16.78/5.25	YES
17.33/5.33	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
17.33/5.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.33/5.33	
17.33/5.33	
17.33/5.33	Termination w.r.t. Q of the given QTRS could be proven:
17.33/5.33	
17.33/5.33	(0) QTRS
17.33/5.33	(1) FlatCCProof [EQUIVALENT, 0 ms]
17.33/5.33	(2) QTRS
17.33/5.33	(3) RootLabelingProof [EQUIVALENT, 0 ms]
17.33/5.33	(4) QTRS
17.33/5.33	(5) DependencyPairsProof [EQUIVALENT, 31 ms]
17.33/5.33	(6) QDP
17.33/5.33	(7) QDPOrderProof [EQUIVALENT, 84 ms]
17.33/5.33	(8) QDP
17.33/5.33	(9) PisEmptyProof [EQUIVALENT, 0 ms]
17.33/5.33	(10) YES
17.33/5.33	
17.33/5.33	
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(0)
17.33/5.33	Obligation:
17.33/5.33	Q restricted rewrite system:
17.33/5.33	The TRS R consists of the following rules:
17.33/5.33	
17.33/5.33	   a(b(b(a(x1)))) -> a(a(a(a(x1))))
17.33/5.33	   b(a(a(a(x1)))) -> a(a(b(b(x1))))
17.33/5.33	   b(b(a(a(x1)))) -> b(a(b(a(x1))))
17.33/5.33	
17.33/5.33	Q is empty.
17.33/5.33	
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(1) FlatCCProof (EQUIVALENT)
17.33/5.33	We used flat context closure [ROOTLAB]
17.33/5.33	As Q is empty the flat context closure was sound AND complete.
17.33/5.33	
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(2)
17.33/5.33	Obligation:
17.33/5.33	Q restricted rewrite system:
17.33/5.33	The TRS R consists of the following rules:
17.33/5.33	
17.33/5.33	   a(b(b(a(x1)))) -> a(a(a(a(x1))))
17.33/5.33	   b(b(a(a(x1)))) -> b(a(b(a(x1))))
17.33/5.33	   a(b(a(a(a(x1))))) -> a(a(a(b(b(x1)))))
17.33/5.33	   b(b(a(a(a(x1))))) -> b(a(a(b(b(x1)))))
17.33/5.33	
17.33/5.33	Q is empty.
17.33/5.33	
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(3) RootLabelingProof (EQUIVALENT)
17.33/5.33	We used plain root labeling [ROOTLAB] with the following heuristic:
17.33/5.33	LabelAll: All function symbols get labeled
17.33/5.33	
17.33/5.33	As Q is empty the root labeling was sound AND complete.
17.33/5.33	
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(4)
17.33/5.33	Obligation:
17.33/5.33	Q restricted rewrite system:
17.33/5.33	The TRS R consists of the following rules:
17.33/5.33	
17.33/5.33	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
17.33/5.33	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
17.33/5.33	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.33/5.33	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.33/5.33	
17.33/5.33	Q is empty.
17.33/5.33	
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(5) DependencyPairsProof (EQUIVALENT)
17.33/5.33	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(6)
17.33/5.33	Obligation:
17.33/5.33	Q DP problem:
17.33/5.33	The TRS P consists of the following rules:
17.33/5.33	
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.33/5.33	
17.33/5.33	The TRS R consists of the following rules:
17.33/5.33	
17.33/5.33	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
17.33/5.33	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
17.33/5.33	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.33/5.33	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.33/5.33	
17.33/5.33	Q is empty.
17.33/5.33	We have to consider all minimal (P,Q,R)-chains.
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(7) QDPOrderProof (EQUIVALENT)
17.33/5.33	We use the reduction pair processor [LPAR04,JAR06].
17.33/5.33	
17.33/5.33	
17.33/5.33	The following pairs can be oriented strictly and are deleted.
17.33/5.33	
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
17.33/5.33	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
17.33/5.33	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
17.33/5.33	The remaining pairs can at least be oriented weakly.
17.33/5.33	Used ordering:  Polynomial interpretation [POLO]:
17.33/5.33	
17.33/5.33	   POL(A_{B_1}(x_1)) = x_1
17.33/5.33	   POL(B_{B_1}(x_1)) = x_1
17.33/5.33	   POL(a_{a_1}(x_1)) = 1 + x_1
17.33/5.33	   POL(a_{b_1}(x_1)) = 1 + x_1
17.33/5.33	   POL(b_{a_1}(x_1)) = 1 + x_1
17.33/5.33	   POL(b_{b_1}(x_1)) = 1 + x_1
17.33/5.33	
17.33/5.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.33/5.33	
17.33/5.33	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
17.33/5.33	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
17.33/5.33	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.33/5.33	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.33/5.33	
17.33/5.33	
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(8)
17.33/5.33	Obligation:
17.33/5.33	Q DP problem:
17.33/5.33	P is empty.
17.33/5.33	The TRS R consists of the following rules:
17.33/5.33	
17.33/5.33	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
17.33/5.33	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
17.33/5.33	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.33/5.33	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
17.33/5.33	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
17.33/5.33	
17.33/5.33	Q is empty.
17.33/5.33	We have to consider all minimal (P,Q,R)-chains.
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(9) PisEmptyProof (EQUIVALENT)
17.33/5.33	The TRS P is empty. Hence, there is no (P,Q,R) chain.
17.33/5.33	----------------------------------------
17.33/5.33	
17.33/5.33	(10)
17.33/5.33	YES
17.58/5.48	EOF