28.67/8.25	YES
28.92/8.33	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
28.92/8.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
28.92/8.33	
28.92/8.33	
28.92/8.33	Termination w.r.t. Q of the given QTRS could be proven:
28.92/8.33	
28.92/8.33	(0) QTRS
28.92/8.33	(1) DependencyPairsProof [EQUIVALENT, 21 ms]
28.92/8.33	(2) QDP
28.92/8.33	(3) QDPOrderProof [EQUIVALENT, 149 ms]
28.92/8.33	(4) QDP
28.92/8.33	(5) QDPOrderProof [EQUIVALENT, 47 ms]
28.92/8.33	(6) QDP
28.92/8.33	(7) QDPOrderProof [EQUIVALENT, 59 ms]
28.92/8.33	(8) QDP
28.92/8.33	(9) PisEmptyProof [EQUIVALENT, 0 ms]
28.92/8.33	(10) YES
28.92/8.33	
28.92/8.33	
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(0)
28.92/8.33	Obligation:
28.92/8.33	Q restricted rewrite system:
28.92/8.33	The TRS R consists of the following rules:
28.92/8.33	
28.92/8.33	   a(x1) -> x1
28.92/8.33	   a(b(b(a(x1)))) -> a(a(b(a(b(b(x1))))))
28.92/8.33	
28.92/8.33	Q is empty.
28.92/8.33	
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(1) DependencyPairsProof (EQUIVALENT)
28.92/8.33	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(2)
28.92/8.33	Obligation:
28.92/8.33	Q DP problem:
28.92/8.33	The TRS P consists of the following rules:
28.92/8.33	
28.92/8.33	   A(b(b(a(x1)))) -> A(a(b(a(b(b(x1))))))
28.92/8.33	   A(b(b(a(x1)))) -> A(b(a(b(b(x1)))))
28.92/8.33	   A(b(b(a(x1)))) -> A(b(b(x1)))
28.92/8.33	
28.92/8.33	The TRS R consists of the following rules:
28.92/8.33	
28.92/8.33	   a(x1) -> x1
28.92/8.33	   a(b(b(a(x1)))) -> a(a(b(a(b(b(x1))))))
28.92/8.33	
28.92/8.33	Q is empty.
28.92/8.33	We have to consider all minimal (P,Q,R)-chains.
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(3) QDPOrderProof (EQUIVALENT)
28.92/8.33	We use the reduction pair processor [LPAR04,JAR06].
28.92/8.33	
28.92/8.33	
28.92/8.33	The following pairs can be oriented strictly and are deleted.
28.92/8.33	
28.92/8.33	   A(b(b(a(x1)))) -> A(b(b(x1)))
28.92/8.33	The remaining pairs can at least be oriented weakly.
28.92/8.33	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
28.92/8.33	
28.92/8.33	   <<<
28.92/8.33	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
28.92/8.33	>>>
28.92/8.33	
28.92/8.33	   <<<
28.92/8.33	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, -I], [-I, 0A, -I]] 	* 	x_1
28.92/8.33	>>>
28.92/8.33	
28.92/8.33	   <<<
28.92/8.33	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, -I], [1A, 1A, -I], [0A, 0A, 0A]] 	* 	x_1
28.92/8.33	>>>
28.92/8.33	
28.92/8.33	
28.92/8.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.92/8.33	
28.92/8.33	   a(x1) -> x1
28.92/8.33	   a(b(b(a(x1)))) -> a(a(b(a(b(b(x1))))))
28.92/8.33	
28.92/8.33	
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(4)
28.92/8.33	Obligation:
28.92/8.33	Q DP problem:
28.92/8.33	The TRS P consists of the following rules:
28.92/8.33	
28.92/8.33	   A(b(b(a(x1)))) -> A(a(b(a(b(b(x1))))))
28.92/8.33	   A(b(b(a(x1)))) -> A(b(a(b(b(x1)))))
28.92/8.33	
28.92/8.33	The TRS R consists of the following rules:
28.92/8.33	
28.92/8.33	   a(x1) -> x1
28.92/8.33	   a(b(b(a(x1)))) -> a(a(b(a(b(b(x1))))))
28.92/8.33	
28.92/8.33	Q is empty.
28.92/8.33	We have to consider all minimal (P,Q,R)-chains.
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(5) QDPOrderProof (EQUIVALENT)
28.92/8.33	We use the reduction pair processor [LPAR04,JAR06].
28.92/8.33	
28.92/8.33	
28.92/8.33	The following pairs can be oriented strictly and are deleted.
28.92/8.33	
28.92/8.33	   A(b(b(a(x1)))) -> A(b(a(b(b(x1)))))
28.92/8.33	The remaining pairs can at least be oriented weakly.
28.92/8.33	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
28.92/8.33	
28.92/8.33	   <<<
28.92/8.33	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
28.92/8.33	>>>
28.92/8.33	
28.92/8.33	   <<<
28.92/8.33	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, -I, -I], [-I, 0A, -I]] 	* 	x_1
28.92/8.33	>>>
28.92/8.33	
28.92/8.33	   <<<
28.92/8.33	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, -I, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
28.92/8.33	>>>
28.92/8.33	
28.92/8.33	
28.92/8.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.92/8.33	
28.92/8.33	   a(x1) -> x1
28.92/8.33	   a(b(b(a(x1)))) -> a(a(b(a(b(b(x1))))))
28.92/8.33	
28.92/8.33	
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(6)
28.92/8.33	Obligation:
28.92/8.33	Q DP problem:
28.92/8.33	The TRS P consists of the following rules:
28.92/8.33	
28.92/8.33	   A(b(b(a(x1)))) -> A(a(b(a(b(b(x1))))))
28.92/8.33	
28.92/8.33	The TRS R consists of the following rules:
28.92/8.33	
28.92/8.33	   a(x1) -> x1
28.92/8.33	   a(b(b(a(x1)))) -> a(a(b(a(b(b(x1))))))
28.92/8.33	
28.92/8.33	Q is empty.
28.92/8.33	We have to consider all minimal (P,Q,R)-chains.
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(7) QDPOrderProof (EQUIVALENT)
28.92/8.33	We use the reduction pair processor [LPAR04,JAR06].
28.92/8.33	
28.92/8.33	
28.92/8.33	The following pairs can be oriented strictly and are deleted.
28.92/8.33	
28.92/8.33	   A(b(b(a(x1)))) -> A(a(b(a(b(b(x1))))))
28.92/8.33	The remaining pairs can at least be oriented weakly.
28.92/8.33	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
28.92/8.33	
28.92/8.33	   <<<
28.92/8.33	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
28.92/8.33	>>>
28.92/8.33	
28.92/8.33	   <<<
28.92/8.33	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, -I], [-I, 0A, -I]] 	* 	x_1
28.92/8.33	>>>
28.92/8.33	
28.92/8.33	   <<<
28.92/8.33	 POL(a(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, -I], [1A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
28.92/8.33	>>>
28.92/8.33	
28.92/8.33	
28.92/8.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.92/8.33	
28.92/8.33	   a(x1) -> x1
28.92/8.33	   a(b(b(a(x1)))) -> a(a(b(a(b(b(x1))))))
28.92/8.33	
28.92/8.33	
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(8)
28.92/8.33	Obligation:
28.92/8.33	Q DP problem:
28.92/8.33	P is empty.
28.92/8.33	The TRS R consists of the following rules:
28.92/8.33	
28.92/8.33	   a(x1) -> x1
28.92/8.33	   a(b(b(a(x1)))) -> a(a(b(a(b(b(x1))))))
28.92/8.33	
28.92/8.33	Q is empty.
28.92/8.33	We have to consider all minimal (P,Q,R)-chains.
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(9) PisEmptyProof (EQUIVALENT)
28.92/8.33	The TRS P is empty. Hence, there is no (P,Q,R) chain.
28.92/8.33	----------------------------------------
28.92/8.33	
28.92/8.33	(10)
28.92/8.33	YES
29.37/8.51	EOF