26.81/7.78 YES 30.80/11.07 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 30.80/11.07 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.80/11.07 30.80/11.07 30.80/11.07 Termination w.r.t. Q of the given QTRS could be proven: 30.80/11.07 30.80/11.07 (0) QTRS 30.80/11.07 (1) QTRS Reverse [EQUIVALENT, 0 ms] 30.80/11.07 (2) QTRS 30.80/11.07 (3) DependencyPairsProof [EQUIVALENT, 10 ms] 30.80/11.07 (4) QDP 30.80/11.07 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 30.80/11.07 (6) QDP 30.80/11.07 (7) QDPOrderProof [EQUIVALENT, 429 ms] 30.80/11.07 (8) QDP 30.80/11.07 (9) PisEmptyProof [EQUIVALENT, 0 ms] 30.80/11.07 (10) YES 30.80/11.07 30.80/11.07 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (0) 30.80/11.07 Obligation: 30.80/11.07 Q restricted rewrite system: 30.80/11.07 The TRS R consists of the following rules: 30.80/11.07 30.80/11.07 a(b(c(x1))) -> c(c(c(b(b(b(a(a(a(x1))))))))) 30.80/11.07 c(b(x1)) -> a(a(a(x1))) 30.80/11.07 a(x1) -> x1 30.80/11.07 b(x1) -> x1 30.80/11.07 c(x1) -> x1 30.80/11.07 30.80/11.07 Q is empty. 30.80/11.07 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (1) QTRS Reverse (EQUIVALENT) 30.80/11.07 We applied the QTRS Reverse Processor [REVERSE]. 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (2) 30.80/11.07 Obligation: 30.80/11.07 Q restricted rewrite system: 30.80/11.07 The TRS R consists of the following rules: 30.80/11.07 30.80/11.07 c(b(a(x1))) -> a(a(a(b(b(b(c(c(c(x1))))))))) 30.80/11.07 b(c(x1)) -> a(a(a(x1))) 30.80/11.07 a(x1) -> x1 30.80/11.07 b(x1) -> x1 30.80/11.07 c(x1) -> x1 30.80/11.07 30.80/11.07 Q is empty. 30.80/11.07 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (3) DependencyPairsProof (EQUIVALENT) 30.80/11.07 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (4) 30.80/11.07 Obligation: 30.80/11.07 Q DP problem: 30.80/11.07 The TRS P consists of the following rules: 30.80/11.07 30.80/11.07 C(b(a(x1))) -> A(a(a(b(b(b(c(c(c(x1))))))))) 30.80/11.07 C(b(a(x1))) -> A(a(b(b(b(c(c(c(x1)))))))) 30.80/11.07 C(b(a(x1))) -> A(b(b(b(c(c(c(x1))))))) 30.80/11.07 C(b(a(x1))) -> B(b(b(c(c(c(x1)))))) 30.80/11.07 C(b(a(x1))) -> B(b(c(c(c(x1))))) 30.80/11.07 C(b(a(x1))) -> B(c(c(c(x1)))) 30.80/11.07 C(b(a(x1))) -> C(c(c(x1))) 30.80/11.07 C(b(a(x1))) -> C(c(x1)) 30.80/11.07 C(b(a(x1))) -> C(x1) 30.80/11.07 B(c(x1)) -> A(a(a(x1))) 30.80/11.07 B(c(x1)) -> A(a(x1)) 30.80/11.07 B(c(x1)) -> A(x1) 30.80/11.07 30.80/11.07 The TRS R consists of the following rules: 30.80/11.07 30.80/11.07 c(b(a(x1))) -> a(a(a(b(b(b(c(c(c(x1))))))))) 30.80/11.07 b(c(x1)) -> a(a(a(x1))) 30.80/11.07 a(x1) -> x1 30.80/11.07 b(x1) -> x1 30.80/11.07 c(x1) -> x1 30.80/11.07 30.80/11.07 Q is empty. 30.80/11.07 We have to consider all minimal (P,Q,R)-chains. 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (5) DependencyGraphProof (EQUIVALENT) 30.80/11.07 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes. 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (6) 30.80/11.07 Obligation: 30.80/11.07 Q DP problem: 30.80/11.07 The TRS P consists of the following rules: 30.80/11.07 30.80/11.07 C(b(a(x1))) -> C(c(x1)) 30.80/11.07 C(b(a(x1))) -> C(c(c(x1))) 30.80/11.07 C(b(a(x1))) -> C(x1) 30.80/11.07 30.80/11.07 The TRS R consists of the following rules: 30.80/11.07 30.80/11.07 c(b(a(x1))) -> a(a(a(b(b(b(c(c(c(x1))))))))) 30.80/11.07 b(c(x1)) -> a(a(a(x1))) 30.80/11.07 a(x1) -> x1 30.80/11.07 b(x1) -> x1 30.80/11.07 c(x1) -> x1 30.80/11.07 30.80/11.07 Q is empty. 30.80/11.07 We have to consider all minimal (P,Q,R)-chains. 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (7) QDPOrderProof (EQUIVALENT) 30.80/11.07 We use the reduction pair processor [LPAR04,JAR06]. 30.80/11.07 30.80/11.07 30.80/11.07 The following pairs can be oriented strictly and are deleted. 30.80/11.07 30.80/11.07 C(b(a(x1))) -> C(c(x1)) 30.80/11.07 C(b(a(x1))) -> C(c(c(x1))) 30.80/11.07 C(b(a(x1))) -> C(x1) 30.80/11.07 The remaining pairs can at least be oriented weakly. 30.80/11.07 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 30.80/11.07 30.80/11.07 <<< 30.80/11.07 POL(C(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 30.80/11.07 >>> 30.80/11.07 30.80/11.07 <<< 30.80/11.07 POL(b(x_1)) = [[-I], [-I], [1A]] + [[0A, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] * x_1 30.80/11.07 >>> 30.80/11.07 30.80/11.07 <<< 30.80/11.07 POL(a(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [-I, 0A, 1A], [-I, -I, 0A]] * x_1 30.80/11.07 >>> 30.80/11.07 30.80/11.07 <<< 30.80/11.07 POL(c(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 1A], [-I, -I, 0A]] * x_1 30.80/11.07 >>> 30.80/11.07 30.80/11.07 30.80/11.07 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 30.80/11.07 30.80/11.07 c(b(a(x1))) -> a(a(a(b(b(b(c(c(c(x1))))))))) 30.80/11.07 c(x1) -> x1 30.80/11.07 b(c(x1)) -> a(a(a(x1))) 30.80/11.07 b(x1) -> x1 30.80/11.07 a(x1) -> x1 30.80/11.07 30.80/11.07 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (8) 30.80/11.07 Obligation: 30.80/11.07 Q DP problem: 30.80/11.07 P is empty. 30.80/11.07 The TRS R consists of the following rules: 30.80/11.07 30.80/11.07 c(b(a(x1))) -> a(a(a(b(b(b(c(c(c(x1))))))))) 30.80/11.07 b(c(x1)) -> a(a(a(x1))) 30.80/11.07 a(x1) -> x1 30.80/11.07 b(x1) -> x1 30.80/11.07 c(x1) -> x1 30.80/11.07 30.80/11.07 Q is empty. 30.80/11.07 We have to consider all minimal (P,Q,R)-chains. 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (9) PisEmptyProof (EQUIVALENT) 30.80/11.07 The TRS P is empty. Hence, there is no (P,Q,R) chain. 30.80/11.07 ---------------------------------------- 30.80/11.07 30.80/11.07 (10) 30.80/11.07 YES 30.83/11.12 EOF