24.59/7.17 YES 24.71/7.21 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 24.71/7.21 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.71/7.21 24.71/7.21 24.71/7.21 Termination w.r.t. Q of the given QTRS could be proven: 24.71/7.21 24.71/7.21 (0) QTRS 24.71/7.21 (1) QTRSRRRProof [EQUIVALENT, 53 ms] 24.71/7.21 (2) QTRS 24.71/7.21 (3) DependencyPairsProof [EQUIVALENT, 3 ms] 24.71/7.21 (4) QDP 24.71/7.21 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 24.71/7.21 (6) QDP 24.71/7.21 (7) QDPOrderProof [EQUIVALENT, 18 ms] 24.71/7.21 (8) QDP 24.71/7.21 (9) PisEmptyProof [EQUIVALENT, 0 ms] 24.71/7.21 (10) YES 24.71/7.21 24.71/7.21 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (0) 24.71/7.21 Obligation: 24.71/7.21 Q restricted rewrite system: 24.71/7.21 The TRS R consists of the following rules: 24.71/7.21 24.71/7.21 b(c(a(x1))) -> a(b(x1)) 24.71/7.21 b(b(b(x1))) -> c(a(c(x1))) 24.71/7.21 c(d(x1)) -> d(c(x1)) 24.71/7.21 c(d(b(x1))) -> d(c(c(x1))) 24.71/7.21 d(c(x1)) -> b(b(b(x1))) 24.71/7.21 c(b(x1)) -> d(a(x1)) 24.71/7.21 d(b(c(x1))) -> a(a(x1)) 24.71/7.21 d(a(x1)) -> b(x1) 24.71/7.21 24.71/7.21 Q is empty. 24.71/7.21 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (1) QTRSRRRProof (EQUIVALENT) 24.71/7.21 Used ordering: 24.71/7.21 Polynomial interpretation [POLO]: 24.71/7.21 24.71/7.21 POL(a(x_1)) = x_1 24.71/7.21 POL(b(x_1)) = 1 + x_1 24.71/7.21 POL(c(x_1)) = 1 + x_1 24.71/7.21 POL(d(x_1)) = 2 + x_1 24.71/7.21 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 24.71/7.21 24.71/7.21 b(c(a(x1))) -> a(b(x1)) 24.71/7.21 b(b(b(x1))) -> c(a(c(x1))) 24.71/7.21 d(b(c(x1))) -> a(a(x1)) 24.71/7.21 d(a(x1)) -> b(x1) 24.71/7.21 24.71/7.21 24.71/7.21 24.71/7.21 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (2) 24.71/7.21 Obligation: 24.71/7.21 Q restricted rewrite system: 24.71/7.21 The TRS R consists of the following rules: 24.71/7.21 24.71/7.21 c(d(x1)) -> d(c(x1)) 24.71/7.21 c(d(b(x1))) -> d(c(c(x1))) 24.71/7.21 d(c(x1)) -> b(b(b(x1))) 24.71/7.21 c(b(x1)) -> d(a(x1)) 24.71/7.21 24.71/7.21 Q is empty. 24.71/7.21 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (3) DependencyPairsProof (EQUIVALENT) 24.71/7.21 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (4) 24.71/7.21 Obligation: 24.71/7.21 Q DP problem: 24.71/7.21 The TRS P consists of the following rules: 24.71/7.21 24.71/7.21 C(d(x1)) -> D(c(x1)) 24.71/7.21 C(d(x1)) -> C(x1) 24.71/7.21 C(d(b(x1))) -> D(c(c(x1))) 24.71/7.21 C(d(b(x1))) -> C(c(x1)) 24.71/7.21 C(d(b(x1))) -> C(x1) 24.71/7.21 C(b(x1)) -> D(a(x1)) 24.71/7.21 24.71/7.21 The TRS R consists of the following rules: 24.71/7.21 24.71/7.21 c(d(x1)) -> d(c(x1)) 24.71/7.21 c(d(b(x1))) -> d(c(c(x1))) 24.71/7.21 d(c(x1)) -> b(b(b(x1))) 24.71/7.21 c(b(x1)) -> d(a(x1)) 24.71/7.21 24.71/7.21 Q is empty. 24.71/7.21 We have to consider all minimal (P,Q,R)-chains. 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (5) DependencyGraphProof (EQUIVALENT) 24.71/7.21 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (6) 24.71/7.21 Obligation: 24.71/7.21 Q DP problem: 24.71/7.21 The TRS P consists of the following rules: 24.71/7.21 24.71/7.21 C(d(b(x1))) -> C(c(x1)) 24.71/7.21 C(d(x1)) -> C(x1) 24.71/7.21 C(d(b(x1))) -> C(x1) 24.71/7.21 24.71/7.21 The TRS R consists of the following rules: 24.71/7.21 24.71/7.21 c(d(x1)) -> d(c(x1)) 24.71/7.21 c(d(b(x1))) -> d(c(c(x1))) 24.71/7.21 d(c(x1)) -> b(b(b(x1))) 24.71/7.21 c(b(x1)) -> d(a(x1)) 24.71/7.21 24.71/7.21 Q is empty. 24.71/7.21 We have to consider all minimal (P,Q,R)-chains. 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (7) QDPOrderProof (EQUIVALENT) 24.71/7.21 We use the reduction pair processor [LPAR04,JAR06]. 24.71/7.21 24.71/7.21 24.71/7.21 The following pairs can be oriented strictly and are deleted. 24.71/7.21 24.71/7.21 C(d(b(x1))) -> C(c(x1)) 24.71/7.21 C(d(x1)) -> C(x1) 24.71/7.21 C(d(b(x1))) -> C(x1) 24.71/7.21 The remaining pairs can at least be oriented weakly. 24.71/7.21 Used ordering: Polynomial interpretation [POLO]: 24.71/7.21 24.71/7.21 POL(C(x_1)) = 4*x_1 24.71/7.21 POL(a(x_1)) = 0 24.71/7.21 POL(b(x_1)) = 1 + 2*x_1 24.71/7.21 POL(c(x_1)) = 1 + 2*x_1 24.71/7.21 POL(d(x_1)) = 3 + 4*x_1 24.71/7.21 24.71/7.21 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 24.71/7.21 24.71/7.21 c(d(x1)) -> d(c(x1)) 24.71/7.21 c(d(b(x1))) -> d(c(c(x1))) 24.71/7.21 c(b(x1)) -> d(a(x1)) 24.71/7.21 d(c(x1)) -> b(b(b(x1))) 24.71/7.21 24.71/7.21 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (8) 24.71/7.21 Obligation: 24.71/7.21 Q DP problem: 24.71/7.21 P is empty. 24.71/7.21 The TRS R consists of the following rules: 24.71/7.21 24.71/7.21 c(d(x1)) -> d(c(x1)) 24.71/7.21 c(d(b(x1))) -> d(c(c(x1))) 24.71/7.21 d(c(x1)) -> b(b(b(x1))) 24.71/7.21 c(b(x1)) -> d(a(x1)) 24.71/7.21 24.71/7.21 Q is empty. 24.71/7.21 We have to consider all minimal (P,Q,R)-chains. 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (9) PisEmptyProof (EQUIVALENT) 24.71/7.21 The TRS P is empty. Hence, there is no (P,Q,R) chain. 24.71/7.21 ---------------------------------------- 24.71/7.21 24.71/7.21 (10) 24.71/7.21 YES 25.02/7.36 EOF