13.04/4.14	YES
13.22/4.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
13.22/4.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.22/4.21	
13.22/4.21	
13.22/4.21	Termination w.r.t. Q of the given QTRS could be proven:
13.22/4.21	
13.22/4.21	(0) QTRS
13.22/4.21	(1) FlatCCProof [EQUIVALENT, 0 ms]
13.22/4.21	(2) QTRS
13.22/4.21	(3) RootLabelingProof [EQUIVALENT, 0 ms]
13.22/4.21	(4) QTRS
13.22/4.21	(5) DependencyPairsProof [EQUIVALENT, 28 ms]
13.22/4.21	(6) QDP
13.22/4.21	(7) QDPOrderProof [EQUIVALENT, 96 ms]
13.22/4.21	(8) QDP
13.22/4.21	(9) PisEmptyProof [EQUIVALENT, 0 ms]
13.22/4.21	(10) YES
13.22/4.21	
13.22/4.21	
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(0)
13.22/4.21	Obligation:
13.22/4.21	Q restricted rewrite system:
13.22/4.21	The TRS R consists of the following rules:
13.22/4.21	
13.22/4.21	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
13.22/4.21	   b(a(b(a(x1)))) -> b(b(b(b(x1))))
13.22/4.21	   a(a(a(b(x1)))) -> a(b(a(a(x1))))
13.22/4.21	
13.22/4.21	Q is empty.
13.22/4.21	
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(1) FlatCCProof (EQUIVALENT)
13.22/4.21	We used flat context closure [ROOTLAB]
13.22/4.21	As Q is empty the flat context closure was sound AND complete.
13.22/4.21	
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(2)
13.22/4.21	Obligation:
13.22/4.21	Q restricted rewrite system:
13.22/4.21	The TRS R consists of the following rules:
13.22/4.21	
13.22/4.21	   b(a(b(a(x1)))) -> b(b(b(b(x1))))
13.22/4.21	   a(a(a(b(x1)))) -> a(b(a(a(x1))))
13.22/4.21	   a(a(b(b(b(x1))))) -> a(b(a(b(a(x1)))))
13.22/4.21	   b(a(b(b(b(x1))))) -> b(b(a(b(a(x1)))))
13.22/4.21	
13.22/4.21	Q is empty.
13.22/4.21	
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(3) RootLabelingProof (EQUIVALENT)
13.22/4.21	We used plain root labeling [ROOTLAB] with the following heuristic:
13.22/4.21	LabelAll: All function symbols get labeled
13.22/4.21	
13.22/4.21	As Q is empty the root labeling was sound AND complete.
13.22/4.21	
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(4)
13.22/4.21	Obligation:
13.22/4.21	Q restricted rewrite system:
13.22/4.21	The TRS R consists of the following rules:
13.22/4.21	
13.22/4.21	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
13.22/4.21	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.22/4.21	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.22/4.21	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
13.22/4.21	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
13.22/4.21	
13.22/4.21	Q is empty.
13.22/4.21	
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(5) DependencyPairsProof (EQUIVALENT)
13.22/4.21	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(6)
13.22/4.21	Obligation:
13.22/4.21	Q DP problem:
13.22/4.21	The TRS P consists of the following rules:
13.22/4.21	
13.22/4.21	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
13.22/4.21	
13.22/4.21	The TRS R consists of the following rules:
13.22/4.21	
13.22/4.21	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
13.22/4.21	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.22/4.21	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.22/4.21	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
13.22/4.21	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
13.22/4.21	
13.22/4.21	Q is empty.
13.22/4.21	We have to consider all minimal (P,Q,R)-chains.
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(7) QDPOrderProof (EQUIVALENT)
13.22/4.21	We use the reduction pair processor [LPAR04,JAR06].
13.22/4.21	
13.22/4.21	
13.22/4.21	The following pairs can be oriented strictly and are deleted.
13.22/4.21	
13.22/4.21	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
13.22/4.21	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
13.22/4.21	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
13.22/4.21	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
13.22/4.21	The remaining pairs can at least be oriented weakly.
13.22/4.21	Used ordering:  Polynomial interpretation [POLO]:
13.22/4.21	
13.22/4.21	   POL(A_{A_1}(x_1)) = x_1
13.22/4.21	   POL(B_{A_1}(x_1)) = x_1
13.22/4.21	   POL(a_{a_1}(x_1)) = 1 + x_1
13.22/4.21	   POL(a_{b_1}(x_1)) = 1 + x_1
13.22/4.21	   POL(b_{a_1}(x_1)) = 1 + x_1
13.22/4.21	   POL(b_{b_1}(x_1)) = 1 + x_1
13.22/4.21	
13.22/4.21	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
13.22/4.21	
13.22/4.21	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
13.22/4.21	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
13.22/4.21	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.22/4.21	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
13.22/4.21	
13.22/4.21	
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(8)
13.22/4.21	Obligation:
13.22/4.21	Q DP problem:
13.22/4.21	P is empty.
13.22/4.21	The TRS R consists of the following rules:
13.22/4.21	
13.22/4.21	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
13.22/4.21	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.22/4.21	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.22/4.21	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
13.22/4.21	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
13.22/4.21	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
13.22/4.21	
13.22/4.21	Q is empty.
13.22/4.21	We have to consider all minimal (P,Q,R)-chains.
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(9) PisEmptyProof (EQUIVALENT)
13.22/4.21	The TRS P is empty. Hence, there is no (P,Q,R) chain.
13.22/4.21	----------------------------------------
13.22/4.21	
13.22/4.21	(10)
13.22/4.21	YES
13.48/4.31	EOF