22.96/7.00	YES
22.96/7.01	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
22.96/7.01	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.96/7.01	
22.96/7.01	
22.96/7.01	Termination w.r.t. Q of the given QTRS could be proven:
22.96/7.01	
22.96/7.01	(0) QTRS
22.96/7.01	(1) QTRS Reverse [EQUIVALENT, 0 ms]
22.96/7.01	(2) QTRS
22.96/7.01	(3) FlatCCProof [EQUIVALENT, 0 ms]
22.96/7.01	(4) QTRS
22.96/7.01	(5) RootLabelingProof [EQUIVALENT, 0 ms]
22.96/7.01	(6) QTRS
22.96/7.01	(7) DependencyPairsProof [EQUIVALENT, 30 ms]
22.96/7.01	(8) QDP
22.96/7.01	(9) QDPOrderProof [EQUIVALENT, 97 ms]
22.96/7.01	(10) QDP
22.96/7.01	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
22.96/7.01	(12) TRUE
22.96/7.01	
22.96/7.01	
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(0)
22.96/7.01	Obligation:
22.96/7.01	Q restricted rewrite system:
22.96/7.01	The TRS R consists of the following rules:
22.96/7.01	
22.96/7.01	   a(a(a(b(x1)))) -> a(b(b(a(x1))))
22.96/7.01	   b(b(b(b(x1)))) -> b(a(b(b(x1))))
22.96/7.01	   a(b(b(a(x1)))) -> b(a(a(a(x1))))
22.96/7.01	   b(a(b(b(x1)))) -> b(b(a(b(x1))))
22.96/7.01	
22.96/7.01	Q is empty.
22.96/7.01	
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(1) QTRS Reverse (EQUIVALENT)
22.96/7.01	We applied the QTRS Reverse Processor [REVERSE].
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(2)
22.96/7.01	Obligation:
22.96/7.01	Q restricted rewrite system:
22.96/7.01	The TRS R consists of the following rules:
22.96/7.01	
22.96/7.01	   b(a(a(a(x1)))) -> a(b(b(a(x1))))
22.96/7.01	   b(b(b(b(x1)))) -> b(b(a(b(x1))))
22.96/7.01	   a(b(b(a(x1)))) -> a(a(a(b(x1))))
22.96/7.01	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
22.96/7.01	
22.96/7.01	Q is empty.
22.96/7.01	
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(3) FlatCCProof (EQUIVALENT)
22.96/7.01	We used flat context closure [ROOTLAB]
22.96/7.01	As Q is empty the flat context closure was sound AND complete.
22.96/7.01	
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(4)
22.96/7.01	Obligation:
22.96/7.01	Q restricted rewrite system:
22.96/7.01	The TRS R consists of the following rules:
22.96/7.01	
22.96/7.01	   b(b(b(b(x1)))) -> b(b(a(b(x1))))
22.96/7.01	   a(b(b(a(x1)))) -> a(a(a(b(x1))))
22.96/7.01	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
22.96/7.01	   b(b(a(a(a(x1))))) -> b(a(b(b(a(x1)))))
22.96/7.01	   a(b(a(a(a(x1))))) -> a(a(b(b(a(x1)))))
22.96/7.01	
22.96/7.01	Q is empty.
22.96/7.01	
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(5) RootLabelingProof (EQUIVALENT)
22.96/7.01	We used plain root labeling [ROOTLAB] with the following heuristic:
22.96/7.01	LabelAll: All function symbols get labeled
22.96/7.01	
22.96/7.01	As Q is empty the root labeling was sound AND complete.
22.96/7.01	
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(6)
22.96/7.01	Obligation:
22.96/7.01	Q restricted rewrite system:
22.96/7.01	The TRS R consists of the following rules:
22.96/7.01	
22.96/7.01	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
22.96/7.01	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
22.96/7.01	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
22.96/7.01	
22.96/7.01	Q is empty.
22.96/7.01	
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(7) DependencyPairsProof (EQUIVALENT)
22.96/7.01	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(8)
22.96/7.01	Obligation:
22.96/7.01	Q DP problem:
22.96/7.01	The TRS P consists of the following rules:
22.96/7.01	
22.96/7.01	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
22.96/7.01	   B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	   B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
22.96/7.01	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
22.96/7.01	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
22.96/7.01	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1)))
22.96/7.01	   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}(a_{a_1}(x1))))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1)))
22.96/7.01	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
22.96/7.01	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1)))
22.96/7.01	   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}(a_{a_1}(x1))))
22.96/7.01	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1)))
22.96/7.01	
22.96/7.01	The TRS R consists of the following rules:
22.96/7.01	
22.96/7.01	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
22.96/7.01	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
22.96/7.01	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
22.96/7.01	
22.96/7.01	Q is empty.
22.96/7.01	We have to consider all minimal (P,Q,R)-chains.
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(9) QDPOrderProof (EQUIVALENT)
22.96/7.01	We use the reduction pair processor [LPAR04,JAR06].
22.96/7.01	
22.96/7.01	
22.96/7.01	The following pairs can be oriented strictly and are deleted.
22.96/7.01	
22.96/7.01	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
22.96/7.01	   B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
22.96/7.01	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
22.96/7.01	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
22.96/7.01	   A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1)))
22.96/7.01	   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}(a_{a_1}(x1))))
22.96/7.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1)))
22.96/7.01	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
22.96/7.01	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1)))
22.96/7.01	   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}(a_{a_1}(x1))))
22.96/7.01	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1)))
22.96/7.01	The remaining pairs can at least be oriented weakly.
22.96/7.01	Used ordering:  Polynomial interpretation [POLO]:
22.96/7.01	
22.96/7.01	   POL(A_{B_1}(x_1)) = x_1
22.96/7.01	   POL(B_{B_1}(x_1)) = x_1
22.96/7.01	   POL(a_{a_1}(x_1)) = 1 + x_1
22.96/7.01	   POL(a_{b_1}(x_1)) = 1 + x_1
22.96/7.01	   POL(b_{a_1}(x_1)) = 1 + x_1
22.96/7.01	   POL(b_{b_1}(x_1)) = 1 + x_1
22.96/7.01	
22.96/7.01	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.96/7.01	
22.96/7.01	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
22.96/7.01	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
22.96/7.01	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
22.96/7.01	
22.96/7.01	
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(10)
22.96/7.01	Obligation:
22.96/7.01	Q DP problem:
22.96/7.01	The TRS P consists of the following rules:
22.96/7.01	
22.96/7.01	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	
22.96/7.01	The TRS R consists of the following rules:
22.96/7.01	
22.96/7.01	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.96/7.01	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
22.96/7.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
22.96/7.01	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
22.96/7.01	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
22.96/7.01	
22.96/7.01	Q is empty.
22.96/7.01	We have to consider all minimal (P,Q,R)-chains.
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(11) DependencyGraphProof (EQUIVALENT)
22.96/7.01	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
22.96/7.01	----------------------------------------
22.96/7.01	
22.96/7.01	(12)
22.96/7.01	TRUE
23.40/7.31	EOF