211.43/54.58	YES
212.56/54.91	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
212.56/54.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
212.56/54.91	
212.56/54.91	
212.56/54.91	Termination w.r.t. Q of the given QTRS could be proven:
212.56/54.91	
212.56/54.91	(0) QTRS
212.56/54.91	(1) QTRS Reverse [EQUIVALENT, 0 ms]
212.56/54.91	(2) QTRS
212.56/54.91	(3) RootLabelingProof [EQUIVALENT, 0 ms]
212.56/54.91	(4) QTRS
212.56/54.91	(5) DependencyPairsProof [EQUIVALENT, 23 ms]
212.56/54.91	(6) QDP
212.56/54.91	(7) DependencyGraphProof [EQUIVALENT, 1 ms]
212.56/54.91	(8) QDP
212.56/54.91	(9) QDPOrderProof [EQUIVALENT, 7446 ms]
212.56/54.91	(10) QDP
212.56/54.91	(11) QDPOrderProof [EQUIVALENT, 4999 ms]
212.56/54.91	(12) QDP
212.56/54.91	(13) PisEmptyProof [EQUIVALENT, 0 ms]
212.56/54.91	(14) YES
212.56/54.91	
212.56/54.91	
212.56/54.91	----------------------------------------
212.56/54.91	
212.56/54.91	(0)
212.56/54.91	Obligation:
212.56/54.91	Q restricted rewrite system:
212.56/54.91	The TRS R consists of the following rules:
212.56/54.91	
212.56/54.91	   b(a(b(b(a(b(b(a(b(a(x1)))))))))) -> a(b(a(b(b(a(b(b(a(b(b(a(x1))))))))))))
212.56/54.91	
212.56/54.91	Q is empty.
212.56/54.91	
212.56/54.91	----------------------------------------
212.56/54.91	
212.56/54.91	(1) QTRS Reverse (EQUIVALENT)
212.56/54.91	We applied the QTRS Reverse Processor [REVERSE].
212.56/54.91	----------------------------------------
212.56/54.91	
212.56/54.91	(2)
212.56/54.91	Obligation:
212.56/54.91	Q restricted rewrite system:
212.56/54.91	The TRS R consists of the following rules:
212.56/54.91	
212.56/54.91	   a(b(a(b(b(a(b(b(a(b(x1)))))))))) -> a(b(b(a(b(b(a(b(b(a(b(a(x1))))))))))))
212.56/54.91	
212.56/54.91	Q is empty.
212.56/54.91	
212.56/54.91	----------------------------------------
212.56/54.91	
212.56/54.91	(3) RootLabelingProof (EQUIVALENT)
212.56/54.91	We used plain root labeling [ROOTLAB] with the following heuristic:
212.56/54.91	LabelAll: All function symbols get labeled
212.56/54.91	
212.56/54.91	As Q is empty the root labeling was sound AND complete.
212.56/54.91	
212.56/54.91	----------------------------------------
212.56/54.91	
212.56/54.91	(4)
212.56/54.91	Obligation:
212.56/54.91	Q restricted rewrite system:
212.56/54.91	The TRS R consists of the following rules:
212.56/54.91	
212.56/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
212.56/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
212.56/54.91	
212.56/54.91	Q is empty.
212.56/54.91	
212.56/54.91	----------------------------------------
212.56/54.91	
212.56/54.91	(5) DependencyPairsProof (EQUIVALENT)
212.56/54.91	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
212.84/54.91	----------------------------------------
212.84/54.91	
212.84/54.91	(6)
212.84/54.91	Obligation:
212.84/54.91	Q DP problem:
212.84/54.91	The TRS P consists of the following rules:
212.84/54.91	
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))))))
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))))))
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(x1)
212.84/54.91	
212.84/54.91	The TRS R consists of the following rules:
212.84/54.91	
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
212.84/54.91	
212.84/54.91	Q is empty.
212.84/54.91	We have to consider all minimal (P,Q,R)-chains.
212.84/54.91	----------------------------------------
212.84/54.91	
212.84/54.91	(7) DependencyGraphProof (EQUIVALENT)
212.84/54.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.
212.84/54.91	----------------------------------------
212.84/54.91	
212.84/54.91	(8)
212.84/54.91	Obligation:
212.84/54.91	Q DP problem:
212.84/54.91	The TRS P consists of the following rules:
212.84/54.91	
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(x1)
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
212.84/54.91	
212.84/54.91	The TRS R consists of the following rules:
212.84/54.91	
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
212.84/54.91	
212.84/54.91	Q is empty.
212.84/54.91	We have to consider all minimal (P,Q,R)-chains.
212.84/54.91	----------------------------------------
212.84/54.91	
212.84/54.91	(9) QDPOrderProof (EQUIVALENT)
212.84/54.91	We use the reduction pair processor [LPAR04,JAR06].
212.84/54.91	
212.84/54.91	
212.84/54.91	The following pairs can be oriented strictly and are deleted.
212.84/54.91	
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(x1)
212.84/54.91	The remaining pairs can at least be oriented weakly.
212.84/54.91	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(A_{B_1}(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, -I]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 0A], [0A, -I, 0A]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(a_{a_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [0A, -I, 0A]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	
212.84/54.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
212.84/54.91	
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
212.84/54.91	
212.84/54.91	
212.84/54.91	----------------------------------------
212.84/54.91	
212.84/54.91	(10)
212.84/54.91	Obligation:
212.84/54.91	Q DP problem:
212.84/54.91	The TRS P consists of the following rules:
212.84/54.91	
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
212.84/54.91	
212.84/54.91	The TRS R consists of the following rules:
212.84/54.91	
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
212.84/54.91	
212.84/54.91	Q is empty.
212.84/54.91	We have to consider all minimal (P,Q,R)-chains.
212.84/54.91	----------------------------------------
212.84/54.91	
212.84/54.91	(11) QDPOrderProof (EQUIVALENT)
212.84/54.91	We use the reduction pair processor [LPAR04,JAR06].
212.84/54.91	
212.84/54.91	
212.84/54.91	The following pairs can be oriented strictly and are deleted.
212.84/54.91	
212.84/54.91	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
212.84/54.91	The remaining pairs can at least be oriented weakly.
212.84/54.91	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(A_{B_1}(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, -I]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	   <<<
212.84/54.91	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [-I, -I, -I], [0A, -I, -I]] 	* 	x_1
212.84/54.91	>>>
212.84/54.91	
212.84/54.91	
212.84/54.91	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
212.84/54.91	
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
212.84/54.91	
212.84/54.91	
212.84/54.91	----------------------------------------
212.84/54.91	
212.84/54.91	(12)
212.84/54.91	Obligation:
212.84/54.91	Q DP problem:
212.84/54.91	P is empty.
212.84/54.91	The TRS R consists of the following rules:
212.84/54.91	
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))))))
212.84/54.91	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))))))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))))))
212.84/54.91	
212.84/54.91	Q is empty.
212.84/54.91	We have to consider all minimal (P,Q,R)-chains.
212.84/54.91	----------------------------------------
212.84/54.91	
212.84/54.91	(13) PisEmptyProof (EQUIVALENT)
212.84/54.91	The TRS P is empty. Hence, there is no (P,Q,R) chain.
212.84/54.91	----------------------------------------
212.84/54.91	
212.84/54.91	(14)
212.84/54.91	YES
212.98/55.00	EOF