33.56/9.47	YES
33.74/9.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
33.74/9.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.74/9.56	
33.74/9.56	
33.74/9.56	Termination w.r.t. Q of the given QTRS could be proven:
33.74/9.56	
33.74/9.56	(0) QTRS
33.74/9.56	(1) QTRS Reverse [EQUIVALENT, 0 ms]
33.74/9.56	(2) QTRS
33.74/9.56	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
33.74/9.56	(4) QDP
33.74/9.56	(5) QDPOrderProof [EQUIVALENT, 19 ms]
33.74/9.56	(6) QDP
33.74/9.56	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
33.74/9.56	(8) AND
33.74/9.56	    (9) QDP
33.74/9.56	        (10) QDPOrderProof [EQUIVALENT, 64 ms]
33.74/9.56	        (11) QDP
33.74/9.56	        (12) PisEmptyProof [EQUIVALENT, 0 ms]
33.74/9.56	        (13) YES
33.74/9.56	    (14) QDP
33.74/9.56	        (15) QDPOrderProof [EQUIVALENT, 73 ms]
33.74/9.56	        (16) QDP
33.74/9.56	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
33.74/9.56	        (18) YES
33.74/9.56	
33.74/9.56	
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(0)
33.74/9.56	Obligation:
33.74/9.56	Q restricted rewrite system:
33.74/9.56	The TRS R consists of the following rules:
33.74/9.56	
33.74/9.56	   a(b(a(b(x1)))) -> a(a(b(b(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> b(b(b(a(x1))))
33.74/9.56	   a(a(a(a(x1)))) -> b(a(b(a(x1))))
33.74/9.56	
33.74/9.56	Q is empty.
33.74/9.56	
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(1) QTRS Reverse (EQUIVALENT)
33.74/9.56	We applied the QTRS Reverse Processor [REVERSE].
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(2)
33.74/9.56	Obligation:
33.74/9.56	Q restricted rewrite system:
33.74/9.56	The TRS R consists of the following rules:
33.74/9.56	
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	
33.74/9.56	Q is empty.
33.74/9.56	
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(3) DependencyPairsProof (EQUIVALENT)
33.74/9.56	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(4)
33.74/9.56	Obligation:
33.74/9.56	Q DP problem:
33.74/9.56	The TRS P consists of the following rules:
33.74/9.56	
33.74/9.56	   B(a(b(a(x1)))) -> B(b(a(a(x1))))
33.74/9.56	   B(a(b(a(x1)))) -> B(a(a(x1)))
33.74/9.56	   B(a(b(a(x1)))) -> A(a(x1))
33.74/9.56	   B(b(b(b(x1)))) -> A(b(b(b(x1))))
33.74/9.56	   A(a(a(a(x1)))) -> A(b(a(b(x1))))
33.74/9.56	   A(a(a(a(x1)))) -> B(a(b(x1)))
33.74/9.56	   A(a(a(a(x1)))) -> A(b(x1))
33.74/9.56	   A(a(a(a(x1)))) -> B(x1)
33.74/9.56	
33.74/9.56	The TRS R consists of the following rules:
33.74/9.56	
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	
33.74/9.56	Q is empty.
33.74/9.56	We have to consider all minimal (P,Q,R)-chains.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(5) QDPOrderProof (EQUIVALENT)
33.74/9.56	We use the reduction pair processor [LPAR04,JAR06].
33.74/9.56	
33.74/9.56	
33.74/9.56	The following pairs can be oriented strictly and are deleted.
33.74/9.56	
33.74/9.56	   B(a(b(a(x1)))) -> B(a(a(x1)))
33.74/9.56	   B(a(b(a(x1)))) -> A(a(x1))
33.74/9.56	   A(a(a(a(x1)))) -> B(a(b(x1)))
33.74/9.56	   A(a(a(a(x1)))) -> A(b(x1))
33.74/9.56	   A(a(a(a(x1)))) -> B(x1)
33.74/9.56	The remaining pairs can at least be oriented weakly.
33.74/9.56	Used ordering:  Polynomial interpretation [POLO]:
33.74/9.56	
33.74/9.56	   POL(A(x_1)) = 1 + x_1
33.74/9.56	   POL(B(x_1)) = 1 + x_1
33.74/9.56	   POL(a(x_1)) = 1 + x_1
33.74/9.56	   POL(b(x_1)) = 1 + x_1
33.74/9.56	
33.74/9.56	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.74/9.56	
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	
33.74/9.56	
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(6)
33.74/9.56	Obligation:
33.74/9.56	Q DP problem:
33.74/9.56	The TRS P consists of the following rules:
33.74/9.56	
33.74/9.56	   B(a(b(a(x1)))) -> B(b(a(a(x1))))
33.74/9.56	   B(b(b(b(x1)))) -> A(b(b(b(x1))))
33.74/9.56	   A(a(a(a(x1)))) -> A(b(a(b(x1))))
33.74/9.56	
33.74/9.56	The TRS R consists of the following rules:
33.74/9.56	
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	
33.74/9.56	Q is empty.
33.74/9.56	We have to consider all minimal (P,Q,R)-chains.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(7) DependencyGraphProof (EQUIVALENT)
33.74/9.56	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(8)
33.74/9.56	Complex Obligation (AND)
33.74/9.56	
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(9)
33.74/9.56	Obligation:
33.74/9.56	Q DP problem:
33.74/9.56	The TRS P consists of the following rules:
33.74/9.56	
33.74/9.56	   A(a(a(a(x1)))) -> A(b(a(b(x1))))
33.74/9.56	
33.74/9.56	The TRS R consists of the following rules:
33.74/9.56	
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	
33.74/9.56	Q is empty.
33.74/9.56	We have to consider all minimal (P,Q,R)-chains.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(10) QDPOrderProof (EQUIVALENT)
33.74/9.56	We use the reduction pair processor [LPAR04,JAR06].
33.74/9.56	
33.74/9.56	
33.74/9.56	The following pairs can be oriented strictly and are deleted.
33.74/9.56	
33.74/9.56	   A(a(a(a(x1)))) -> A(b(a(b(x1))))
33.74/9.56	The remaining pairs can at least be oriented weakly.
33.74/9.56	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.74/9.56	
33.74/9.56	   <<<
33.74/9.56	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
33.74/9.56	>>>
33.74/9.56	
33.74/9.56	   <<<
33.74/9.56	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 1A, -I], [0A, 0A, 0A]] 	* 	x_1
33.74/9.56	>>>
33.74/9.56	
33.74/9.56	   <<<
33.74/9.56	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[1A, 0A, 0A], [0A, 0A, -I], [0A, 1A, -I]] 	* 	x_1
33.74/9.56	>>>
33.74/9.56	
33.74/9.56	
33.74/9.56	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.74/9.56	
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	
33.74/9.56	
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(11)
33.74/9.56	Obligation:
33.74/9.56	Q DP problem:
33.74/9.56	P is empty.
33.74/9.56	The TRS R consists of the following rules:
33.74/9.56	
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	
33.74/9.56	Q is empty.
33.74/9.56	We have to consider all minimal (P,Q,R)-chains.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(12) PisEmptyProof (EQUIVALENT)
33.74/9.56	The TRS P is empty. Hence, there is no (P,Q,R) chain.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(13)
33.74/9.56	YES
33.74/9.56	
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(14)
33.74/9.56	Obligation:
33.74/9.56	Q DP problem:
33.74/9.56	The TRS P consists of the following rules:
33.74/9.56	
33.74/9.56	   B(a(b(a(x1)))) -> B(b(a(a(x1))))
33.74/9.56	
33.74/9.56	The TRS R consists of the following rules:
33.74/9.56	
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	
33.74/9.56	Q is empty.
33.74/9.56	We have to consider all minimal (P,Q,R)-chains.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(15) QDPOrderProof (EQUIVALENT)
33.74/9.56	We use the reduction pair processor [LPAR04,JAR06].
33.74/9.56	
33.74/9.56	
33.74/9.56	The following pairs can be oriented strictly and are deleted.
33.74/9.56	
33.74/9.56	   B(a(b(a(x1)))) -> B(b(a(a(x1))))
33.74/9.56	The remaining pairs can at least be oriented weakly.
33.74/9.56	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.74/9.56	
33.74/9.56	   <<<
33.74/9.56	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
33.74/9.56	>>>
33.74/9.56	
33.74/9.56	   <<<
33.74/9.56	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [0A, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
33.74/9.56	>>>
33.74/9.56	
33.74/9.56	   <<<
33.74/9.56	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 1A, 0A], [-I, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
33.74/9.56	>>>
33.74/9.56	
33.74/9.56	
33.74/9.56	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.74/9.56	
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	
33.74/9.56	
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(16)
33.74/9.56	Obligation:
33.74/9.56	Q DP problem:
33.74/9.56	P is empty.
33.74/9.56	The TRS R consists of the following rules:
33.74/9.56	
33.74/9.56	   b(a(b(a(x1)))) -> b(b(a(a(x1))))
33.74/9.56	   b(b(b(b(x1)))) -> a(b(b(b(x1))))
33.74/9.56	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
33.74/9.56	
33.74/9.56	Q is empty.
33.74/9.56	We have to consider all minimal (P,Q,R)-chains.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(17) PisEmptyProof (EQUIVALENT)
33.74/9.56	The TRS P is empty. Hence, there is no (P,Q,R) chain.
33.74/9.56	----------------------------------------
33.74/9.56	
33.74/9.56	(18)
33.74/9.56	YES
34.26/9.64	EOF