24.93/7.34 YES 25.64/7.46 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 25.64/7.46 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.64/7.46 25.64/7.46 25.64/7.46 Termination w.r.t. Q of the given QTRS could be proven: 25.64/7.46 25.64/7.46 (0) QTRS 25.64/7.46 (1) DependencyPairsProof [EQUIVALENT, 1 ms] 25.64/7.46 (2) QDP 25.64/7.46 (3) DependencyGraphProof [EQUIVALENT, 1 ms] 25.64/7.46 (4) QDP 25.64/7.46 (5) QDPOrderProof [EQUIVALENT, 100 ms] 25.64/7.46 (6) QDP 25.64/7.46 (7) UsableRulesProof [EQUIVALENT, 0 ms] 25.64/7.46 (8) QDP 25.64/7.46 (9) MNOCProof [EQUIVALENT, 1 ms] 25.64/7.46 (10) QDP 25.64/7.46 (11) MRRProof [EQUIVALENT, 0 ms] 25.64/7.46 (12) QDP 25.64/7.46 (13) PisEmptyProof [EQUIVALENT, 0 ms] 25.64/7.46 (14) YES 25.64/7.46 25.64/7.46 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (0) 25.64/7.46 Obligation: 25.64/7.46 Q restricted rewrite system: 25.64/7.46 The TRS R consists of the following rules: 25.64/7.46 25.64/7.46 a(x1) -> b(x1) 25.64/7.46 a(a(b(x1))) -> a(b(a(a(c(x1))))) 25.64/7.46 c(b(x1)) -> x1 25.64/7.46 25.64/7.46 Q is empty. 25.64/7.46 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (1) DependencyPairsProof (EQUIVALENT) 25.64/7.46 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (2) 25.64/7.46 Obligation: 25.64/7.46 Q DP problem: 25.64/7.46 The TRS P consists of the following rules: 25.64/7.46 25.64/7.46 A(a(b(x1))) -> A(b(a(a(c(x1))))) 25.64/7.46 A(a(b(x1))) -> A(a(c(x1))) 25.64/7.46 A(a(b(x1))) -> A(c(x1)) 25.64/7.46 A(a(b(x1))) -> C(x1) 25.64/7.46 25.64/7.46 The TRS R consists of the following rules: 25.64/7.46 25.64/7.46 a(x1) -> b(x1) 25.64/7.46 a(a(b(x1))) -> a(b(a(a(c(x1))))) 25.64/7.46 c(b(x1)) -> x1 25.64/7.46 25.64/7.46 Q is empty. 25.64/7.46 We have to consider all minimal (P,Q,R)-chains. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (3) DependencyGraphProof (EQUIVALENT) 25.64/7.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (4) 25.64/7.46 Obligation: 25.64/7.46 Q DP problem: 25.64/7.46 The TRS P consists of the following rules: 25.64/7.46 25.64/7.46 A(a(b(x1))) -> A(c(x1)) 25.64/7.46 A(a(b(x1))) -> A(a(c(x1))) 25.64/7.46 25.64/7.46 The TRS R consists of the following rules: 25.64/7.46 25.64/7.46 a(x1) -> b(x1) 25.64/7.46 a(a(b(x1))) -> a(b(a(a(c(x1))))) 25.64/7.46 c(b(x1)) -> x1 25.64/7.46 25.64/7.46 Q is empty. 25.64/7.46 We have to consider all minimal (P,Q,R)-chains. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (5) QDPOrderProof (EQUIVALENT) 25.64/7.46 We use the reduction pair processor [LPAR04,JAR06]. 25.64/7.46 25.64/7.46 25.64/7.46 The following pairs can be oriented strictly and are deleted. 25.64/7.46 25.64/7.46 A(a(b(x1))) -> A(a(c(x1))) 25.64/7.46 The remaining pairs can at least be oriented weakly. 25.64/7.46 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 25.64/7.46 25.64/7.46 <<< 25.64/7.46 POL(A(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 25.64/7.46 >>> 25.64/7.46 25.64/7.46 <<< 25.64/7.46 POL(a(x_1)) = [[1A], [0A], [0A]] + [[1A, 0A, 1A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 25.64/7.46 >>> 25.64/7.46 25.64/7.46 <<< 25.64/7.46 POL(b(x_1)) = [[1A], [0A], [-I]] + [[-I, 0A, 1A], [0A, -I, 0A], [0A, -I, 0A]] * x_1 25.64/7.46 >>> 25.64/7.46 25.64/7.46 <<< 25.64/7.46 POL(c(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, 0A], [0A, 0A, 1A], [-I, -I, 0A]] * x_1 25.64/7.46 >>> 25.64/7.46 25.64/7.46 25.64/7.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 25.64/7.46 25.64/7.46 c(b(x1)) -> x1 25.64/7.46 a(x1) -> b(x1) 25.64/7.46 a(a(b(x1))) -> a(b(a(a(c(x1))))) 25.64/7.46 25.64/7.46 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (6) 25.64/7.46 Obligation: 25.64/7.46 Q DP problem: 25.64/7.46 The TRS P consists of the following rules: 25.64/7.46 25.64/7.46 A(a(b(x1))) -> A(c(x1)) 25.64/7.46 25.64/7.46 The TRS R consists of the following rules: 25.64/7.46 25.64/7.46 a(x1) -> b(x1) 25.64/7.46 a(a(b(x1))) -> a(b(a(a(c(x1))))) 25.64/7.46 c(b(x1)) -> x1 25.64/7.46 25.64/7.46 Q is empty. 25.64/7.46 We have to consider all minimal (P,Q,R)-chains. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (7) UsableRulesProof (EQUIVALENT) 25.64/7.46 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. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (8) 25.64/7.46 Obligation: 25.64/7.46 Q DP problem: 25.64/7.46 The TRS P consists of the following rules: 25.64/7.46 25.64/7.46 A(a(b(x1))) -> A(c(x1)) 25.64/7.46 25.64/7.46 The TRS R consists of the following rules: 25.64/7.46 25.64/7.46 c(b(x1)) -> x1 25.64/7.46 25.64/7.46 Q is empty. 25.64/7.46 We have to consider all minimal (P,Q,R)-chains. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (9) MNOCProof (EQUIVALENT) 25.64/7.46 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (10) 25.64/7.46 Obligation: 25.64/7.46 Q DP problem: 25.64/7.46 The TRS P consists of the following rules: 25.64/7.46 25.64/7.46 A(a(b(x1))) -> A(c(x1)) 25.64/7.46 25.64/7.46 The TRS R consists of the following rules: 25.64/7.46 25.64/7.46 c(b(x1)) -> x1 25.64/7.46 25.64/7.46 The set Q consists of the following terms: 25.64/7.46 25.64/7.46 c(b(x0)) 25.64/7.46 25.64/7.46 We have to consider all minimal (P,Q,R)-chains. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (11) MRRProof (EQUIVALENT) 25.64/7.46 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. 25.64/7.46 25.64/7.46 Strictly oriented dependency pairs: 25.64/7.46 25.64/7.46 A(a(b(x1))) -> A(c(x1)) 25.64/7.46 25.64/7.46 Strictly oriented rules of the TRS R: 25.64/7.46 25.64/7.46 c(b(x1)) -> x1 25.64/7.46 25.64/7.46 Used ordering: Polynomial interpretation [POLO]: 25.64/7.46 25.64/7.46 POL(A(x_1)) = 3*x_1 25.64/7.46 POL(a(x_1)) = 3 + x_1 25.64/7.46 POL(b(x_1)) = 2 + 3*x_1 25.64/7.46 POL(c(x_1)) = x_1 25.64/7.46 25.64/7.46 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (12) 25.64/7.46 Obligation: 25.64/7.46 Q DP problem: 25.64/7.46 P is empty. 25.64/7.46 R is empty. 25.64/7.46 The set Q consists of the following terms: 25.64/7.46 25.64/7.46 c(b(x0)) 25.64/7.46 25.64/7.46 We have to consider all minimal (P,Q,R)-chains. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (13) PisEmptyProof (EQUIVALENT) 25.64/7.46 The TRS P is empty. Hence, there is no (P,Q,R) chain. 25.64/7.46 ---------------------------------------- 25.64/7.46 25.64/7.46 (14) 25.64/7.46 YES 25.87/7.56 EOF