29.52/8.44 YES 29.57/8.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 29.57/8.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.57/8.49 29.57/8.49 29.57/8.49 Termination w.r.t. Q of the given QTRS could be proven: 29.57/8.49 29.57/8.49 (0) QTRS 29.57/8.49 (1) QTRS Reverse [EQUIVALENT, 0 ms] 29.57/8.49 (2) QTRS 29.57/8.49 (3) DependencyPairsProof [EQUIVALENT, 43 ms] 29.57/8.49 (4) QDP 29.57/8.49 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 29.57/8.49 (6) QDP 29.57/8.49 (7) MRRProof [EQUIVALENT, 117 ms] 29.57/8.49 (8) QDP 29.57/8.49 (9) QDPOrderProof [EQUIVALENT, 0 ms] 29.57/8.49 (10) QDP 29.57/8.49 (11) PisEmptyProof [EQUIVALENT, 0 ms] 29.57/8.49 (12) YES 29.57/8.49 29.57/8.49 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (0) 29.57/8.49 Obligation: 29.57/8.49 Q restricted rewrite system: 29.57/8.49 The TRS R consists of the following rules: 29.57/8.49 29.57/8.49 a(a(b(b(x1)))) -> b(b(c(c(a(a(x1)))))) 29.57/8.49 b(b(c(c(x1)))) -> c(c(b(b(b(b(x1)))))) 29.57/8.49 a(a(c(c(x1)))) -> c(c(a(a(b(b(x1)))))) 29.57/8.49 29.57/8.49 Q is empty. 29.57/8.49 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (1) QTRS Reverse (EQUIVALENT) 29.57/8.49 We applied the QTRS Reverse Processor [REVERSE]. 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (2) 29.57/8.49 Obligation: 29.57/8.49 Q restricted rewrite system: 29.57/8.49 The TRS R consists of the following rules: 29.57/8.49 29.57/8.49 b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) 29.57/8.49 c(c(b(b(x1)))) -> b(b(b(b(c(c(x1)))))) 29.57/8.49 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 29.57/8.49 29.57/8.49 Q is empty. 29.57/8.49 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (3) DependencyPairsProof (EQUIVALENT) 29.57/8.49 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (4) 29.57/8.49 Obligation: 29.57/8.49 Q DP problem: 29.57/8.49 The TRS P consists of the following rules: 29.57/8.49 29.57/8.49 B(b(a(a(x1)))) -> C(c(b(b(x1)))) 29.57/8.49 B(b(a(a(x1)))) -> C(b(b(x1))) 29.57/8.49 B(b(a(a(x1)))) -> B(b(x1)) 29.57/8.49 B(b(a(a(x1)))) -> B(x1) 29.57/8.49 C(c(b(b(x1)))) -> B(b(b(b(c(c(x1)))))) 29.57/8.49 C(c(b(b(x1)))) -> B(b(b(c(c(x1))))) 29.57/8.49 C(c(b(b(x1)))) -> B(b(c(c(x1)))) 29.57/8.49 C(c(b(b(x1)))) -> B(c(c(x1))) 29.57/8.49 C(c(b(b(x1)))) -> C(c(x1)) 29.57/8.49 C(c(b(b(x1)))) -> C(x1) 29.57/8.49 C(c(a(a(x1)))) -> B(b(a(a(c(c(x1)))))) 29.57/8.49 C(c(a(a(x1)))) -> B(a(a(c(c(x1))))) 29.57/8.49 C(c(a(a(x1)))) -> C(c(x1)) 29.57/8.49 C(c(a(a(x1)))) -> C(x1) 29.57/8.49 29.57/8.49 The TRS R consists of the following rules: 29.57/8.49 29.57/8.49 b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) 29.57/8.49 c(c(b(b(x1)))) -> b(b(b(b(c(c(x1)))))) 29.57/8.49 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 29.57/8.49 29.57/8.49 Q is empty. 29.57/8.49 We have to consider all minimal (P,Q,R)-chains. 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (5) DependencyGraphProof (EQUIVALENT) 29.57/8.49 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (6) 29.57/8.49 Obligation: 29.57/8.49 Q DP problem: 29.57/8.49 The TRS P consists of the following rules: 29.57/8.49 29.57/8.49 C(c(b(b(x1)))) -> B(b(b(b(c(c(x1)))))) 29.57/8.49 B(b(a(a(x1)))) -> C(c(b(b(x1)))) 29.57/8.49 C(c(b(b(x1)))) -> B(b(b(c(c(x1))))) 29.57/8.49 B(b(a(a(x1)))) -> B(b(x1)) 29.57/8.49 B(b(a(a(x1)))) -> B(x1) 29.57/8.49 C(c(b(b(x1)))) -> B(b(c(c(x1)))) 29.57/8.49 C(c(b(b(x1)))) -> B(c(c(x1))) 29.57/8.49 C(c(b(b(x1)))) -> C(c(x1)) 29.57/8.49 C(c(b(b(x1)))) -> C(x1) 29.57/8.49 C(c(a(a(x1)))) -> B(b(a(a(c(c(x1)))))) 29.57/8.49 C(c(a(a(x1)))) -> C(c(x1)) 29.57/8.49 C(c(a(a(x1)))) -> C(x1) 29.57/8.49 29.57/8.49 The TRS R consists of the following rules: 29.57/8.49 29.57/8.49 b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) 29.57/8.49 c(c(b(b(x1)))) -> b(b(b(b(c(c(x1)))))) 29.57/8.49 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 29.57/8.49 29.57/8.49 Q is empty. 29.57/8.49 We have to consider all minimal (P,Q,R)-chains. 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (7) MRRProof (EQUIVALENT) 29.57/8.49 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. 29.57/8.49 29.57/8.49 Strictly oriented dependency pairs: 29.57/8.49 29.57/8.49 C(c(b(b(x1)))) -> B(b(b(b(c(c(x1)))))) 29.57/8.49 B(b(a(a(x1)))) -> C(c(b(b(x1)))) 29.57/8.49 C(c(b(b(x1)))) -> B(b(b(c(c(x1))))) 29.57/8.49 B(b(a(a(x1)))) -> B(b(x1)) 29.57/8.49 B(b(a(a(x1)))) -> B(x1) 29.57/8.49 C(c(b(b(x1)))) -> B(b(c(c(x1)))) 29.57/8.49 C(c(b(b(x1)))) -> B(c(c(x1))) 29.57/8.49 C(c(a(a(x1)))) -> B(b(a(a(c(c(x1)))))) 29.57/8.49 C(c(a(a(x1)))) -> C(c(x1)) 29.57/8.49 C(c(a(a(x1)))) -> C(x1) 29.57/8.49 29.57/8.49 29.57/8.49 Used ordering: Polynomial interpretation [POLO]: 29.57/8.49 29.57/8.49 POL(B(x_1)) = 2*x_1 29.57/8.49 POL(C(x_1)) = 1 + 3*x_1 29.57/8.49 POL(a(x_1)) = 2 + 2*x_1 29.57/8.49 POL(b(x_1)) = x_1 29.57/8.49 POL(c(x_1)) = x_1 29.57/8.49 29.57/8.49 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (8) 29.57/8.49 Obligation: 29.57/8.49 Q DP problem: 29.57/8.49 The TRS P consists of the following rules: 29.57/8.49 29.57/8.49 C(c(b(b(x1)))) -> C(c(x1)) 29.57/8.49 C(c(b(b(x1)))) -> C(x1) 29.57/8.49 29.57/8.49 The TRS R consists of the following rules: 29.57/8.49 29.57/8.49 b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) 29.57/8.49 c(c(b(b(x1)))) -> b(b(b(b(c(c(x1)))))) 29.57/8.49 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 29.57/8.49 29.57/8.49 Q is empty. 29.57/8.49 We have to consider all minimal (P,Q,R)-chains. 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (9) QDPOrderProof (EQUIVALENT) 29.57/8.49 We use the reduction pair processor [LPAR04,JAR06]. 29.57/8.49 29.57/8.49 29.57/8.49 The following pairs can be oriented strictly and are deleted. 29.57/8.49 29.57/8.49 C(c(b(b(x1)))) -> C(c(x1)) 29.57/8.49 C(c(b(b(x1)))) -> C(x1) 29.57/8.49 The remaining pairs can at least be oriented weakly. 29.57/8.49 Used ordering: Polynomial interpretation [POLO]: 29.57/8.49 29.57/8.49 POL(C(x_1)) = 2*x_1 29.57/8.49 POL(a(x_1)) = 4 29.57/8.49 POL(b(x_1)) = 1 + x_1 29.57/8.49 POL(c(x_1)) = 2*x_1 29.57/8.49 29.57/8.49 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 29.57/8.49 29.57/8.49 c(c(b(b(x1)))) -> b(b(b(b(c(c(x1)))))) 29.57/8.49 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 29.57/8.49 b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) 29.57/8.49 29.57/8.49 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (10) 29.57/8.49 Obligation: 29.57/8.49 Q DP problem: 29.57/8.49 P is empty. 29.57/8.49 The TRS R consists of the following rules: 29.57/8.49 29.57/8.49 b(b(a(a(x1)))) -> a(a(c(c(b(b(x1)))))) 29.57/8.49 c(c(b(b(x1)))) -> b(b(b(b(c(c(x1)))))) 29.57/8.49 c(c(a(a(x1)))) -> b(b(a(a(c(c(x1)))))) 29.57/8.49 29.57/8.49 Q is empty. 29.57/8.49 We have to consider all minimal (P,Q,R)-chains. 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (11) PisEmptyProof (EQUIVALENT) 29.57/8.49 The TRS P is empty. Hence, there is no (P,Q,R) chain. 29.57/8.49 ---------------------------------------- 29.57/8.49 29.57/8.49 (12) 29.57/8.49 YES 29.88/8.61 EOF