106.88/28.07	YES
106.88/28.10	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
106.88/28.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
106.88/28.10	
106.88/28.10	
106.88/28.10	Termination w.r.t. Q of the given QTRS could be proven:
106.88/28.10	
106.88/28.10	(0) QTRS
106.88/28.10	(1) QTRS Reverse [EQUIVALENT, 0 ms]
106.88/28.10	(2) QTRS
106.88/28.10	(3) RootLabelingProof [EQUIVALENT, 0 ms]
106.88/28.10	(4) QTRS
106.88/28.10	(5) DependencyPairsProof [EQUIVALENT, 16 ms]
106.88/28.10	(6) QDP
106.88/28.10	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
106.88/28.10	(8) QDP
106.88/28.10	(9) QDPOrderProof [EQUIVALENT, 6573 ms]
106.88/28.10	(10) QDP
106.88/28.10	(11) PisEmptyProof [EQUIVALENT, 0 ms]
106.88/28.10	(12) YES
106.88/28.10	
106.88/28.10	
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(0)
106.88/28.10	Obligation:
106.88/28.10	Q restricted rewrite system:
106.88/28.10	The TRS R consists of the following rules:
106.88/28.10	
106.88/28.10	   a(b(a(b(a(b(a(a(b(x1))))))))) -> a(a(b(a(a(b(a(b(a(b(a(b(x1))))))))))))
106.88/28.10	
106.88/28.10	Q is empty.
106.88/28.10	
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(1) QTRS Reverse (EQUIVALENT)
106.88/28.10	We applied the QTRS Reverse Processor [REVERSE].
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(2)
106.88/28.10	Obligation:
106.88/28.10	Q restricted rewrite system:
106.88/28.10	The TRS R consists of the following rules:
106.88/28.10	
106.88/28.10	   b(a(a(b(a(b(a(b(a(x1))))))))) -> b(a(b(a(b(a(b(a(a(b(a(a(x1))))))))))))
106.88/28.10	
106.88/28.10	Q is empty.
106.88/28.10	
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(3) RootLabelingProof (EQUIVALENT)
106.88/28.10	We used plain root labeling [ROOTLAB] with the following heuristic:
106.88/28.10	LabelAll: All function symbols get labeled
106.88/28.10	
106.88/28.10	As Q is empty the root labeling was sound AND complete.
106.88/28.10	
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(4)
106.88/28.10	Obligation:
106.88/28.10	Q restricted rewrite system:
106.88/28.10	The TRS R consists of the following rules:
106.88/28.10	
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))))))
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))))))
106.88/28.10	
106.88/28.10	Q is empty.
106.88/28.10	
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(5) DependencyPairsProof (EQUIVALENT)
106.88/28.10	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(6)
106.88/28.10	Obligation:
106.88/28.10	Q DP problem:
106.88/28.10	The TRS P consists of the following rules:
106.88/28.10	
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))))))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))))))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
106.88/28.10	
106.88/28.10	The TRS R consists of the following rules:
106.88/28.10	
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))))))
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))))))
106.88/28.10	
106.88/28.10	Q is empty.
106.88/28.10	We have to consider all minimal (P,Q,R)-chains.
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(7) DependencyGraphProof (EQUIVALENT)
106.88/28.10	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 8 less nodes.
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(8)
106.88/28.10	Obligation:
106.88/28.10	Q DP problem:
106.88/28.10	The TRS P consists of the following rules:
106.88/28.10	
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
106.88/28.10	
106.88/28.10	The TRS R consists of the following rules:
106.88/28.10	
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))))))
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))))))
106.88/28.10	
106.88/28.10	Q is empty.
106.88/28.10	We have to consider all minimal (P,Q,R)-chains.
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(9) QDPOrderProof (EQUIVALENT)
106.88/28.10	We use the reduction pair processor [LPAR04,JAR06].
106.88/28.10	
106.88/28.10	
106.88/28.10	The following pairs can be oriented strictly and are deleted.
106.88/28.10	
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
106.88/28.10	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
106.88/28.10	The remaining pairs can at least be oriented weakly.
106.88/28.10	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
106.88/28.10	
106.88/28.10	   <<<
106.88/28.10	 POL(B_{A_1}(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
106.88/28.10	>>>
106.88/28.10	
106.88/28.10	   <<<
106.88/28.10	 POL(a_{a_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [-I, -I, -I], [0A, 0A, 0A]] 	* 	x_1
106.88/28.10	>>>
106.88/28.10	
106.88/28.10	   <<<
106.88/28.10	 POL(a_{b_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [-I, 0A, 0A]] 	* 	x_1
106.88/28.10	>>>
106.88/28.10	
106.88/28.10	   <<<
106.88/28.10	 POL(b_{a_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
106.88/28.10	>>>
106.88/28.10	
106.88/28.10	
106.88/28.10	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
106.88/28.10	
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))))))
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))))))
106.88/28.10	
106.88/28.10	
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(10)
106.88/28.10	Obligation:
106.88/28.10	Q DP problem:
106.88/28.10	P is empty.
106.88/28.10	The TRS R consists of the following rules:
106.88/28.10	
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))))))))
106.88/28.10	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))))))))))
106.88/28.10	
106.88/28.10	Q is empty.
106.88/28.10	We have to consider all minimal (P,Q,R)-chains.
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(11) PisEmptyProof (EQUIVALENT)
106.88/28.10	The TRS P is empty. Hence, there is no (P,Q,R) chain.
106.88/28.10	----------------------------------------
106.88/28.10	
106.88/28.10	(12)
106.88/28.10	YES
107.12/29.68	EOF