62.01/16.72	YES
62.18/16.76	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
62.18/16.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
62.18/16.76	
62.18/16.76	
62.18/16.76	Termination w.r.t. Q of the given QTRS could be proven:
62.18/16.76	
62.18/16.76	(0) QTRS
62.18/16.76	(1) QTRS Reverse [EQUIVALENT, 0 ms]
62.18/16.76	(2) QTRS
62.18/16.76	(3) FlatCCProof [EQUIVALENT, 0 ms]
62.18/16.76	(4) QTRS
62.18/16.76	(5) RootLabelingProof [EQUIVALENT, 0 ms]
62.18/16.76	(6) QTRS
62.18/16.76	(7) DependencyPairsProof [EQUIVALENT, 31 ms]
62.18/16.76	(8) QDP
62.18/16.76	(9) QDPOrderProof [EQUIVALENT, 96 ms]
62.18/16.76	(10) QDP
62.18/16.76	(11) QDPOrderProof [EQUIVALENT, 115 ms]
62.18/16.76	(12) QDP
62.18/16.76	(13) QDPOrderProof [EQUIVALENT, 325 ms]
62.18/16.76	(14) QDP
62.18/16.76	(15) PisEmptyProof [EQUIVALENT, 0 ms]
62.18/16.76	(16) YES
62.18/16.76	
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(0)
62.18/16.76	Obligation:
62.18/16.76	Q restricted rewrite system:
62.18/16.76	The TRS R consists of the following rules:
62.18/16.76	
62.18/16.76	   a(b(a(a(x1)))) -> a(a(b(a(x1))))
62.18/16.76	   b(a(a(b(x1)))) -> b(b(b(b(x1))))
62.18/16.76	   b(a(a(b(x1)))) -> a(a(a(b(x1))))
62.18/16.76	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
62.18/16.76	
62.18/16.76	Q is empty.
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(1) QTRS Reverse (EQUIVALENT)
62.18/16.76	We applied the QTRS Reverse Processor [REVERSE].
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(2)
62.18/16.76	Obligation:
62.18/16.76	Q restricted rewrite system:
62.18/16.76	The TRS R consists of the following rules:
62.18/16.76	
62.18/16.76	   a(a(b(a(x1)))) -> a(b(a(a(x1))))
62.18/16.76	   b(a(a(b(x1)))) -> b(b(b(b(x1))))
62.18/16.76	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
62.18/16.76	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
62.18/16.76	
62.18/16.76	Q is empty.
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(3) FlatCCProof (EQUIVALENT)
62.18/16.76	We used flat context closure [ROOTLAB]
62.18/16.76	As Q is empty the flat context closure was sound AND complete.
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(4)
62.18/16.76	Obligation:
62.18/16.76	Q restricted rewrite system:
62.18/16.76	The TRS R consists of the following rules:
62.18/16.76	
62.18/16.76	   a(a(b(a(x1)))) -> a(b(a(a(x1))))
62.18/16.76	   b(a(a(b(x1)))) -> b(b(b(b(x1))))
62.18/16.76	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
62.18/16.76	   a(a(b(b(a(x1))))) -> a(b(a(a(b(x1)))))
62.18/16.76	   b(a(b(b(a(x1))))) -> b(b(a(a(b(x1)))))
62.18/16.76	
62.18/16.76	Q is empty.
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(5) RootLabelingProof (EQUIVALENT)
62.18/16.76	We used plain root labeling [ROOTLAB] with the following heuristic:
62.18/16.76	LabelAll: All function symbols get labeled
62.18/16.76	
62.18/16.76	As Q is empty the root labeling was sound AND complete.
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(6)
62.18/16.76	Obligation:
62.18/16.76	Q restricted rewrite system:
62.18/16.76	The TRS R consists of the following rules:
62.18/16.76	
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	
62.18/16.76	Q is empty.
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(7) DependencyPairsProof (EQUIVALENT)
62.18/16.76	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(8)
62.18/16.76	Obligation:
62.18/16.76	Q DP problem:
62.18/16.76	The TRS P consists of the following rules:
62.18/16.76	
62.18/16.76	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
62.18/16.76	
62.18/16.76	The TRS R consists of the following rules:
62.18/16.76	
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	
62.18/16.76	Q is empty.
62.18/16.76	We have to consider all minimal (P,Q,R)-chains.
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(9) QDPOrderProof (EQUIVALENT)
62.18/16.76	We use the reduction pair processor [LPAR04,JAR06].
62.18/16.76	
62.18/16.76	
62.18/16.76	The following pairs can be oriented strictly and are deleted.
62.18/16.76	
62.18/16.76	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
62.18/16.76	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
62.18/16.76	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
62.18/16.76	The remaining pairs can at least be oriented weakly.
62.18/16.76	Used ordering:  Polynomial interpretation [POLO]:
62.18/16.76	
62.18/16.76	   POL(A_{A_1}(x_1)) = x_1
62.18/16.76	   POL(B_{A_1}(x_1)) = x_1
62.18/16.76	   POL(a_{a_1}(x_1)) = 1 + x_1
62.18/16.76	   POL(a_{b_1}(x_1)) = 1 + x_1
62.18/16.76	   POL(b_{a_1}(x_1)) = 1 + x_1
62.18/16.76	   POL(b_{b_1}(x_1)) = 1 + x_1
62.18/16.76	
62.18/16.76	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
62.18/16.76	
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(10)
62.18/16.76	Obligation:
62.18/16.76	Q DP problem:
62.18/16.76	The TRS P consists of the following rules:
62.18/16.76	
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	
62.18/16.76	The TRS R consists of the following rules:
62.18/16.76	
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	
62.18/16.76	Q is empty.
62.18/16.76	We have to consider all minimal (P,Q,R)-chains.
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(11) QDPOrderProof (EQUIVALENT)
62.18/16.76	We use the reduction pair processor [LPAR04,JAR06].
62.18/16.76	
62.18/16.76	
62.18/16.76	The following pairs can be oriented strictly and are deleted.
62.18/16.76	
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	The remaining pairs can at least be oriented weakly.
62.18/16.76	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(B_{A_1}(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, 0A], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [1A, 0A, 1A], [0A, 1A, 0A]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [1A, -I, -I], [-I, 0A, 0A]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	
62.18/16.76	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
62.18/16.76	
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
62.18/16.76	
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(12)
62.18/16.76	Obligation:
62.18/16.76	Q DP problem:
62.18/16.76	The TRS P consists of the following rules:
62.18/16.76	
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	
62.18/16.76	The TRS R consists of the following rules:
62.18/16.76	
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	
62.18/16.76	Q is empty.
62.18/16.76	We have to consider all minimal (P,Q,R)-chains.
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(13) QDPOrderProof (EQUIVALENT)
62.18/16.76	We use the reduction pair processor [LPAR04,JAR06].
62.18/16.76	
62.18/16.76	
62.18/16.76	The following pairs can be oriented strictly and are deleted.
62.18/16.76	
62.18/16.76	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	The remaining pairs can at least be oriented weakly.
62.18/16.76	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(B_{A_1}(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(a_{a_1}(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, -I], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(a_{b_1}(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, -I], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(b_{b_1}(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 1A, 0A], [-I, -I, 1A], [-I, -I, -I]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	   <<<
62.18/16.76	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
62.18/16.76	>>>
62.18/16.76	
62.18/16.76	
62.18/16.76	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
62.18/16.76	
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
62.18/16.76	
62.18/16.76	
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(14)
62.18/16.76	Obligation:
62.18/16.76	Q DP problem:
62.18/16.76	P is empty.
62.18/16.76	The TRS R consists of the following rules:
62.18/16.76	
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
62.18/16.76	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
62.18/16.76	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
62.18/16.76	
62.18/16.76	Q is empty.
62.18/16.76	We have to consider all minimal (P,Q,R)-chains.
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(15) PisEmptyProof (EQUIVALENT)
62.18/16.76	The TRS P is empty. Hence, there is no (P,Q,R) chain.
62.18/16.76	----------------------------------------
62.18/16.76	
62.18/16.76	(16)
62.18/16.76	YES
62.37/16.83	EOF