27.09/7.83	YES
27.87/8.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.87/8.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.87/8.05	
27.87/8.05	
27.87/8.05	Termination w.r.t. Q of the given QTRS could be proven:
27.87/8.05	
27.87/8.05	(0) QTRS
27.87/8.05	(1) FlatCCProof [EQUIVALENT, 0 ms]
27.87/8.05	(2) QTRS
27.87/8.05	(3) RootLabelingProof [EQUIVALENT, 0 ms]
27.87/8.05	(4) QTRS
27.87/8.05	(5) DependencyPairsProof [EQUIVALENT, 1 ms]
27.87/8.05	(6) QDP
27.87/8.05	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
27.87/8.05	(8) QDP
27.87/8.05	(9) QDPOrderProof [EQUIVALENT, 134 ms]
27.87/8.05	(10) QDP
27.87/8.05	(11) PisEmptyProof [EQUIVALENT, 0 ms]
27.87/8.05	(12) YES
27.87/8.05	
27.87/8.05	
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(0)
27.87/8.05	Obligation:
27.87/8.05	Q restricted rewrite system:
27.87/8.05	The TRS R consists of the following rules:
27.87/8.05	
27.87/8.05	   a(a(b(a(x1)))) -> a(a(a(a(x1))))
27.87/8.05	   a(a(a(b(x1)))) -> b(a(b(a(x1))))
27.87/8.05	
27.87/8.05	Q is empty.
27.87/8.05	
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(1) FlatCCProof (EQUIVALENT)
27.87/8.05	We used flat context closure [ROOTLAB]
27.87/8.05	As Q is empty the flat context closure was sound AND complete.
27.87/8.05	
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(2)
27.87/8.05	Obligation:
27.87/8.05	Q restricted rewrite system:
27.87/8.05	The TRS R consists of the following rules:
27.87/8.05	
27.87/8.05	   a(a(b(a(x1)))) -> a(a(a(a(x1))))
27.87/8.05	   a(a(a(a(b(x1))))) -> a(b(a(b(a(x1)))))
27.87/8.05	   b(a(a(a(b(x1))))) -> b(b(a(b(a(x1)))))
27.87/8.05	
27.87/8.05	Q is empty.
27.87/8.05	
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(3) RootLabelingProof (EQUIVALENT)
27.87/8.05	We used plain root labeling [ROOTLAB] with the following heuristic:
27.87/8.05	LabelAll: All function symbols get labeled
27.87/8.05	
27.87/8.05	As Q is empty the root labeling was sound AND complete.
27.87/8.05	
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(4)
27.87/8.05	Obligation:
27.87/8.05	Q restricted rewrite system:
27.87/8.05	The TRS R consists of the following rules:
27.87/8.05	
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	
27.87/8.05	Q is empty.
27.87/8.05	
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(5) DependencyPairsProof (EQUIVALENT)
27.87/8.05	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(6)
27.87/8.05	Obligation:
27.87/8.05	Q DP problem:
27.87/8.05	The TRS P consists of the following rules:
27.87/8.05	
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
27.87/8.05	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
27.87/8.05	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
27.87/8.05	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
27.87/8.05	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
27.87/8.05	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
27.87/8.05	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
27.87/8.05	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
27.87/8.05	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
27.87/8.05	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
27.87/8.05	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
27.87/8.05	
27.87/8.05	The TRS R consists of the following rules:
27.87/8.05	
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	
27.87/8.05	Q is empty.
27.87/8.05	We have to consider all minimal (P,Q,R)-chains.
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(7) DependencyGraphProof (EQUIVALENT)
27.87/8.05	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(8)
27.87/8.05	Obligation:
27.87/8.05	Q DP problem:
27.87/8.05	The TRS P consists of the following rules:
27.87/8.05	
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
27.87/8.05	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
27.87/8.05	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
27.87/8.05	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
27.87/8.05	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
27.87/8.05	
27.87/8.05	The TRS R consists of the following rules:
27.87/8.05	
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	
27.87/8.05	Q is empty.
27.87/8.05	We have to consider all minimal (P,Q,R)-chains.
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(9) QDPOrderProof (EQUIVALENT)
27.87/8.05	We use the reduction pair processor [LPAR04,JAR06].
27.87/8.05	
27.87/8.05	
27.87/8.05	The following pairs can be oriented strictly and are deleted.
27.87/8.05	
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
27.87/8.05	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
27.87/8.05	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
27.87/8.05	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
27.87/8.05	   B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
27.87/8.05	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
27.87/8.05	The remaining pairs can at least be oriented weakly.
27.87/8.05	Used ordering:  Polynomial interpretation [POLO]:
27.87/8.05	
27.87/8.05	   POL(A_{A_1}(x_1)) = 4*x_1
27.87/8.05	   POL(B_{A_1}(x_1)) = 2*x_1
27.87/8.05	   POL(a_{a_1}(x_1)) = 4*x_1
27.87/8.05	   POL(a_{b_1}(x_1)) = 4 + 4*x_1
27.87/8.05	   POL(b_{a_1}(x_1)) = 4*x_1
27.87/8.05	   POL(b_{b_1}(x_1)) = 4 + 4*x_1
27.87/8.05	
27.87/8.05	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.87/8.05	
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	
27.87/8.05	
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(10)
27.87/8.05	Obligation:
27.87/8.05	Q DP problem:
27.87/8.05	P is empty.
27.87/8.05	The TRS R consists of the following rules:
27.87/8.05	
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
27.87/8.05	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
27.87/8.05	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
27.87/8.05	
27.87/8.05	Q is empty.
27.87/8.05	We have to consider all minimal (P,Q,R)-chains.
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(11) PisEmptyProof (EQUIVALENT)
27.87/8.05	The TRS P is empty. Hence, there is no (P,Q,R) chain.
27.87/8.05	----------------------------------------
27.87/8.05	
27.87/8.05	(12)
27.87/8.05	YES
28.13/8.18	EOF