30.64/8.76	YES
30.64/8.79	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
30.64/8.79	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.64/8.79	
30.64/8.79	
30.64/8.79	Termination w.r.t. Q of the given QTRS could be proven:
30.64/8.79	
30.64/8.79	(0) QTRS
30.64/8.79	(1) DependencyPairsProof [EQUIVALENT, 31 ms]
30.64/8.79	(2) QDP
30.64/8.79	(3) QDPOrderProof [EQUIVALENT, 227 ms]
30.64/8.79	(4) QDP
30.64/8.79	(5) QDPOrderProof [EQUIVALENT, 0 ms]
30.64/8.79	(6) QDP
30.64/8.79	(7) QDPOrderProof [EQUIVALENT, 58 ms]
30.64/8.79	(8) QDP
30.64/8.79	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
30.64/8.79	(10) TRUE
30.64/8.79	
30.64/8.79	
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(0)
30.64/8.79	Obligation:
30.64/8.79	Q restricted rewrite system:
30.64/8.79	The TRS R consists of the following rules:
30.64/8.79	
30.64/8.79	   a(x1) -> b(x1)
30.64/8.79	   b(a(a(b(x1)))) -> a(a(b(b(a(x1)))))
30.64/8.79	
30.64/8.79	Q is empty.
30.64/8.79	
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(1) DependencyPairsProof (EQUIVALENT)
30.64/8.79	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(2)
30.64/8.79	Obligation:
30.64/8.79	Q DP problem:
30.64/8.79	The TRS P consists of the following rules:
30.64/8.79	
30.64/8.79	   A(x1) -> B(x1)
30.64/8.79	   B(a(a(b(x1)))) -> A(a(b(b(a(x1)))))
30.64/8.79	   B(a(a(b(x1)))) -> A(b(b(a(x1))))
30.64/8.79	   B(a(a(b(x1)))) -> B(b(a(x1)))
30.64/8.79	   B(a(a(b(x1)))) -> B(a(x1))
30.64/8.79	   B(a(a(b(x1)))) -> A(x1)
30.64/8.79	
30.64/8.79	The TRS R consists of the following rules:
30.64/8.79	
30.64/8.79	   a(x1) -> b(x1)
30.64/8.79	   b(a(a(b(x1)))) -> a(a(b(b(a(x1)))))
30.64/8.79	
30.64/8.79	Q is empty.
30.64/8.79	We have to consider all minimal (P,Q,R)-chains.
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(3) QDPOrderProof (EQUIVALENT)
30.64/8.79	We use the reduction pair processor [LPAR04,JAR06].
30.64/8.79	
30.64/8.79	
30.64/8.79	The following pairs can be oriented strictly and are deleted.
30.64/8.79	
30.64/8.79	   B(a(a(b(x1)))) -> B(b(a(x1)))
30.64/8.79	   B(a(a(b(x1)))) -> B(a(x1))
30.64/8.79	The remaining pairs can at least be oriented weakly.
30.64/8.79	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 1A, -I]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 1A], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [0A, 0A, -I]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	
30.64/8.79	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.64/8.79	
30.64/8.79	   b(a(a(b(x1)))) -> a(a(b(b(a(x1)))))
30.64/8.79	   a(x1) -> b(x1)
30.64/8.79	
30.64/8.79	
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(4)
30.64/8.79	Obligation:
30.64/8.79	Q DP problem:
30.64/8.79	The TRS P consists of the following rules:
30.64/8.79	
30.64/8.79	   A(x1) -> B(x1)
30.64/8.79	   B(a(a(b(x1)))) -> A(a(b(b(a(x1)))))
30.64/8.79	   B(a(a(b(x1)))) -> A(b(b(a(x1))))
30.64/8.79	   B(a(a(b(x1)))) -> A(x1)
30.64/8.79	
30.64/8.79	The TRS R consists of the following rules:
30.64/8.79	
30.64/8.79	   a(x1) -> b(x1)
30.64/8.79	   b(a(a(b(x1)))) -> a(a(b(b(a(x1)))))
30.64/8.79	
30.64/8.79	Q is empty.
30.64/8.79	We have to consider all minimal (P,Q,R)-chains.
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(5) QDPOrderProof (EQUIVALENT)
30.64/8.79	We use the reduction pair processor [LPAR04,JAR06].
30.64/8.79	
30.64/8.79	
30.64/8.79	The following pairs can be oriented strictly and are deleted.
30.64/8.79	
30.64/8.79	   B(a(a(b(x1)))) -> A(b(b(a(x1))))
30.64/8.79	   B(a(a(b(x1)))) -> A(x1)
30.64/8.79	The remaining pairs can at least be oriented weakly.
30.64/8.79	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [1A, 1A, -I]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [1A, 1A, -I]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	
30.64/8.79	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.64/8.79	
30.64/8.79	   b(a(a(b(x1)))) -> a(a(b(b(a(x1)))))
30.64/8.79	   a(x1) -> b(x1)
30.64/8.79	
30.64/8.79	
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(6)
30.64/8.79	Obligation:
30.64/8.79	Q DP problem:
30.64/8.79	The TRS P consists of the following rules:
30.64/8.79	
30.64/8.79	   A(x1) -> B(x1)
30.64/8.79	   B(a(a(b(x1)))) -> A(a(b(b(a(x1)))))
30.64/8.79	
30.64/8.79	The TRS R consists of the following rules:
30.64/8.79	
30.64/8.79	   a(x1) -> b(x1)
30.64/8.79	   b(a(a(b(x1)))) -> a(a(b(b(a(x1)))))
30.64/8.79	
30.64/8.79	Q is empty.
30.64/8.79	We have to consider all minimal (P,Q,R)-chains.
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(7) QDPOrderProof (EQUIVALENT)
30.64/8.79	We use the reduction pair processor [LPAR04,JAR06].
30.64/8.79	
30.64/8.79	
30.64/8.79	The following pairs can be oriented strictly and are deleted.
30.64/8.79	
30.64/8.79	   A(x1) -> B(x1)
30.64/8.79	The remaining pairs can at least be oriented weakly.
30.64/8.79	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(A(x_1)) =  	[[-I]] 	 +  	[[1A, 0A, 1A]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	   <<<
30.64/8.79	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, 0A, -I]] 	* 	x_1
30.64/8.79	>>>
30.64/8.79	
30.64/8.79	
30.64/8.79	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.64/8.79	
30.64/8.79	   b(a(a(b(x1)))) -> a(a(b(b(a(x1)))))
30.64/8.79	   a(x1) -> b(x1)
30.64/8.79	
30.64/8.79	
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(8)
30.64/8.79	Obligation:
30.64/8.79	Q DP problem:
30.64/8.79	The TRS P consists of the following rules:
30.64/8.79	
30.64/8.79	   B(a(a(b(x1)))) -> A(a(b(b(a(x1)))))
30.64/8.79	
30.64/8.79	The TRS R consists of the following rules:
30.64/8.79	
30.64/8.79	   a(x1) -> b(x1)
30.64/8.79	   b(a(a(b(x1)))) -> a(a(b(b(a(x1)))))
30.64/8.79	
30.64/8.79	Q is empty.
30.64/8.79	We have to consider all minimal (P,Q,R)-chains.
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(9) DependencyGraphProof (EQUIVALENT)
30.64/8.79	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
30.64/8.79	----------------------------------------
30.64/8.79	
30.64/8.79	(10)
30.64/8.79	TRUE
31.16/8.90	EOF