20.89/6.21	YES
21.11/6.23	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
21.11/6.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.11/6.23	
21.11/6.23	
21.11/6.23	Termination w.r.t. Q of the given QTRS could be proven:
21.11/6.23	
21.11/6.23	(0) QTRS
21.11/6.23	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
21.11/6.23	(2) QDP
21.11/6.23	(3) QDPOrderProof [EQUIVALENT, 132 ms]
21.11/6.23	(4) QDP
21.11/6.23	(5) QDPOrderProof [EQUIVALENT, 30 ms]
21.11/6.23	(6) QDP
21.11/6.23	(7) PisEmptyProof [EQUIVALENT, 0 ms]
21.11/6.23	(8) YES
21.11/6.23	
21.11/6.23	
21.11/6.23	----------------------------------------
21.11/6.23	
21.11/6.23	(0)
21.11/6.23	Obligation:
21.11/6.23	Q restricted rewrite system:
21.11/6.23	The TRS R consists of the following rules:
21.11/6.23	
21.11/6.23	   a(b(a(x1))) -> b(b(b(b(x1))))
21.11/6.23	   a(b(b(x1))) -> b(b(a(a(x1))))
21.11/6.23	
21.11/6.23	Q is empty.
21.11/6.23	
21.11/6.23	----------------------------------------
21.11/6.23	
21.11/6.23	(1) DependencyPairsProof (EQUIVALENT)
21.11/6.23	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
21.11/6.23	----------------------------------------
21.11/6.23	
21.11/6.23	(2)
21.11/6.23	Obligation:
21.11/6.23	Q DP problem:
21.11/6.23	The TRS P consists of the following rules:
21.11/6.23	
21.11/6.23	   A(b(b(x1))) -> A(a(x1))
21.11/6.23	   A(b(b(x1))) -> A(x1)
21.11/6.23	
21.11/6.23	The TRS R consists of the following rules:
21.11/6.23	
21.11/6.23	   a(b(a(x1))) -> b(b(b(b(x1))))
21.11/6.23	   a(b(b(x1))) -> b(b(a(a(x1))))
21.11/6.23	
21.11/6.23	Q is empty.
21.11/6.23	We have to consider all minimal (P,Q,R)-chains.
21.11/6.23	----------------------------------------
21.11/6.23	
21.11/6.23	(3) QDPOrderProof (EQUIVALENT)
21.11/6.23	We use the reduction pair processor [LPAR04,JAR06].
21.11/6.23	
21.11/6.23	
21.11/6.23	The following pairs can be oriented strictly and are deleted.
21.11/6.23	
21.11/6.23	   A(b(b(x1))) -> A(x1)
21.11/6.23	The remaining pairs can at least be oriented weakly.
21.11/6.23	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
21.11/6.23	
21.11/6.23	   <<<
21.11/6.23	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
21.11/6.23	>>>
21.11/6.23	
21.11/6.23	   <<<
21.11/6.23	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 1A, -I]] 	* 	x_1
21.11/6.23	>>>
21.11/6.23	
21.11/6.23	   <<<
21.11/6.23	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [1A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
21.11/6.23	>>>
21.11/6.23	
21.11/6.23	
21.11/6.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.11/6.23	
21.11/6.23	   a(b(a(x1))) -> b(b(b(b(x1))))
21.11/6.23	   a(b(b(x1))) -> b(b(a(a(x1))))
21.11/6.23	
21.11/6.23	
21.11/6.23	----------------------------------------
21.11/6.23	
21.11/6.23	(4)
21.11/6.23	Obligation:
21.11/6.23	Q DP problem:
21.11/6.23	The TRS P consists of the following rules:
21.11/6.23	
21.11/6.23	   A(b(b(x1))) -> A(a(x1))
21.11/6.23	
21.11/6.23	The TRS R consists of the following rules:
21.11/6.23	
21.11/6.23	   a(b(a(x1))) -> b(b(b(b(x1))))
21.11/6.23	   a(b(b(x1))) -> b(b(a(a(x1))))
21.11/6.23	
21.11/6.23	Q is empty.
21.11/6.23	We have to consider all minimal (P,Q,R)-chains.
21.11/6.23	----------------------------------------
21.11/6.23	
21.11/6.23	(5) QDPOrderProof (EQUIVALENT)
21.11/6.23	We use the reduction pair processor [LPAR04,JAR06].
21.11/6.23	
21.11/6.23	
21.11/6.23	The following pairs can be oriented strictly and are deleted.
21.11/6.23	
21.11/6.23	   A(b(b(x1))) -> A(a(x1))
21.11/6.23	The remaining pairs can at least be oriented weakly.
21.11/6.23	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
21.11/6.23	
21.11/6.23	   <<<
21.11/6.23	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
21.11/6.23	>>>
21.11/6.23	
21.11/6.23	   <<<
21.11/6.23	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 1A, 0A], [0A, -I, -I], [-I, -I, 0A]] 	* 	x_1
21.11/6.23	>>>
21.11/6.23	
21.11/6.23	   <<<
21.11/6.23	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
21.11/6.23	>>>
21.11/6.23	
21.11/6.23	
21.11/6.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.11/6.23	
21.11/6.23	   a(b(a(x1))) -> b(b(b(b(x1))))
21.11/6.23	   a(b(b(x1))) -> b(b(a(a(x1))))
21.11/6.23	
21.11/6.23	
21.11/6.23	----------------------------------------
21.11/6.23	
21.11/6.23	(6)
21.11/6.23	Obligation:
21.11/6.23	Q DP problem:
21.11/6.23	P is empty.
21.11/6.23	The TRS R consists of the following rules:
21.11/6.23	
21.11/6.23	   a(b(a(x1))) -> b(b(b(b(x1))))
21.11/6.23	   a(b(b(x1))) -> b(b(a(a(x1))))
21.11/6.23	
21.11/6.23	Q is empty.
21.11/6.23	We have to consider all minimal (P,Q,R)-chains.
21.11/6.23	----------------------------------------
21.11/6.23	
21.11/6.23	(7) PisEmptyProof (EQUIVALENT)
21.11/6.23	The TRS P is empty. Hence, there is no (P,Q,R) chain.
21.11/6.23	----------------------------------------
21.11/6.23	
21.11/6.23	(8)
21.11/6.23	YES
21.24/6.29	EOF