22.29/6.64	YES
22.41/6.66	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
22.41/6.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.41/6.66	
22.41/6.66	
22.41/6.66	Termination w.r.t. Q of the given QTRS could be proven:
22.41/6.66	
22.41/6.66	(0) QTRS
22.41/6.66	(1) DependencyPairsProof [EQUIVALENT, 27 ms]
22.41/6.66	(2) QDP
22.41/6.66	(3) QDPOrderProof [EQUIVALENT, 132 ms]
22.41/6.66	(4) QDP
22.41/6.66	(5) QDPOrderProof [EQUIVALENT, 0 ms]
22.41/6.66	(6) QDP
22.41/6.66	(7) PisEmptyProof [EQUIVALENT, 0 ms]
22.41/6.66	(8) YES
22.41/6.66	
22.41/6.66	
22.41/6.66	----------------------------------------
22.41/6.66	
22.41/6.66	(0)
22.41/6.66	Obligation:
22.41/6.66	Q restricted rewrite system:
22.41/6.66	The TRS R consists of the following rules:
22.41/6.66	
22.41/6.66	   a(a(x1)) -> x1
22.41/6.66	   a(b(b(b(x1)))) -> b(b(b(a(b(a(x1))))))
22.41/6.66	
22.41/6.66	Q is empty.
22.41/6.66	
22.41/6.66	----------------------------------------
22.41/6.66	
22.41/6.66	(1) DependencyPairsProof (EQUIVALENT)
22.41/6.66	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
22.41/6.66	----------------------------------------
22.41/6.66	
22.41/6.66	(2)
22.41/6.66	Obligation:
22.41/6.66	Q DP problem:
22.41/6.66	The TRS P consists of the following rules:
22.41/6.66	
22.41/6.66	   A(b(b(b(x1)))) -> A(b(a(x1)))
22.41/6.66	   A(b(b(b(x1)))) -> A(x1)
22.41/6.66	
22.41/6.66	The TRS R consists of the following rules:
22.41/6.66	
22.41/6.66	   a(a(x1)) -> x1
22.41/6.66	   a(b(b(b(x1)))) -> b(b(b(a(b(a(x1))))))
22.41/6.66	
22.41/6.66	Q is empty.
22.41/6.66	We have to consider all minimal (P,Q,R)-chains.
22.41/6.66	----------------------------------------
22.41/6.66	
22.41/6.66	(3) QDPOrderProof (EQUIVALENT)
22.41/6.66	We use the reduction pair processor [LPAR04,JAR06].
22.41/6.66	
22.41/6.66	
22.41/6.66	The following pairs can be oriented strictly and are deleted.
22.41/6.66	
22.41/6.66	   A(b(b(b(x1)))) -> A(x1)
22.41/6.66	The remaining pairs can at least be oriented weakly.
22.41/6.66	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
22.41/6.66	
22.41/6.66	   <<<
22.41/6.66	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
22.41/6.66	>>>
22.41/6.66	
22.41/6.66	   <<<
22.41/6.66	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, -I, -I], [0A, 1A, -I]] 	* 	x_1
22.41/6.66	>>>
22.41/6.66	
22.41/6.66	   <<<
22.41/6.66	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
22.41/6.66	>>>
22.41/6.66	
22.41/6.66	
22.41/6.66	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.41/6.66	
22.41/6.66	   a(a(x1)) -> x1
22.41/6.66	   a(b(b(b(x1)))) -> b(b(b(a(b(a(x1))))))
22.41/6.66	
22.41/6.66	
22.41/6.66	----------------------------------------
22.41/6.66	
22.41/6.66	(4)
22.41/6.66	Obligation:
22.41/6.66	Q DP problem:
22.41/6.66	The TRS P consists of the following rules:
22.41/6.66	
22.41/6.66	   A(b(b(b(x1)))) -> A(b(a(x1)))
22.41/6.66	
22.41/6.66	The TRS R consists of the following rules:
22.41/6.66	
22.41/6.66	   a(a(x1)) -> x1
22.41/6.66	   a(b(b(b(x1)))) -> b(b(b(a(b(a(x1))))))
22.41/6.66	
22.41/6.66	Q is empty.
22.41/6.66	We have to consider all minimal (P,Q,R)-chains.
22.41/6.66	----------------------------------------
22.41/6.66	
22.41/6.66	(5) QDPOrderProof (EQUIVALENT)
22.41/6.66	We use the reduction pair processor [LPAR04,JAR06].
22.41/6.66	
22.41/6.66	
22.41/6.66	The following pairs can be oriented strictly and are deleted.
22.41/6.66	
22.41/6.66	   A(b(b(b(x1)))) -> A(b(a(x1)))
22.41/6.66	The remaining pairs can at least be oriented weakly.
22.41/6.66	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
22.41/6.66	
22.41/6.66	   <<<
22.41/6.66	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
22.41/6.66	>>>
22.41/6.66	
22.41/6.66	   <<<
22.41/6.66	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 1A, 0A], [-I, -I, 0A], [0A, 0A, -I]] 	* 	x_1
22.41/6.66	>>>
22.41/6.66	
22.41/6.66	   <<<
22.41/6.66	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
22.41/6.66	>>>
22.41/6.66	
22.41/6.66	
22.41/6.66	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.41/6.66	
22.41/6.66	   a(a(x1)) -> x1
22.41/6.66	   a(b(b(b(x1)))) -> b(b(b(a(b(a(x1))))))
22.41/6.66	
22.41/6.66	
22.41/6.66	----------------------------------------
22.41/6.66	
22.41/6.66	(6)
22.41/6.66	Obligation:
22.41/6.66	Q DP problem:
22.41/6.66	P is empty.
22.41/6.66	The TRS R consists of the following rules:
22.41/6.66	
22.41/6.66	   a(a(x1)) -> x1
22.41/6.66	   a(b(b(b(x1)))) -> b(b(b(a(b(a(x1))))))
22.41/6.66	
22.41/6.66	Q is empty.
22.41/6.66	We have to consider all minimal (P,Q,R)-chains.
22.41/6.66	----------------------------------------
22.41/6.66	
22.41/6.66	(7) PisEmptyProof (EQUIVALENT)
22.41/6.66	The TRS P is empty. Hence, there is no (P,Q,R) chain.
22.41/6.66	----------------------------------------
22.41/6.66	
22.41/6.66	(8)
22.41/6.66	YES
22.41/6.70	EOF