0A, 1A], [1A, 0A, 0A]] * x_1 25.59/7.36 >>> 25.59/7.36 25.59/7.36 <<< 25.59/7.36 POL(B(x_1)) = [[-I]] + [[0A, 0A, -I]] * x_1 25.59/7.36 >>> 25.59/7.36 25.59/7.36 <<< 25.59/7.36 POL(c(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, 0A, -I]] * x_1 25.59/7.36 >>> 25.59/7.36 25.59/7.36 <<< 25.59/7.36 POL(b(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, -I], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 25.59/7.36 >>> 25.59/7.36 25.59/7.36 25.59/7.36 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.59/7.36 25.59/7.36 b(c(b(x1))) -> a(b(c(x1))) 25.59/7.36 a(a(x1)) -> b(b(b(c(x1)))) 25.59/7.36 25.59/7.36 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (6) 25.59/7.36 Obligation: 25.59/7.36 Q DP problem: 25.59/7.36 The TRS P consists of the following rules: 25.59/7.36 25.59/7.36 B(c(b(x1))) -> A(b(c(x1))) 25.59/7.36 B(c(b(x1))) -> B(c(x1)) 25.59/7.36 25.59/7.36 The TRS R consists of the following rules: 25.59/7.36 25.59/7.36 a(a(x1)) -> b(b(b(c(x1)))) 25.59/7.36 b(c(b(x1))) -> a(b(c(x1))) 25.59/7.36 25.59/7.36 Q is empty. 25.59/7.36 We have to consider all minimal (P,Q,R)-chains. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (7) DependencyGraphProof (EQUIVALENT) 25.59/7.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (8) 25.59/7.36 Obligation: 25.59/7.36 Q DP problem: 25.59/7.36 The TRS P consists of the following rules: 25.59/7.36 25.59/7.36 B(c(b(x1))) -> B(c(x1)) 25.59/7.36 25.59/7.36 The TRS R consists of the following rules: 25.59/7.36 25.59/7.36 a(a(x1)) -> b(b(b(c(x1)))) 25.59/7.36 b(c(b(x1))) -> a(b(c(x1))) 25.59/7.36 25.59/7.36 Q is empty. 25.59/7.36 We have to consider all minimal (P,Q,R)-chains. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (9) UsableRulesProof (EQUIVALENT) 25.59/7.36 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (10) 25.59/7.36 Obligation: 25.59/7.36 Q DP problem: 25.59/7.36 The TRS P consists of the following rules: 25.59/7.36 25.59/7.36 B(c(b(x1))) -> B(c(x1)) 25.59/7.36 25.59/7.36 R is empty. 25.59/7.36 Q is empty. 25.59/7.36 We have to consider all minimal (P,Q,R)-chains. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (11) QDPOrderProof (EQUIVALENT) 25.59/7.36 We use the reduction pair processor [LPAR04,JAR06]. 25.59/7.36 25.59/7.36 25.59/7.36 The following pairs can be oriented strictly and are deleted. 25.59/7.36 25.59/7.36 B(c(b(x1))) -> B(c(x1)) 25.59/7.36 The remaining pairs can at least be oriented weakly. 25.59/7.36 Used ordering: Polynomial interpretation [POLO]: 25.59/7.36 25.59/7.36 POL(B(x_1)) = x_1 25.59/7.36 POL(b(x_1)) = 1 + x_1 25.59/7.36 POL(c(x_1)) = x_1 25.59/7.36 25.59/7.36 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.59/7.36 none 25.59/7.36 25.59/7.36 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (12) 25.59/7.36 Obligation: 25.59/7.36 Q DP problem: 25.59/7.36 P is empty. 25.59/7.36 R is empty. 25.59/7.36 Q is empty. 25.59/7.36 We have to consider all minimal (P,Q,R)-chains. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (13) PisEmptyProof (EQUIVALENT) 25.59/7.36 The TRS P is empty. Hence, there is no (P,Q,R) chain. 25.59/7.36 ---------------------------------------- 25.59/7.36 25.59/7.36 (14) 25.59/7.36 YES 25.92/7.42 EOF