46.30/12.79	YES
46.30/12.85	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
46.30/12.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
46.30/12.85	
46.30/12.85	
46.30/12.85	Termination w.r.t. Q of the given QTRS could be proven:
46.30/12.85	
46.30/12.85	(0) QTRS
46.30/12.85	(1) DependencyPairsProof [EQUIVALENT, 34 ms]
46.30/12.85	(2) QDP
46.30/12.85	(3) QDPOrderProof [EQUIVALENT, 76 ms]
46.30/12.85	(4) QDP
46.30/12.85	(5) QDPOrderProof [EQUIVALENT, 99 ms]
46.30/12.85	(6) QDP
46.30/12.85	(7) DependencyGraphProof [EQUIVALENT, 1 ms]
46.30/12.85	(8) AND
46.30/12.85	    (9) QDP
46.30/12.85	        (10) QDPOrderProof [EQUIVALENT, 217 ms]
46.30/12.85	        (11) QDP
46.30/12.85	        (12) PisEmptyProof [EQUIVALENT, 0 ms]
46.30/12.85	        (13) YES
46.30/12.85	    (14) QDP
46.30/12.85	        (15) QDPOrderProof [EQUIVALENT, 92 ms]
46.30/12.85	        (16) QDP
46.30/12.85	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
46.30/12.85	        (18) YES
46.30/12.85	
46.30/12.85	
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(0)
46.30/12.85	Obligation:
46.30/12.85	Q restricted rewrite system:
46.30/12.85	The TRS R consists of the following rules:
46.30/12.85	
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	Q is empty.
46.30/12.85	
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(1) DependencyPairsProof (EQUIVALENT)
46.30/12.85	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(2)
46.30/12.85	Obligation:
46.30/12.85	Q DP problem:
46.30/12.85	The TRS P consists of the following rules:
46.30/12.85	
46.30/12.85	   A(b(b(b(x1)))) -> B(a(b(a(x1))))
46.30/12.85	   A(b(b(b(x1)))) -> A(b(a(x1)))
46.30/12.85	   A(b(b(b(x1)))) -> B(a(x1))
46.30/12.85	   A(b(b(b(x1)))) -> A(x1)
46.30/12.85	   A(b(b(b(x1)))) -> A(a(a(b(x1))))
46.30/12.85	   A(b(b(b(x1)))) -> A(a(b(x1)))
46.30/12.85	   A(b(b(b(x1)))) -> A(b(x1))
46.30/12.85	   B(a(a(b(x1)))) -> B(b(a(b(x1))))
46.30/12.85	   B(a(a(b(x1)))) -> B(a(b(x1)))
46.30/12.85	   B(a(a(a(x1)))) -> A(a(a(b(x1))))
46.30/12.85	   B(a(a(a(x1)))) -> A(a(b(x1)))
46.30/12.85	   B(a(a(a(x1)))) -> A(b(x1))
46.30/12.85	   B(a(a(a(x1)))) -> B(x1)
46.30/12.85	
46.30/12.85	The TRS R consists of the following rules:
46.30/12.85	
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	Q is empty.
46.30/12.85	We have to consider all minimal (P,Q,R)-chains.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(3) QDPOrderProof (EQUIVALENT)
46.30/12.85	We use the reduction pair processor [LPAR04,JAR06].
46.30/12.85	
46.30/12.85	
46.30/12.85	The following pairs can be oriented strictly and are deleted.
46.30/12.85	
46.30/12.85	   A(b(b(b(x1)))) -> A(b(a(x1)))
46.30/12.85	   A(b(b(b(x1)))) -> B(a(x1))
46.30/12.85	   A(b(b(b(x1)))) -> A(x1)
46.30/12.85	   A(b(b(b(x1)))) -> A(a(b(x1)))
46.30/12.85	   A(b(b(b(x1)))) -> A(b(x1))
46.30/12.85	   B(a(a(b(x1)))) -> B(a(b(x1)))
46.30/12.85	   B(a(a(a(x1)))) -> A(a(b(x1)))
46.30/12.85	   B(a(a(a(x1)))) -> A(b(x1))
46.30/12.85	   B(a(a(a(x1)))) -> B(x1)
46.30/12.85	The remaining pairs can at least be oriented weakly.
46.30/12.85	Used ordering:  Polynomial interpretation [POLO]:
46.30/12.85	
46.30/12.85	   POL(A(x_1)) = 1 + x_1
46.30/12.85	   POL(B(x_1)) = 1 + x_1
46.30/12.85	   POL(a(x_1)) = 1 + x_1
46.30/12.85	   POL(b(x_1)) = 1 + x_1
46.30/12.85	
46.30/12.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
46.30/12.85	
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(4)
46.30/12.85	Obligation:
46.30/12.85	Q DP problem:
46.30/12.85	The TRS P consists of the following rules:
46.30/12.85	
46.30/12.85	   A(b(b(b(x1)))) -> B(a(b(a(x1))))
46.30/12.85	   A(b(b(b(x1)))) -> A(a(a(b(x1))))
46.30/12.85	   B(a(a(b(x1)))) -> B(b(a(b(x1))))
46.30/12.85	   B(a(a(a(x1)))) -> A(a(a(b(x1))))
46.30/12.85	
46.30/12.85	The TRS R consists of the following rules:
46.30/12.85	
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	Q is empty.
46.30/12.85	We have to consider all minimal (P,Q,R)-chains.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(5) QDPOrderProof (EQUIVALENT)
46.30/12.85	We use the reduction pair processor [LPAR04,JAR06].
46.30/12.85	
46.30/12.85	
46.30/12.85	The following pairs can be oriented strictly and are deleted.
46.30/12.85	
46.30/12.85	   A(b(b(b(x1)))) -> B(a(b(a(x1))))
46.30/12.85	The remaining pairs can at least be oriented weakly.
46.30/12.85	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 1A, 0A]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 1A, 1A], [0A, 0A, 0A]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	
46.30/12.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
46.30/12.85	
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(6)
46.30/12.85	Obligation:
46.30/12.85	Q DP problem:
46.30/12.85	The TRS P consists of the following rules:
46.30/12.85	
46.30/12.85	   A(b(b(b(x1)))) -> A(a(a(b(x1))))
46.30/12.85	   B(a(a(b(x1)))) -> B(b(a(b(x1))))
46.30/12.85	   B(a(a(a(x1)))) -> A(a(a(b(x1))))
46.30/12.85	
46.30/12.85	The TRS R consists of the following rules:
46.30/12.85	
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	Q is empty.
46.30/12.85	We have to consider all minimal (P,Q,R)-chains.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(7) DependencyGraphProof (EQUIVALENT)
46.30/12.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(8)
46.30/12.85	Complex Obligation (AND)
46.30/12.85	
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(9)
46.30/12.85	Obligation:
46.30/12.85	Q DP problem:
46.30/12.85	The TRS P consists of the following rules:
46.30/12.85	
46.30/12.85	   A(b(b(b(x1)))) -> A(a(a(b(x1))))
46.30/12.85	
46.30/12.85	The TRS R consists of the following rules:
46.30/12.85	
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	Q is empty.
46.30/12.85	We have to consider all minimal (P,Q,R)-chains.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(10) QDPOrderProof (EQUIVALENT)
46.30/12.85	We use the reduction pair processor [LPAR04,JAR06].
46.30/12.85	
46.30/12.85	
46.30/12.85	The following pairs can be oriented strictly and are deleted.
46.30/12.85	
46.30/12.85	   A(b(b(b(x1)))) -> A(a(a(b(x1))))
46.30/12.85	The remaining pairs can at least be oriented weakly.
46.30/12.85	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [0A, -I, 1A], [1A, 0A, 1A]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(a(x_1)) =  	[[1A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [0A, -I, 0A], [1A, 0A, -I]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	
46.30/12.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
46.30/12.85	
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(11)
46.30/12.85	Obligation:
46.30/12.85	Q DP problem:
46.30/12.85	P is empty.
46.30/12.85	The TRS R consists of the following rules:
46.30/12.85	
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	Q is empty.
46.30/12.85	We have to consider all minimal (P,Q,R)-chains.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(12) PisEmptyProof (EQUIVALENT)
46.30/12.85	The TRS P is empty. Hence, there is no (P,Q,R) chain.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(13)
46.30/12.85	YES
46.30/12.85	
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(14)
46.30/12.85	Obligation:
46.30/12.85	Q DP problem:
46.30/12.85	The TRS P consists of the following rules:
46.30/12.85	
46.30/12.85	   B(a(a(b(x1)))) -> B(b(a(b(x1))))
46.30/12.85	
46.30/12.85	The TRS R consists of the following rules:
46.30/12.85	
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	Q is empty.
46.30/12.85	We have to consider all minimal (P,Q,R)-chains.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(15) QDPOrderProof (EQUIVALENT)
46.30/12.85	We use the reduction pair processor [LPAR04,JAR06].
46.30/12.85	
46.30/12.85	
46.30/12.85	The following pairs can be oriented strictly and are deleted.
46.30/12.85	
46.30/12.85	   B(a(a(b(x1)))) -> B(b(a(b(x1))))
46.30/12.85	The remaining pairs can at least be oriented weakly.
46.30/12.85	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-I, 0A, 1A], [0A, -I, -I], [-I, 0A, -I]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	   <<<
46.30/12.85	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 1A, 0A], [-I, 0A, 0A]] 	* 	x_1
46.30/12.85	>>>
46.30/12.85	
46.30/12.85	
46.30/12.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
46.30/12.85	
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(16)
46.30/12.85	Obligation:
46.30/12.85	Q DP problem:
46.30/12.85	P is empty.
46.30/12.85	The TRS R consists of the following rules:
46.30/12.85	
46.30/12.85	   a(b(b(b(x1)))) -> b(a(b(a(x1))))
46.30/12.85	   a(b(b(b(x1)))) -> a(a(a(b(x1))))
46.30/12.85	   b(a(a(b(x1)))) -> b(b(a(b(x1))))
46.30/12.85	   b(a(a(a(x1)))) -> a(a(a(b(x1))))
46.30/12.85	
46.30/12.85	Q is empty.
46.30/12.85	We have to consider all minimal (P,Q,R)-chains.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(17) PisEmptyProof (EQUIVALENT)
46.30/12.85	The TRS P is empty. Hence, there is no (P,Q,R) chain.
46.30/12.85	----------------------------------------
46.30/12.85	
46.30/12.85	(18)
46.30/12.85	YES
46.88/12.93	EOF