92.84/24.64	YES
93.37/24.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
93.37/24.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
93.37/24.68	
93.37/24.68	
93.37/24.68	Termination w.r.t. Q of the given QTRS could be proven:
93.37/24.68	
93.37/24.68	(0) QTRS
93.37/24.68	(1) QTRS Reverse [EQUIVALENT, 0 ms]
93.37/24.68	(2) QTRS
93.37/24.68	(3) RootLabelingProof [EQUIVALENT, 0 ms]
93.37/24.68	(4) QTRS
93.37/24.68	(5) DependencyPairsProof [EQUIVALENT, 37 ms]
93.37/24.68	(6) QDP
93.37/24.68	(7) QDPOrderProof [EQUIVALENT, 2076 ms]
93.37/24.68	(8) QDP
93.37/24.68	(9) QDPOrderProof [EQUIVALENT, 639 ms]
93.37/24.68	(10) QDP
93.37/24.68	(11) QDPOrderProof [EQUIVALENT, 10 ms]
93.37/24.68	(12) QDP
93.37/24.68	(13) PisEmptyProof [EQUIVALENT, 0 ms]
93.37/24.68	(14) YES
93.37/24.68	
93.37/24.68	
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(0)
93.37/24.68	Obligation:
93.37/24.68	Q restricted rewrite system:
93.37/24.68	The TRS R consists of the following rules:
93.37/24.68	
93.37/24.68	   b(a(b(b(x1)))) -> b(b(b(a(b(x1)))))
93.37/24.68	   b(a(a(b(b(x1))))) -> b(a(b(b(a(a(b(x1)))))))
93.37/24.68	   b(a(a(a(b(b(x1)))))) -> b(a(a(b(b(a(a(a(b(x1)))))))))
93.37/24.68	
93.37/24.68	Q is empty.
93.37/24.68	
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(1) QTRS Reverse (EQUIVALENT)
93.37/24.68	We applied the QTRS Reverse Processor [REVERSE].
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(2)
93.37/24.68	Obligation:
93.37/24.68	Q restricted rewrite system:
93.37/24.68	The TRS R consists of the following rules:
93.37/24.68	
93.37/24.68	   b(b(a(b(x1)))) -> b(a(b(b(b(x1)))))
93.37/24.68	   b(b(a(a(b(x1))))) -> b(a(a(b(b(a(b(x1)))))))
93.37/24.68	   b(b(a(a(a(b(x1)))))) -> b(a(a(a(b(b(a(a(b(x1)))))))))
93.37/24.68	
93.37/24.68	Q is empty.
93.37/24.68	
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(3) RootLabelingProof (EQUIVALENT)
93.37/24.68	We used plain root labeling [ROOTLAB] with the following heuristic:
93.37/24.68	LabelAll: All function symbols get labeled
93.37/24.68	
93.37/24.68	As Q is empty the root labeling was sound AND complete.
93.37/24.68	
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(4)
93.37/24.68	Obligation:
93.37/24.68	Q restricted rewrite system:
93.37/24.68	The TRS R consists of the following rules:
93.37/24.68	
93.37/24.68	   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}(b_{b_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))
93.37/24.68	
93.37/24.68	Q is empty.
93.37/24.68	
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(5) DependencyPairsProof (EQUIVALENT)
93.37/24.68	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(6)
93.37/24.68	Obligation:
93.37/24.68	Q DP problem:
93.37/24.68	The TRS P consists of the following rules:
93.37/24.68	
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
93.37/24.68	
93.37/24.68	The TRS R consists of the following rules:
93.37/24.68	
93.37/24.68	   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}(b_{b_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))
93.37/24.68	
93.37/24.68	Q is empty.
93.37/24.68	We have to consider all minimal (P,Q,R)-chains.
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(7) QDPOrderProof (EQUIVALENT)
93.37/24.68	We use the reduction pair processor [LPAR04,JAR06].
93.37/24.68	
93.37/24.68	
93.37/24.68	The following pairs can be oriented strictly and are deleted.
93.37/24.68	
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
93.37/24.68	The remaining pairs can at least be oriented weakly.
93.37/24.68	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(B_{B_1}(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(b_{a_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(a_{b_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, 0A]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(b_{b_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(a_{a_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	
93.37/24.68	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
93.37/24.68	
93.37/24.68	   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}(b_{b_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))
93.37/24.68	
93.37/24.68	
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(8)
93.37/24.68	Obligation:
93.37/24.68	Q DP problem:
93.37/24.68	The TRS P consists of the following rules:
93.37/24.68	
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
93.37/24.68	
93.37/24.68	The TRS R consists of the following rules:
93.37/24.68	
93.37/24.68	   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}(b_{b_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))
93.37/24.68	
93.37/24.68	Q is empty.
93.37/24.68	We have to consider all minimal (P,Q,R)-chains.
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(9) QDPOrderProof (EQUIVALENT)
93.37/24.68	We use the reduction pair processor [LPAR04,JAR06].
93.37/24.68	
93.37/24.68	
93.37/24.68	The following pairs can be oriented strictly and are deleted.
93.37/24.68	
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
93.37/24.68	The remaining pairs can at least be oriented weakly.
93.37/24.68	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(B_{B_1}(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	   <<<
93.37/24.68	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, -I, 1A], [-I, -I, -I]] 	* 	x_1
93.37/24.68	>>>
93.37/24.68	
93.37/24.68	
93.37/24.68	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
93.37/24.68	
93.37/24.68	   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}(b_{b_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))
93.37/24.68	
93.37/24.68	
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(10)
93.37/24.68	Obligation:
93.37/24.68	Q DP problem:
93.37/24.68	The TRS P consists of the following rules:
93.37/24.68	
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
93.37/24.68	
93.37/24.68	The TRS R consists of the following rules:
93.37/24.68	
93.37/24.68	   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}(b_{b_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))
93.37/24.68	
93.37/24.68	Q is empty.
93.37/24.68	We have to consider all minimal (P,Q,R)-chains.
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(11) QDPOrderProof (EQUIVALENT)
93.37/24.68	We use the reduction pair processor [LPAR04,JAR06].
93.37/24.68	
93.37/24.68	
93.37/24.68	The following pairs can be oriented strictly and are deleted.
93.37/24.68	
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
93.37/24.68	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
93.37/24.68	The remaining pairs can at least be oriented weakly.
93.37/24.68	Used ordering:  Polynomial interpretation [POLO]:
93.37/24.68	
93.37/24.68	   POL(B_{B_1}(x_1)) = x_1
93.37/24.68	   POL(a_{a_1}(x_1)) = 0
93.37/24.68	   POL(a_{b_1}(x_1)) = x_1
93.37/24.68	   POL(b_{a_1}(x_1)) = 1 + x_1
93.37/24.68	   POL(b_{b_1}(x_1)) = x_1
93.37/24.68	
93.37/24.68	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
93.37/24.68	
93.37/24.68	   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}(b_{b_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))
93.37/24.68	
93.37/24.68	
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(12)
93.37/24.68	Obligation:
93.37/24.68	Q DP problem:
93.37/24.68	P is empty.
93.37/24.68	The TRS R consists of the following rules:
93.37/24.68	
93.37/24.68	   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}(b_{b_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))
93.37/24.68	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))
93.37/24.68	
93.37/24.68	Q is empty.
93.37/24.68	We have to consider all minimal (P,Q,R)-chains.
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(13) PisEmptyProof (EQUIVALENT)
93.37/24.68	The TRS P is empty. Hence, there is no (P,Q,R) chain.
93.37/24.68	----------------------------------------
93.37/24.68	
93.37/24.68	(14)
93.37/24.68	YES
93.68/24.83	EOF