30.08/8.56	YES
30.25/8.61	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
30.25/8.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.25/8.61	
30.25/8.61	
30.25/8.61	Termination w.r.t. Q of the given QTRS could be proven:
30.25/8.61	
30.25/8.61	(0) QTRS
30.25/8.61	(1) DependencyPairsProof [EQUIVALENT, 15 ms]
30.25/8.61	(2) QDP
30.25/8.61	(3) DependencyGraphProof [EQUIVALENT, 5 ms]
30.25/8.61	(4) AND
30.25/8.61	    (5) QDP
30.25/8.61	        (6) QDPSizeChangeProof [EQUIVALENT, 0 ms]
30.25/8.61	        (7) YES
30.25/8.61	    (8) QDP
30.25/8.61	        (9) QDPOrderProof [EQUIVALENT, 447 ms]
30.25/8.61	        (10) QDP
30.25/8.61	        (11) PisEmptyProof [EQUIVALENT, 0 ms]
30.25/8.61	        (12) YES
30.25/8.61	
30.25/8.61	
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(0)
30.25/8.61	Obligation:
30.25/8.61	Q restricted rewrite system:
30.25/8.61	The TRS R consists of the following rules:
30.25/8.61	
30.25/8.61	   b(b(x1)) -> b(a(b(a(a(b(x1))))))
30.25/8.61	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
30.25/8.61	   b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1))))))
30.25/8.61	   b(b(a(a(b(x1))))) -> b(a(a(b(b(x1)))))
30.25/8.61	
30.25/8.61	Q is empty.
30.25/8.61	
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(1) DependencyPairsProof (EQUIVALENT)
30.25/8.61	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(2)
30.25/8.61	Obligation:
30.25/8.61	Q DP problem:
30.25/8.61	The TRS P consists of the following rules:
30.25/8.61	
30.25/8.61	   B(b(x1)) -> B(a(b(a(a(b(x1))))))
30.25/8.61	   B(b(x1)) -> B(a(a(b(x1))))
30.25/8.61	   B(b(a(b(x1)))) -> B(a(b(b(x1))))
30.25/8.61	   B(b(a(b(x1)))) -> B(b(x1))
30.25/8.61	   B(a(a(a(b(a(a(b(x1)))))))) -> B(a(a(a(b(b(x1))))))
30.25/8.61	   B(a(a(a(b(a(a(b(x1)))))))) -> B(b(x1))
30.25/8.61	   B(b(a(a(b(x1))))) -> B(a(a(b(b(x1)))))
30.25/8.61	   B(b(a(a(b(x1))))) -> B(b(x1))
30.25/8.61	
30.25/8.61	The TRS R consists of the following rules:
30.25/8.61	
30.25/8.61	   b(b(x1)) -> b(a(b(a(a(b(x1))))))
30.25/8.61	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
30.25/8.61	   b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1))))))
30.25/8.61	   b(b(a(a(b(x1))))) -> b(a(a(b(b(x1)))))
30.25/8.61	
30.25/8.61	Q is empty.
30.25/8.61	We have to consider all minimal (P,Q,R)-chains.
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(3) DependencyGraphProof (EQUIVALENT)
30.25/8.61	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(4)
30.25/8.61	Complex Obligation (AND)
30.25/8.61	
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(5)
30.25/8.61	Obligation:
30.25/8.61	Q DP problem:
30.25/8.61	The TRS P consists of the following rules:
30.25/8.61	
30.25/8.61	   B(b(a(a(b(x1))))) -> B(b(x1))
30.25/8.61	   B(b(a(b(x1)))) -> B(b(x1))
30.25/8.61	
30.25/8.61	The TRS R consists of the following rules:
30.25/8.61	
30.25/8.61	   b(b(x1)) -> b(a(b(a(a(b(x1))))))
30.25/8.61	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
30.25/8.61	   b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1))))))
30.25/8.61	   b(b(a(a(b(x1))))) -> b(a(a(b(b(x1)))))
30.25/8.61	
30.25/8.61	Q is empty.
30.25/8.61	We have to consider all minimal (P,Q,R)-chains.
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(6) QDPSizeChangeProof (EQUIVALENT)
30.25/8.61	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
30.25/8.61	
30.25/8.61	From the DPs we obtained the following set of size-change graphs:
30.25/8.61	*B(b(a(a(b(x1))))) -> B(b(x1))
30.25/8.61	The graph contains the following edges 1 > 1
30.25/8.61	
30.25/8.61	
30.25/8.61	*B(b(a(b(x1)))) -> B(b(x1))
30.25/8.61	The graph contains the following edges 1 > 1
30.25/8.61	
30.25/8.61	
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(7)
30.25/8.61	YES
30.25/8.61	
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(8)
30.25/8.61	Obligation:
30.25/8.61	Q DP problem:
30.25/8.61	The TRS P consists of the following rules:
30.25/8.61	
30.25/8.61	   B(a(a(a(b(a(a(b(x1)))))))) -> B(a(a(a(b(b(x1))))))
30.25/8.61	
30.25/8.61	The TRS R consists of the following rules:
30.25/8.61	
30.25/8.61	   b(b(x1)) -> b(a(b(a(a(b(x1))))))
30.25/8.61	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
30.25/8.61	   b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1))))))
30.25/8.61	   b(b(a(a(b(x1))))) -> b(a(a(b(b(x1)))))
30.25/8.61	
30.25/8.61	Q is empty.
30.25/8.61	We have to consider all minimal (P,Q,R)-chains.
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(9) QDPOrderProof (EQUIVALENT)
30.25/8.61	We use the reduction pair processor [LPAR04,JAR06].
30.25/8.61	
30.25/8.61	
30.25/8.61	The following pairs can be oriented strictly and are deleted.
30.25/8.61	
30.25/8.61	   B(a(a(a(b(a(a(b(x1)))))))) -> B(a(a(a(b(b(x1))))))
30.25/8.61	The remaining pairs can at least be oriented weakly.
30.25/8.61	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.25/8.61	
30.25/8.61	   <<<
30.25/8.61	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
30.25/8.61	>>>
30.25/8.61	
30.25/8.61	   <<<
30.25/8.61	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [-I, 0A, 0A], [0A, -I, -I]] 	* 	x_1
30.25/8.61	>>>
30.25/8.61	
30.25/8.61	   <<<
30.25/8.61	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
30.25/8.61	>>>
30.25/8.61	
30.25/8.61	
30.25/8.61	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.25/8.61	
30.25/8.61	   b(b(x1)) -> b(a(b(a(a(b(x1))))))
30.25/8.61	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
30.25/8.61	   b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1))))))
30.25/8.61	   b(b(a(a(b(x1))))) -> b(a(a(b(b(x1)))))
30.25/8.61	
30.25/8.61	
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(10)
30.25/8.61	Obligation:
30.25/8.61	Q DP problem:
30.25/8.61	P is empty.
30.25/8.61	The TRS R consists of the following rules:
30.25/8.61	
30.25/8.61	   b(b(x1)) -> b(a(b(a(a(b(x1))))))
30.25/8.61	   b(b(a(b(x1)))) -> b(a(b(b(x1))))
30.25/8.61	   b(a(a(a(b(a(a(b(x1)))))))) -> b(a(a(a(b(b(x1))))))
30.25/8.61	   b(b(a(a(b(x1))))) -> b(a(a(b(b(x1)))))
30.25/8.61	
30.25/8.61	Q is empty.
30.25/8.61	We have to consider all minimal (P,Q,R)-chains.
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(11) PisEmptyProof (EQUIVALENT)
30.25/8.61	The TRS P is empty. Hence, there is no (P,Q,R) chain.
30.25/8.61	----------------------------------------
30.25/8.61	
30.25/8.61	(12)
30.25/8.61	YES
30.37/8.66	EOF