39.50/10.99	YES
39.50/11.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
39.50/11.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
39.50/11.05	
39.50/11.05	
39.50/11.05	Termination w.r.t. Q of the given QTRS could be proven:
39.50/11.05	
39.50/11.05	(0) QTRS
39.50/11.05	(1) DependencyPairsProof [EQUIVALENT, 33 ms]
39.50/11.05	(2) QDP
39.50/11.05	(3) QDPOrderProof [EQUIVALENT, 194 ms]
39.50/11.05	(4) QDP
39.50/11.05	(5) QDPOrderProof [EQUIVALENT, 117 ms]
39.50/11.05	(6) QDP
39.50/11.05	(7) QDPOrderProof [EQUIVALENT, 85 ms]
39.50/11.05	(8) QDP
39.50/11.05	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
39.50/11.05	(10) TRUE
39.50/11.05	
39.50/11.05	
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(0)
39.50/11.05	Obligation:
39.50/11.05	Q restricted rewrite system:
39.50/11.05	The TRS R consists of the following rules:
39.50/11.05	
39.50/11.05	   a(x1) -> x1
39.50/11.05	   a(b(a(x1))) -> a(x1)
39.50/11.05	   b(a(a(b(x1)))) -> a(a(a(a(b(b(b(x1)))))))
39.50/11.05	   a(a(a(a(x1)))) -> b(x1)
39.50/11.05	
39.50/11.05	Q is empty.
39.50/11.05	
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(1) DependencyPairsProof (EQUIVALENT)
39.50/11.05	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(2)
39.50/11.05	Obligation:
39.50/11.05	Q DP problem:
39.50/11.05	The TRS P consists of the following rules:
39.50/11.05	
39.50/11.05	   B(a(a(b(x1)))) -> A(a(a(a(b(b(b(x1)))))))
39.50/11.05	   B(a(a(b(x1)))) -> A(a(a(b(b(b(x1))))))
39.50/11.05	   B(a(a(b(x1)))) -> A(a(b(b(b(x1)))))
39.50/11.05	   B(a(a(b(x1)))) -> A(b(b(b(x1))))
39.50/11.05	   B(a(a(b(x1)))) -> B(b(b(x1)))
39.50/11.05	   B(a(a(b(x1)))) -> B(b(x1))
39.50/11.05	   A(a(a(a(x1)))) -> B(x1)
39.50/11.05	
39.50/11.05	The TRS R consists of the following rules:
39.50/11.05	
39.50/11.05	   a(x1) -> x1
39.50/11.05	   a(b(a(x1))) -> a(x1)
39.50/11.05	   b(a(a(b(x1)))) -> a(a(a(a(b(b(b(x1)))))))
39.50/11.05	   a(a(a(a(x1)))) -> b(x1)
39.50/11.05	
39.50/11.05	Q is empty.
39.50/11.05	We have to consider all minimal (P,Q,R)-chains.
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(3) QDPOrderProof (EQUIVALENT)
39.50/11.05	We use the reduction pair processor [LPAR04,JAR06].
39.50/11.05	
39.50/11.05	
39.50/11.05	The following pairs can be oriented strictly and are deleted.
39.50/11.05	
39.50/11.05	   B(a(a(b(x1)))) -> A(a(b(b(b(x1)))))
39.50/11.05	   B(a(a(b(x1)))) -> A(b(b(b(x1))))
39.50/11.05	The remaining pairs can at least be oriented weakly.
39.50/11.05	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	
39.50/11.05	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.50/11.05	
39.50/11.05	   a(a(a(a(x1)))) -> b(x1)
39.50/11.05	   b(a(a(b(x1)))) -> a(a(a(a(b(b(b(x1)))))))
39.50/11.05	   a(b(a(x1))) -> a(x1)
39.50/11.05	   a(x1) -> x1
39.50/11.05	
39.50/11.05	
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(4)
39.50/11.05	Obligation:
39.50/11.05	Q DP problem:
39.50/11.05	The TRS P consists of the following rules:
39.50/11.05	
39.50/11.05	   B(a(a(b(x1)))) -> A(a(a(a(b(b(b(x1)))))))
39.50/11.05	   B(a(a(b(x1)))) -> A(a(a(b(b(b(x1))))))
39.50/11.05	   B(a(a(b(x1)))) -> B(b(b(x1)))
39.50/11.05	   B(a(a(b(x1)))) -> B(b(x1))
39.50/11.05	   A(a(a(a(x1)))) -> B(x1)
39.50/11.05	
39.50/11.05	The TRS R consists of the following rules:
39.50/11.05	
39.50/11.05	   a(x1) -> x1
39.50/11.05	   a(b(a(x1))) -> a(x1)
39.50/11.05	   b(a(a(b(x1)))) -> a(a(a(a(b(b(b(x1)))))))
39.50/11.05	   a(a(a(a(x1)))) -> b(x1)
39.50/11.05	
39.50/11.05	Q is empty.
39.50/11.05	We have to consider all minimal (P,Q,R)-chains.
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(5) QDPOrderProof (EQUIVALENT)
39.50/11.05	We use the reduction pair processor [LPAR04,JAR06].
39.50/11.05	
39.50/11.05	
39.50/11.05	The following pairs can be oriented strictly and are deleted.
39.50/11.05	
39.50/11.05	   B(a(a(b(x1)))) -> B(b(b(x1)))
39.50/11.05	   B(a(a(b(x1)))) -> B(b(x1))
39.50/11.05	The remaining pairs can at least be oriented weakly.
39.50/11.05	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(B(x_1)) =  	[[-I]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	
39.50/11.05	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.50/11.05	
39.50/11.05	   a(a(a(a(x1)))) -> b(x1)
39.50/11.05	   b(a(a(b(x1)))) -> a(a(a(a(b(b(b(x1)))))))
39.50/11.05	   a(b(a(x1))) -> a(x1)
39.50/11.05	   a(x1) -> x1
39.50/11.05	
39.50/11.05	
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(6)
39.50/11.05	Obligation:
39.50/11.05	Q DP problem:
39.50/11.05	The TRS P consists of the following rules:
39.50/11.05	
39.50/11.05	   B(a(a(b(x1)))) -> A(a(a(a(b(b(b(x1)))))))
39.50/11.05	   B(a(a(b(x1)))) -> A(a(a(b(b(b(x1))))))
39.50/11.05	   A(a(a(a(x1)))) -> B(x1)
39.50/11.05	
39.50/11.05	The TRS R consists of the following rules:
39.50/11.05	
39.50/11.05	   a(x1) -> x1
39.50/11.05	   a(b(a(x1))) -> a(x1)
39.50/11.05	   b(a(a(b(x1)))) -> a(a(a(a(b(b(b(x1)))))))
39.50/11.05	   a(a(a(a(x1)))) -> b(x1)
39.50/11.05	
39.50/11.05	Q is empty.
39.50/11.05	We have to consider all minimal (P,Q,R)-chains.
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(7) QDPOrderProof (EQUIVALENT)
39.50/11.05	We use the reduction pair processor [LPAR04,JAR06].
39.50/11.05	
39.50/11.05	
39.50/11.05	The following pairs can be oriented strictly and are deleted.
39.50/11.05	
39.50/11.05	   A(a(a(a(x1)))) -> B(x1)
39.50/11.05	The remaining pairs can at least be oriented weakly.
39.50/11.05	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [1A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 1A], [-I, -I, 1A], [-I, -I, 0A]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	   <<<
39.50/11.05	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
39.50/11.05	>>>
39.50/11.05	
39.50/11.05	
39.50/11.05	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
39.50/11.05	
39.50/11.05	   a(a(a(a(x1)))) -> b(x1)
39.50/11.05	   b(a(a(b(x1)))) -> a(a(a(a(b(b(b(x1)))))))
39.50/11.05	   a(b(a(x1))) -> a(x1)
39.50/11.05	   a(x1) -> x1
39.50/11.05	
39.50/11.05	
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(8)
39.50/11.05	Obligation:
39.50/11.05	Q DP problem:
39.50/11.05	The TRS P consists of the following rules:
39.50/11.05	
39.50/11.05	   B(a(a(b(x1)))) -> A(a(a(a(b(b(b(x1)))))))
39.50/11.05	   B(a(a(b(x1)))) -> A(a(a(b(b(b(x1))))))
39.50/11.05	
39.50/11.05	The TRS R consists of the following rules:
39.50/11.05	
39.50/11.05	   a(x1) -> x1
39.50/11.05	   a(b(a(x1))) -> a(x1)
39.50/11.05	   b(a(a(b(x1)))) -> a(a(a(a(b(b(b(x1)))))))
39.50/11.05	   a(a(a(a(x1)))) -> b(x1)
39.50/11.05	
39.50/11.05	Q is empty.
39.50/11.05	We have to consider all minimal (P,Q,R)-chains.
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(9) DependencyGraphProof (EQUIVALENT)
39.50/11.05	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
39.50/11.05	----------------------------------------
39.50/11.05	
39.50/11.05	(10)
39.50/11.05	TRUE
39.98/11.14	EOF