34.44/9.71 YES 34.44/9.75 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 34.44/9.75 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 34.44/9.75 34.44/9.75 34.44/9.75 Termination w.r.t. Q of the given QTRS could be proven: 34.44/9.75 34.44/9.75 (0) QTRS 34.44/9.75 (1) DependencyPairsProof [EQUIVALENT, 51 ms] 34.44/9.75 (2) QDP 34.44/9.75 (3) QDPOrderProof [EQUIVALENT, 175 ms] 34.44/9.75 (4) QDP 34.44/9.75 (5) MRRProof [EQUIVALENT, 109 ms] 34.44/9.75 (6) QDP 34.44/9.75 (7) DependencyGraphProof [EQUIVALENT, 1 ms] 34.44/9.75 (8) QDP 34.44/9.75 (9) MRRProof [EQUIVALENT, 0 ms] 34.44/9.75 (10) QDP 34.44/9.75 (11) MRRProof [EQUIVALENT, 12 ms] 34.44/9.75 (12) QDP 34.44/9.75 (13) PisEmptyProof [EQUIVALENT, 0 ms] 34.44/9.75 (14) YES 34.44/9.75 34.44/9.75 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (0) 34.44/9.75 Obligation: 34.44/9.75 Q restricted rewrite system: 34.44/9.75 The TRS R consists of the following rules: 34.44/9.75 34.44/9.75 a(a(a(a(x1)))) -> b(b(x1)) 34.44/9.75 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 34.44/9.75 b(b(b(b(c(c(x1)))))) -> c(c(a(a(x1)))) 34.44/9.75 b(b(b(b(x1)))) -> a(a(a(a(a(a(x1)))))) 34.44/9.75 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 34.44/9.75 34.44/9.75 Q is empty. 34.44/9.75 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (1) DependencyPairsProof (EQUIVALENT) 34.44/9.75 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (2) 34.44/9.75 Obligation: 34.44/9.75 Q DP problem: 34.44/9.75 The TRS P consists of the following rules: 34.44/9.75 34.44/9.75 A(a(a(a(x1)))) -> B(b(x1)) 34.44/9.75 A(a(a(a(x1)))) -> B(x1) 34.44/9.75 B(b(a(a(x1)))) -> A(a(b(b(x1)))) 34.44/9.75 B(b(a(a(x1)))) -> A(b(b(x1))) 34.44/9.75 B(b(a(a(x1)))) -> B(b(x1)) 34.44/9.75 B(b(a(a(x1)))) -> B(x1) 34.44/9.75 B(b(b(b(c(c(x1)))))) -> C(c(a(a(x1)))) 34.44/9.75 B(b(b(b(c(c(x1)))))) -> C(a(a(x1))) 34.44/9.75 B(b(b(b(c(c(x1)))))) -> A(a(x1)) 34.44/9.75 B(b(b(b(c(c(x1)))))) -> A(x1) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(a(a(a(x1)))))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(a(a(x1))))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(a(x1)))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(x1))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(x1)) 34.44/9.75 B(b(b(b(x1)))) -> A(x1) 34.44/9.75 C(c(a(a(x1)))) -> B(b(a(a(c(c(x1)))))) 34.44/9.75 C(c(a(a(x1)))) -> B(a(a(c(c(x1))))) 34.44/9.75 C(c(a(a(x1)))) -> A(a(c(c(x1)))) 34.44/9.75 C(c(a(a(x1)))) -> A(c(c(x1))) 34.44/9.75 C(c(a(a(x1)))) -> C(c(x1)) 34.44/9.75 C(c(a(a(x1)))) -> C(x1) 34.44/9.75 34.44/9.75 The TRS R consists of the following rules: 34.44/9.75 34.44/9.75 a(a(a(a(x1)))) -> b(b(x1)) 34.44/9.75 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 34.44/9.75 b(b(b(b(c(c(x1)))))) -> c(c(a(a(x1)))) 34.44/9.75 b(b(b(b(x1)))) -> a(a(a(a(a(a(x1)))))) 34.44/9.75 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 34.44/9.75 34.44/9.75 Q is empty. 34.44/9.75 We have to consider all minimal (P,Q,R)-chains. 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (3) QDPOrderProof (EQUIVALENT) 34.44/9.75 We use the reduction pair processor [LPAR04,JAR06]. 34.44/9.75 34.44/9.75 34.44/9.75 The following pairs can be oriented strictly and are deleted. 34.44/9.75 34.44/9.75 B(b(b(b(c(c(x1)))))) -> C(a(a(x1))) 34.44/9.75 B(b(b(b(c(c(x1)))))) -> A(a(x1)) 34.44/9.75 B(b(b(b(c(c(x1)))))) -> A(x1) 34.44/9.75 C(c(a(a(x1)))) -> C(x1) 34.44/9.75 The remaining pairs can at least be oriented weakly. 34.44/9.75 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 34.44/9.75 34.44/9.75 POL( A_1(x_1) ) = x_1 + 1 34.44/9.75 POL( B_1(x_1) ) = x_1 + 1 34.44/9.75 POL( C_1(x_1) ) = x_1 + 2 34.44/9.75 POL( b_1(x_1) ) = x_1 34.44/9.75 POL( a_1(x_1) ) = x_1 34.44/9.75 POL( c_1(x_1) ) = x_1 + 1 34.44/9.75 34.44/9.75 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 34.44/9.75 34.44/9.75 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 34.44/9.75 a(a(a(a(x1)))) -> b(b(x1)) 34.44/9.75 b(b(b(b(c(c(x1)))))) -> c(c(a(a(x1)))) 34.44/9.75 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 34.44/9.75 b(b(b(b(x1)))) -> a(a(a(a(a(a(x1)))))) 34.44/9.75 34.44/9.75 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (4) 34.44/9.75 Obligation: 34.44/9.75 Q DP problem: 34.44/9.75 The TRS P consists of the following rules: 34.44/9.75 34.44/9.75 A(a(a(a(x1)))) -> B(b(x1)) 34.44/9.75 A(a(a(a(x1)))) -> B(x1) 34.44/9.75 B(b(a(a(x1)))) -> A(a(b(b(x1)))) 34.44/9.75 B(b(a(a(x1)))) -> A(b(b(x1))) 34.44/9.75 B(b(a(a(x1)))) -> B(b(x1)) 34.44/9.75 B(b(a(a(x1)))) -> B(x1) 34.44/9.75 B(b(b(b(c(c(x1)))))) -> C(c(a(a(x1)))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(a(a(a(x1)))))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(a(a(x1))))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(a(x1)))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(x1))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(x1)) 34.44/9.75 B(b(b(b(x1)))) -> A(x1) 34.44/9.75 C(c(a(a(x1)))) -> B(b(a(a(c(c(x1)))))) 34.44/9.75 C(c(a(a(x1)))) -> B(a(a(c(c(x1))))) 34.44/9.75 C(c(a(a(x1)))) -> A(a(c(c(x1)))) 34.44/9.75 C(c(a(a(x1)))) -> A(c(c(x1))) 34.44/9.75 C(c(a(a(x1)))) -> C(c(x1)) 34.44/9.75 34.44/9.75 The TRS R consists of the following rules: 34.44/9.75 34.44/9.75 a(a(a(a(x1)))) -> b(b(x1)) 34.44/9.75 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 34.44/9.75 b(b(b(b(c(c(x1)))))) -> c(c(a(a(x1)))) 34.44/9.75 b(b(b(b(x1)))) -> a(a(a(a(a(a(x1)))))) 34.44/9.75 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 34.44/9.75 34.44/9.75 Q is empty. 34.44/9.75 We have to consider all minimal (P,Q,R)-chains. 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (5) MRRProof (EQUIVALENT) 34.44/9.75 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. 34.44/9.75 34.44/9.75 Strictly oriented dependency pairs: 34.44/9.75 34.44/9.75 A(a(a(a(x1)))) -> B(x1) 34.44/9.75 B(b(a(a(x1)))) -> A(b(b(x1))) 34.44/9.75 B(b(a(a(x1)))) -> B(b(x1)) 34.44/9.75 B(b(a(a(x1)))) -> B(x1) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(a(a(a(x1)))))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(a(a(x1))))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(a(x1)))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(a(x1))) 34.44/9.75 B(b(b(b(x1)))) -> A(a(x1)) 34.44/9.75 B(b(b(b(x1)))) -> A(x1) 34.44/9.75 C(c(a(a(x1)))) -> B(b(a(a(c(c(x1)))))) 34.44/9.75 C(c(a(a(x1)))) -> B(a(a(c(c(x1))))) 34.44/9.75 C(c(a(a(x1)))) -> A(a(c(c(x1)))) 34.44/9.75 C(c(a(a(x1)))) -> A(c(c(x1))) 34.44/9.75 C(c(a(a(x1)))) -> C(c(x1)) 34.44/9.75 34.44/9.75 Strictly oriented rules of the TRS R: 34.44/9.75 34.44/9.75 b(b(b(b(x1)))) -> a(a(a(a(a(a(x1)))))) 34.44/9.75 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 34.44/9.75 34.44/9.75 Used ordering: Polynomial interpretation [POLO]: 34.44/9.75 34.44/9.75 POL(A(x_1)) = 1 + x_1 34.44/9.75 POL(B(x_1)) = 2 + x_1 34.44/9.75 POL(C(x_1)) = 2*x_1 34.44/9.75 POL(a(x_1)) = 1 + x_1 34.44/9.75 POL(b(x_1)) = 2 + x_1 34.44/9.75 POL(c(x_1)) = 2*x_1 34.44/9.75 34.44/9.75 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (6) 34.44/9.75 Obligation: 34.44/9.75 Q DP problem: 34.44/9.75 The TRS P consists of the following rules: 34.44/9.75 34.44/9.75 A(a(a(a(x1)))) -> B(b(x1)) 34.44/9.75 B(b(a(a(x1)))) -> A(a(b(b(x1)))) 34.44/9.75 B(b(b(b(c(c(x1)))))) -> C(c(a(a(x1)))) 34.44/9.75 34.44/9.75 The TRS R consists of the following rules: 34.44/9.75 34.44/9.75 a(a(a(a(x1)))) -> b(b(x1)) 34.44/9.75 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 34.44/9.75 b(b(b(b(c(c(x1)))))) -> c(c(a(a(x1)))) 34.44/9.75 34.44/9.75 Q is empty. 34.44/9.75 We have to consider all minimal (P,Q,R)-chains. 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (7) DependencyGraphProof (EQUIVALENT) 34.44/9.75 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (8) 34.44/9.75 Obligation: 34.44/9.75 Q DP problem: 34.44/9.75 The TRS P consists of the following rules: 34.44/9.75 34.44/9.75 B(b(a(a(x1)))) -> A(a(b(b(x1)))) 34.44/9.75 A(a(a(a(x1)))) -> B(b(x1)) 34.44/9.75 34.44/9.75 The TRS R consists of the following rules: 34.44/9.75 34.44/9.75 a(a(a(a(x1)))) -> b(b(x1)) 34.44/9.75 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 34.44/9.75 b(b(b(b(c(c(x1)))))) -> c(c(a(a(x1)))) 34.44/9.75 34.44/9.75 Q is empty. 34.44/9.75 We have to consider all minimal (P,Q,R)-chains. 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (9) MRRProof (EQUIVALENT) 34.44/9.75 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. 34.44/9.75 34.44/9.75 34.44/9.75 Strictly oriented rules of the TRS R: 34.44/9.75 34.44/9.75 b(b(b(b(c(c(x1)))))) -> c(c(a(a(x1)))) 34.44/9.75 34.44/9.75 Used ordering: Polynomial interpretation [POLO]: 34.44/9.75 34.44/9.75 POL(A(x_1)) = x_1 34.44/9.75 POL(B(x_1)) = 1 + x_1 34.44/9.75 POL(a(x_1)) = 1 + x_1 34.44/9.75 POL(b(x_1)) = 2 + x_1 34.44/9.75 POL(c(x_1)) = x_1 34.44/9.75 34.44/9.75 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (10) 34.44/9.75 Obligation: 34.44/9.75 Q DP problem: 34.44/9.75 The TRS P consists of the following rules: 34.44/9.75 34.44/9.75 B(b(a(a(x1)))) -> A(a(b(b(x1)))) 34.44/9.75 A(a(a(a(x1)))) -> B(b(x1)) 34.44/9.75 34.44/9.75 The TRS R consists of the following rules: 34.44/9.75 34.44/9.75 a(a(a(a(x1)))) -> b(b(x1)) 34.44/9.75 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 34.44/9.75 34.44/9.75 Q is empty. 34.44/9.75 We have to consider all minimal (P,Q,R)-chains. 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (11) MRRProof (EQUIVALENT) 34.44/9.75 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. 34.44/9.75 34.44/9.75 Strictly oriented dependency pairs: 34.44/9.75 34.44/9.75 B(b(a(a(x1)))) -> A(a(b(b(x1)))) 34.44/9.75 A(a(a(a(x1)))) -> B(b(x1)) 34.44/9.75 34.44/9.75 Strictly oriented rules of the TRS R: 34.44/9.75 34.44/9.75 a(a(a(a(x1)))) -> b(b(x1)) 34.44/9.75 34.44/9.75 Used ordering: Polynomial interpretation [POLO]: 34.44/9.75 34.44/9.75 POL(A(x_1)) = 1 + x_1 34.44/9.75 POL(B(x_1)) = 3 + x_1 34.44/9.75 POL(a(x_1)) = 1 + x_1 34.44/9.75 POL(b(x_1)) = x_1 34.44/9.75 34.44/9.75 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (12) 34.44/9.75 Obligation: 34.44/9.75 Q DP problem: 34.44/9.75 P is empty. 34.44/9.75 The TRS R consists of the following rules: 34.44/9.75 34.44/9.75 b(b(a(a(x1)))) -> a(a(b(b(x1)))) 34.44/9.75 34.44/9.75 Q is empty. 34.44/9.75 We have to consider all minimal (P,Q,R)-chains. 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (13) PisEmptyProof (EQUIVALENT) 34.44/9.75 The TRS P is empty. Hence, there is no (P,Q,R) chain. 34.44/9.75 ---------------------------------------- 34.44/9.75 34.44/9.75 (14) 34.44/9.75 YES 34.76/9.81 EOF