22.06/6.49	YES
22.06/6.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
22.06/6.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.06/6.51	
22.06/6.51	
22.06/6.51	Termination w.r.t. Q of the given QTRS could be proven:
22.06/6.51	
22.06/6.51	(0) QTRS
22.06/6.51	(1) QTRS Reverse [EQUIVALENT, 0 ms]
22.06/6.51	(2) QTRS
22.06/6.51	(3) FlatCCProof [EQUIVALENT, 0 ms]
22.06/6.51	(4) QTRS
22.06/6.51	(5) RootLabelingProof [EQUIVALENT, 2 ms]
22.06/6.51	(6) QTRS
22.06/6.51	(7) DependencyPairsProof [EQUIVALENT, 14 ms]
22.06/6.51	(8) QDP
22.06/6.51	(9) DependencyGraphProof [EQUIVALENT, 8 ms]
22.06/6.51	(10) QDP
22.06/6.51	(11) QDPOrderProof [EQUIVALENT, 82 ms]
22.06/6.51	(12) QDP
22.06/6.51	(13) QDPOrderProof [EQUIVALENT, 71 ms]
22.06/6.51	(14) QDP
22.06/6.51	(15) PisEmptyProof [EQUIVALENT, 0 ms]
22.06/6.51	(16) YES
22.06/6.51	
22.06/6.51	
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(0)
22.06/6.51	Obligation:
22.06/6.51	Q restricted rewrite system:
22.06/6.51	The TRS R consists of the following rules:
22.06/6.51	
22.06/6.51	   a(a(a(a(x1)))) -> b(a(a(a(x1))))
22.06/6.51	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
22.06/6.51	   a(a(b(a(x1)))) -> a(a(a(b(x1))))
22.06/6.51	
22.06/6.51	Q is empty.
22.06/6.51	
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(1) QTRS Reverse (EQUIVALENT)
22.06/6.51	We applied the QTRS Reverse Processor [REVERSE].
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(2)
22.06/6.51	Obligation:
22.06/6.51	Q restricted rewrite system:
22.06/6.51	The TRS R consists of the following rules:
22.06/6.51	
22.06/6.51	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
22.06/6.51	   b(a(b(b(x1)))) -> b(b(a(a(x1))))
22.06/6.51	   a(b(a(a(x1)))) -> b(a(a(a(x1))))
22.06/6.51	
22.06/6.51	Q is empty.
22.06/6.51	
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(3) FlatCCProof (EQUIVALENT)
22.06/6.51	We used flat context closure [ROOTLAB]
22.06/6.51	As Q is empty the flat context closure was sound AND complete.
22.06/6.51	
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(4)
22.06/6.51	Obligation:
22.06/6.51	Q restricted rewrite system:
22.06/6.51	The TRS R consists of the following rules:
22.06/6.51	
22.06/6.51	   a(a(a(a(x1)))) -> a(a(a(b(x1))))
22.06/6.51	   b(a(b(b(x1)))) -> b(b(a(a(x1))))
22.06/6.51	   a(a(b(a(a(x1))))) -> a(b(a(a(a(x1)))))
22.06/6.51	   b(a(b(a(a(x1))))) -> b(b(a(a(a(x1)))))
22.06/6.51	
22.06/6.51	Q is empty.
22.06/6.51	
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(5) RootLabelingProof (EQUIVALENT)
22.06/6.51	We used plain root labeling [ROOTLAB] with the following heuristic:
22.06/6.51	LabelAll: All function symbols get labeled
22.06/6.51	
22.06/6.51	As Q is empty the root labeling was sound AND complete.
22.06/6.51	
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(6)
22.06/6.51	Obligation:
22.06/6.51	Q restricted rewrite system:
22.06/6.51	The TRS R consists of the following rules:
22.06/6.51	
22.06/6.51	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.51	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.51	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.51	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.51	
22.06/6.51	Q is empty.
22.06/6.51	
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(7) DependencyPairsProof (EQUIVALENT)
22.06/6.51	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(8)
22.06/6.51	Obligation:
22.06/6.51	Q DP problem:
22.06/6.51	The TRS P consists of the following rules:
22.06/6.51	
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
22.06/6.51	
22.06/6.51	The TRS R consists of the following rules:
22.06/6.51	
22.06/6.51	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.51	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.51	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.51	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.51	
22.06/6.51	Q is empty.
22.06/6.51	We have to consider all minimal (P,Q,R)-chains.
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(9) DependencyGraphProof (EQUIVALENT)
22.06/6.51	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(10)
22.06/6.51	Obligation:
22.06/6.51	Q DP problem:
22.06/6.51	The TRS P consists of the following rules:
22.06/6.51	
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
22.06/6.51	
22.06/6.51	The TRS R consists of the following rules:
22.06/6.51	
22.06/6.51	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.51	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.51	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.51	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.51	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.51	
22.06/6.51	Q is empty.
22.06/6.51	We have to consider all minimal (P,Q,R)-chains.
22.06/6.51	----------------------------------------
22.06/6.51	
22.06/6.51	(11) QDPOrderProof (EQUIVALENT)
22.06/6.51	We use the reduction pair processor [LPAR04,JAR06].
22.06/6.51	
22.06/6.51	
22.06/6.51	The following pairs can be oriented strictly and are deleted.
22.06/6.51	
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
22.06/6.51	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
22.06/6.51	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.51	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
22.06/6.51	The remaining pairs can at least be oriented weakly.
22.06/6.52	Used ordering:  Polynomial interpretation [POLO]:
22.06/6.52	
22.06/6.52	   POL(A_{A_1}(x_1)) = x_1
22.06/6.52	   POL(B_{A_1}(x_1)) = x_1
22.06/6.52	   POL(a_{a_1}(x_1)) = 1 + x_1
22.06/6.52	   POL(a_{b_1}(x_1)) = 1 + x_1
22.06/6.52	   POL(b_{a_1}(x_1)) = 1 + x_1
22.06/6.52	   POL(b_{b_1}(x_1)) = 1 + x_1
22.06/6.52	
22.06/6.52	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.06/6.52	
22.06/6.52	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.52	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.52	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.06/6.52	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.52	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.52	
22.06/6.52	
22.06/6.52	----------------------------------------
22.06/6.52	
22.06/6.52	(12)
22.06/6.52	Obligation:
22.06/6.52	Q DP problem:
22.06/6.52	The TRS P consists of the following rules:
22.06/6.52	
22.06/6.52	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.52	
22.06/6.52	The TRS R consists of the following rules:
22.06/6.52	
22.06/6.52	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.52	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.52	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.52	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.52	
22.06/6.52	Q is empty.
22.06/6.52	We have to consider all minimal (P,Q,R)-chains.
22.06/6.52	----------------------------------------
22.06/6.52	
22.06/6.52	(13) QDPOrderProof (EQUIVALENT)
22.06/6.52	We use the reduction pair processor [LPAR04,JAR06].
22.06/6.52	
22.06/6.52	
22.06/6.52	The following pairs can be oriented strictly and are deleted.
22.06/6.52	
22.06/6.52	   A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.52	The remaining pairs can at least be oriented weakly.
22.06/6.52	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
22.06/6.52	
22.06/6.52	POL( A_{A_1}_1(x_1) ) = max{0, x_1 - 2}
22.06/6.52	POL( a_{b_1}_1(x_1) ) = max{0, 2x_1 - 2}
22.06/6.52	POL( b_{a_1}_1(x_1) ) = 0
22.06/6.52	POL( b_{b_1}_1(x_1) ) = max{0, -2}
22.06/6.52	POL( a_{a_1}_1(x_1) ) = x_1 + 1
22.06/6.52	
22.06/6.52	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.06/6.52	
22.06/6.52	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.52	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.52	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.52	
22.06/6.52	
22.06/6.52	----------------------------------------
22.06/6.52	
22.06/6.52	(14)
22.06/6.52	Obligation:
22.06/6.52	Q DP problem:
22.06/6.52	P is empty.
22.06/6.52	The TRS R consists of the following rules:
22.06/6.52	
22.06/6.52	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
22.06/6.52	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
22.06/6.52	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.52	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
22.06/6.52	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
22.06/6.52	
22.06/6.52	Q is empty.
22.06/6.52	We have to consider all minimal (P,Q,R)-chains.
22.06/6.52	----------------------------------------
22.06/6.52	
22.06/6.52	(15) PisEmptyProof (EQUIVALENT)
22.06/6.52	The TRS P is empty. Hence, there is no (P,Q,R) chain.
22.06/6.52	----------------------------------------
22.06/6.52	
22.06/6.52	(16)
22.06/6.52	YES
22.43/6.58	EOF