22.58/6.61	YES
22.64/6.63	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
22.64/6.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
22.64/6.63	
22.64/6.63	
22.64/6.63	Termination w.r.t. Q of the given QTRS could be proven:
22.64/6.63	
22.64/6.63	(0) QTRS
22.64/6.63	(1) DependencyPairsProof [EQUIVALENT, 31 ms]
22.64/6.63	(2) QDP
22.64/6.63	(3) QDPOrderProof [EQUIVALENT, 137 ms]
22.64/6.63	(4) QDP
22.64/6.63	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
22.64/6.63	(6) QDP
22.64/6.63	(7) QDPOrderProof [EQUIVALENT, 0 ms]
22.64/6.63	(8) QDP
22.64/6.63	(9) PisEmptyProof [EQUIVALENT, 0 ms]
22.64/6.63	(10) YES
22.64/6.63	
22.64/6.63	
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(0)
22.64/6.63	Obligation:
22.64/6.63	Q restricted rewrite system:
22.64/6.63	The TRS R consists of the following rules:
22.64/6.63	
22.64/6.63	   a(b(a(x1))) -> b(b(a(x1)))
22.64/6.63	   b(b(b(x1))) -> b(a(x1))
22.64/6.63	   b(b(x1)) -> a(a(a(x1)))
22.64/6.63	
22.64/6.63	Q is empty.
22.64/6.63	
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(1) DependencyPairsProof (EQUIVALENT)
22.64/6.63	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(2)
22.64/6.63	Obligation:
22.64/6.63	Q DP problem:
22.64/6.63	The TRS P consists of the following rules:
22.64/6.63	
22.64/6.63	   A(b(a(x1))) -> B(b(a(x1)))
22.64/6.63	   B(b(b(x1))) -> B(a(x1))
22.64/6.63	   B(b(b(x1))) -> A(x1)
22.64/6.63	   B(b(x1)) -> A(a(a(x1)))
22.64/6.63	   B(b(x1)) -> A(a(x1))
22.64/6.63	   B(b(x1)) -> A(x1)
22.64/6.63	
22.64/6.63	The TRS R consists of the following rules:
22.64/6.63	
22.64/6.63	   a(b(a(x1))) -> b(b(a(x1)))
22.64/6.63	   b(b(b(x1))) -> b(a(x1))
22.64/6.63	   b(b(x1)) -> a(a(a(x1)))
22.64/6.63	
22.64/6.63	Q is empty.
22.64/6.63	We have to consider all minimal (P,Q,R)-chains.
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(3) QDPOrderProof (EQUIVALENT)
22.64/6.63	We use the reduction pair processor [LPAR04,JAR06].
22.64/6.63	
22.64/6.63	
22.64/6.63	The following pairs can be oriented strictly and are deleted.
22.64/6.63	
22.64/6.63	   A(b(a(x1))) -> B(b(a(x1)))
22.64/6.63	The remaining pairs can at least be oriented weakly.
22.64/6.63	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
22.64/6.63	
22.64/6.63	   <<<
22.64/6.63	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 1A]] 	* 	x_1
22.64/6.63	>>>
22.64/6.63	
22.64/6.63	   <<<
22.64/6.63	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 1A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
22.64/6.63	>>>
22.64/6.63	
22.64/6.63	   <<<
22.64/6.63	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 1A], [-I, -I, -I], [-I, -I, 0A]] 	* 	x_1
22.64/6.63	>>>
22.64/6.63	
22.64/6.63	   <<<
22.64/6.63	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
22.64/6.63	>>>
22.64/6.63	
22.64/6.63	
22.64/6.63	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.64/6.63	
22.64/6.63	   b(b(b(x1))) -> b(a(x1))
22.64/6.63	   b(b(x1)) -> a(a(a(x1)))
22.64/6.63	   a(b(a(x1))) -> b(b(a(x1)))
22.64/6.63	
22.64/6.63	
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(4)
22.64/6.63	Obligation:
22.64/6.63	Q DP problem:
22.64/6.63	The TRS P consists of the following rules:
22.64/6.63	
22.64/6.63	   B(b(b(x1))) -> B(a(x1))
22.64/6.63	   B(b(b(x1))) -> A(x1)
22.64/6.63	   B(b(x1)) -> A(a(a(x1)))
22.64/6.63	   B(b(x1)) -> A(a(x1))
22.64/6.63	   B(b(x1)) -> A(x1)
22.64/6.63	
22.64/6.63	The TRS R consists of the following rules:
22.64/6.63	
22.64/6.63	   a(b(a(x1))) -> b(b(a(x1)))
22.64/6.63	   b(b(b(x1))) -> b(a(x1))
22.64/6.63	   b(b(x1)) -> a(a(a(x1)))
22.64/6.63	
22.64/6.63	Q is empty.
22.64/6.63	We have to consider all minimal (P,Q,R)-chains.
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(5) DependencyGraphProof (EQUIVALENT)
22.64/6.63	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(6)
22.64/6.63	Obligation:
22.64/6.63	Q DP problem:
22.64/6.63	The TRS P consists of the following rules:
22.64/6.63	
22.64/6.63	   B(b(b(x1))) -> B(a(x1))
22.64/6.63	
22.64/6.63	The TRS R consists of the following rules:
22.64/6.63	
22.64/6.63	   a(b(a(x1))) -> b(b(a(x1)))
22.64/6.63	   b(b(b(x1))) -> b(a(x1))
22.64/6.63	   b(b(x1)) -> a(a(a(x1)))
22.64/6.63	
22.64/6.63	Q is empty.
22.64/6.63	We have to consider all minimal (P,Q,R)-chains.
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(7) QDPOrderProof (EQUIVALENT)
22.64/6.63	We use the reduction pair processor [LPAR04,JAR06].
22.64/6.63	
22.64/6.63	
22.64/6.63	The following pairs can be oriented strictly and are deleted.
22.64/6.63	
22.64/6.63	   B(b(b(x1))) -> B(a(x1))
22.64/6.63	The remaining pairs can at least be oriented weakly.
22.64/6.63	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
22.64/6.63	
22.64/6.63	   <<<
22.64/6.63	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
22.64/6.63	>>>
22.64/6.63	
22.64/6.63	   <<<
22.64/6.63	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [1A, -I, 0A], [-I, 0A, -I]] 	* 	x_1
22.64/6.63	>>>
22.64/6.63	
22.64/6.63	   <<<
22.64/6.63	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
22.64/6.63	>>>
22.64/6.63	
22.64/6.63	
22.64/6.63	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
22.64/6.63	
22.64/6.63	   b(b(b(x1))) -> b(a(x1))
22.64/6.63	   b(b(x1)) -> a(a(a(x1)))
22.64/6.63	   a(b(a(x1))) -> b(b(a(x1)))
22.64/6.63	
22.64/6.63	
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(8)
22.64/6.63	Obligation:
22.64/6.63	Q DP problem:
22.64/6.63	P is empty.
22.64/6.63	The TRS R consists of the following rules:
22.64/6.63	
22.64/6.63	   a(b(a(x1))) -> b(b(a(x1)))
22.64/6.63	   b(b(b(x1))) -> b(a(x1))
22.64/6.63	   b(b(x1)) -> a(a(a(x1)))
22.64/6.63	
22.64/6.63	Q is empty.
22.64/6.63	We have to consider all minimal (P,Q,R)-chains.
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(9) PisEmptyProof (EQUIVALENT)
22.64/6.63	The TRS P is empty. Hence, there is no (P,Q,R) chain.
22.64/6.63	----------------------------------------
22.64/6.63	
22.64/6.63	(10)
22.64/6.63	YES
22.93/6.73	EOF