26.70/7.68	YES
26.98/7.74	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
26.98/7.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.98/7.74	
26.98/7.74	
26.98/7.74	Termination w.r.t. Q of the given QTRS could be proven:
26.98/7.74	
26.98/7.74	(0) QTRS
26.98/7.74	(1) QTRS Reverse [EQUIVALENT, 0 ms]
26.98/7.74	(2) QTRS
26.98/7.74	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
26.98/7.74	(4) QDP
26.98/7.74	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
26.98/7.74	(6) QDP
26.98/7.74	(7) QDPOrderProof [EQUIVALENT, 151 ms]
26.98/7.74	(8) QDP
26.98/7.74	(9) QDPOrderProof [EQUIVALENT, 55 ms]
26.98/7.74	(10) QDP
26.98/7.74	(11) QDPOrderProof [EQUIVALENT, 55 ms]
26.98/7.74	(12) QDP
26.98/7.74	(13) PisEmptyProof [EQUIVALENT, 0 ms]
26.98/7.74	(14) YES
26.98/7.74	
26.98/7.74	
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(0)
26.98/7.74	Obligation:
26.98/7.74	Q restricted rewrite system:
26.98/7.74	The TRS R consists of the following rules:
26.98/7.74	
26.98/7.74	   a(x1) -> x1
26.98/7.74	   a(a(b(b(x1)))) -> b(b(a(a(b(a(x1))))))
26.98/7.74	
26.98/7.74	Q is empty.
26.98/7.74	
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(1) QTRS Reverse (EQUIVALENT)
26.98/7.74	We applied the QTRS Reverse Processor [REVERSE].
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(2)
26.98/7.74	Obligation:
26.98/7.74	Q restricted rewrite system:
26.98/7.74	The TRS R consists of the following rules:
26.98/7.74	
26.98/7.74	   a(x1) -> x1
26.98/7.74	   b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
26.98/7.74	
26.98/7.74	Q is empty.
26.98/7.74	
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(3) DependencyPairsProof (EQUIVALENT)
26.98/7.74	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(4)
26.98/7.74	Obligation:
26.98/7.74	Q DP problem:
26.98/7.74	The TRS P consists of the following rules:
26.98/7.74	
26.98/7.74	   B(b(a(a(x1)))) -> A(b(a(a(b(b(x1))))))
26.98/7.74	   B(b(a(a(x1)))) -> B(a(a(b(b(x1)))))
26.98/7.74	   B(b(a(a(x1)))) -> A(a(b(b(x1))))
26.98/7.74	   B(b(a(a(x1)))) -> A(b(b(x1)))
26.98/7.74	   B(b(a(a(x1)))) -> B(b(x1))
26.98/7.74	   B(b(a(a(x1)))) -> B(x1)
26.98/7.74	
26.98/7.74	The TRS R consists of the following rules:
26.98/7.74	
26.98/7.74	   a(x1) -> x1
26.98/7.74	   b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
26.98/7.74	
26.98/7.74	Q is empty.
26.98/7.74	We have to consider all minimal (P,Q,R)-chains.
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(5) DependencyGraphProof (EQUIVALENT)
26.98/7.74	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(6)
26.98/7.74	Obligation:
26.98/7.74	Q DP problem:
26.98/7.74	The TRS P consists of the following rules:
26.98/7.74	
26.98/7.74	   B(b(a(a(x1)))) -> B(b(x1))
26.98/7.74	   B(b(a(a(x1)))) -> B(a(a(b(b(x1)))))
26.98/7.74	   B(b(a(a(x1)))) -> B(x1)
26.98/7.74	
26.98/7.74	The TRS R consists of the following rules:
26.98/7.74	
26.98/7.74	   a(x1) -> x1
26.98/7.74	   b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
26.98/7.74	
26.98/7.74	Q is empty.
26.98/7.74	We have to consider all minimal (P,Q,R)-chains.
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(7) QDPOrderProof (EQUIVALENT)
26.98/7.74	We use the reduction pair processor [LPAR04,JAR06].
26.98/7.74	
26.98/7.74	
26.98/7.74	The following pairs can be oriented strictly and are deleted.
26.98/7.74	
26.98/7.74	   B(b(a(a(x1)))) -> B(x1)
26.98/7.74	The remaining pairs can at least be oriented weakly.
26.98/7.74	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.98/7.74	
26.98/7.74	   <<<
26.98/7.74	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
26.98/7.74	>>>
26.98/7.74	
26.98/7.74	   <<<
26.98/7.74	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
26.98/7.74	>>>
26.98/7.74	
26.98/7.74	   <<<
26.98/7.74	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, 0A, -I], [1A, -I, 0A]] 	* 	x_1
26.98/7.74	>>>
26.98/7.74	
26.98/7.74	
26.98/7.74	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.98/7.74	
26.98/7.74	   b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
26.98/7.74	   a(x1) -> x1
26.98/7.74	
26.98/7.74	
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(8)
26.98/7.74	Obligation:
26.98/7.74	Q DP problem:
26.98/7.74	The TRS P consists of the following rules:
26.98/7.74	
26.98/7.74	   B(b(a(a(x1)))) -> B(b(x1))
26.98/7.74	   B(b(a(a(x1)))) -> B(a(a(b(b(x1)))))
26.98/7.74	
26.98/7.74	The TRS R consists of the following rules:
26.98/7.74	
26.98/7.74	   a(x1) -> x1
26.98/7.74	   b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
26.98/7.74	
26.98/7.74	Q is empty.
26.98/7.74	We have to consider all minimal (P,Q,R)-chains.
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(9) QDPOrderProof (EQUIVALENT)
26.98/7.74	We use the reduction pair processor [LPAR04,JAR06].
26.98/7.74	
26.98/7.74	
26.98/7.74	The following pairs can be oriented strictly and are deleted.
26.98/7.74	
26.98/7.74	   B(b(a(a(x1)))) -> B(b(x1))
26.98/7.74	The remaining pairs can at least be oriented weakly.
26.98/7.74	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.98/7.74	
26.98/7.74	   <<<
26.98/7.74	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
26.98/7.74	>>>
26.98/7.74	
26.98/7.74	   <<<
26.98/7.74	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
26.98/7.74	>>>
26.98/7.74	
26.98/7.74	   <<<
26.98/7.74	 POL(a(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [1A, 0A, 0A]] 	* 	x_1
26.98/7.74	>>>
26.98/7.74	
26.98/7.74	
26.98/7.74	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.98/7.74	
26.98/7.74	   b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
26.98/7.74	   a(x1) -> x1
26.98/7.74	
26.98/7.74	
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(10)
26.98/7.74	Obligation:
26.98/7.74	Q DP problem:
26.98/7.74	The TRS P consists of the following rules:
26.98/7.74	
26.98/7.74	   B(b(a(a(x1)))) -> B(a(a(b(b(x1)))))
26.98/7.74	
26.98/7.74	The TRS R consists of the following rules:
26.98/7.74	
26.98/7.74	   a(x1) -> x1
26.98/7.74	   b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
26.98/7.74	
26.98/7.74	Q is empty.
26.98/7.74	We have to consider all minimal (P,Q,R)-chains.
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(11) QDPOrderProof (EQUIVALENT)
26.98/7.74	We use the reduction pair processor [LPAR04,JAR06].
26.98/7.74	
26.98/7.74	
26.98/7.74	The following pairs can be oriented strictly and are deleted.
26.98/7.74	
26.98/7.74	   B(b(a(a(x1)))) -> B(a(a(b(b(x1)))))
26.98/7.74	The remaining pairs can at least be oriented weakly.
26.98/7.74	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.98/7.74	
26.98/7.74	   <<<
26.98/7.74	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
26.98/7.74	>>>
26.98/7.74	
26.98/7.74	   <<<
26.98/7.74	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
26.98/7.74	>>>
26.98/7.74	
26.98/7.74	   <<<
26.98/7.74	 POL(a(x_1)) =  	[[-I], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
26.98/7.74	>>>
26.98/7.74	
26.98/7.74	
26.98/7.74	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.98/7.74	
26.98/7.74	   b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
26.98/7.74	   a(x1) -> x1
26.98/7.74	
26.98/7.74	
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(12)
26.98/7.74	Obligation:
26.98/7.74	Q DP problem:
26.98/7.74	P is empty.
26.98/7.74	The TRS R consists of the following rules:
26.98/7.74	
26.98/7.74	   a(x1) -> x1
26.98/7.74	   b(b(a(a(x1)))) -> a(b(a(a(b(b(x1))))))
26.98/7.74	
26.98/7.74	Q is empty.
26.98/7.74	We have to consider all minimal (P,Q,R)-chains.
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(13) PisEmptyProof (EQUIVALENT)
26.98/7.74	The TRS P is empty. Hence, there is no (P,Q,R) chain.
26.98/7.74	----------------------------------------
26.98/7.74	
26.98/7.74	(14)
26.98/7.74	YES
27.72/7.96	EOF