29.55/8.54	YES
29.55/8.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
29.55/8.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
29.55/8.58	
29.55/8.58	
29.55/8.58	Termination w.r.t. Q of the given QTRS could be proven:
29.55/8.58	
29.55/8.58	(0) QTRS
29.55/8.58	(1) DependencyPairsProof [EQUIVALENT, 16 ms]
29.55/8.58	(2) QDP
29.55/8.58	(3) QDPOrderProof [EQUIVALENT, 120 ms]
29.55/8.58	(4) QDP
29.55/8.58	(5) QDPOrderProof [EQUIVALENT, 251 ms]
29.55/8.58	(6) QDP
29.55/8.58	(7) PisEmptyProof [EQUIVALENT, 0 ms]
29.55/8.58	(8) YES
29.55/8.58	
29.55/8.58	
29.55/8.58	----------------------------------------
29.55/8.58	
29.55/8.58	(0)
29.55/8.58	Obligation:
29.55/8.58	Q restricted rewrite system:
29.55/8.58	The TRS R consists of the following rules:
29.55/8.58	
29.55/8.58	   b(b(a(b(x1)))) -> a(b(b(b(x1))))
29.55/8.58	   a(a(a(b(x1)))) -> a(b(b(a(x1))))
29.55/8.58	   b(a(a(b(x1)))) -> a(a(a(b(x1))))
29.55/8.58	
29.55/8.58	Q is empty.
29.55/8.58	
29.55/8.58	----------------------------------------
29.55/8.58	
29.55/8.58	(1) DependencyPairsProof (EQUIVALENT)
29.55/8.58	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
29.55/8.58	----------------------------------------
29.55/8.58	
29.55/8.58	(2)
29.55/8.58	Obligation:
29.55/8.58	Q DP problem:
29.55/8.58	The TRS P consists of the following rules:
29.55/8.58	
29.55/8.58	   B(b(a(b(x1)))) -> A(b(b(b(x1))))
29.55/8.58	   B(b(a(b(x1)))) -> B(b(b(x1)))
29.55/8.58	   B(b(a(b(x1)))) -> B(b(x1))
29.55/8.58	   A(a(a(b(x1)))) -> A(b(b(a(x1))))
29.55/8.58	   A(a(a(b(x1)))) -> B(b(a(x1)))
29.55/8.58	   A(a(a(b(x1)))) -> B(a(x1))
29.55/8.58	   A(a(a(b(x1)))) -> A(x1)
29.55/8.58	   B(a(a(b(x1)))) -> A(a(a(b(x1))))
29.55/8.58	
29.55/8.58	The TRS R consists of the following rules:
29.55/8.58	
29.55/8.58	   b(b(a(b(x1)))) -> a(b(b(b(x1))))
29.55/8.58	   a(a(a(b(x1)))) -> a(b(b(a(x1))))
29.55/8.58	   b(a(a(b(x1)))) -> a(a(a(b(x1))))
29.55/8.58	
29.55/8.58	Q is empty.
29.55/8.58	We have to consider all minimal (P,Q,R)-chains.
29.55/8.58	----------------------------------------
29.55/8.58	
29.55/8.58	(3) QDPOrderProof (EQUIVALENT)
29.55/8.58	We use the reduction pair processor [LPAR04,JAR06].
29.55/8.58	
29.55/8.58	
29.55/8.58	The following pairs can be oriented strictly and are deleted.
29.55/8.58	
29.55/8.58	   B(b(a(b(x1)))) -> A(b(b(b(x1))))
29.55/8.58	   B(b(a(b(x1)))) -> B(b(b(x1)))
29.55/8.58	   B(b(a(b(x1)))) -> B(b(x1))
29.55/8.58	   A(a(a(b(x1)))) -> B(b(a(x1)))
29.55/8.58	   A(a(a(b(x1)))) -> B(a(x1))
29.55/8.58	   A(a(a(b(x1)))) -> A(x1)
29.55/8.58	   B(a(a(b(x1)))) -> A(a(a(b(x1))))
29.55/8.58	The remaining pairs can at least be oriented weakly.
29.55/8.58	Used ordering:  Polynomial interpretation [POLO]:
29.55/8.58	
29.55/8.58	   POL(A(x_1)) = 4 + 2*x_1
29.55/8.58	   POL(B(x_1)) = 1 + 4*x_1
29.55/8.58	   POL(a(x_1)) = 2 + 2*x_1
29.55/8.58	   POL(b(x_1)) = 2 + 2*x_1
29.55/8.58	
29.55/8.58	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.55/8.58	
29.55/8.58	   b(b(a(b(x1)))) -> a(b(b(b(x1))))
29.55/8.58	   b(a(a(b(x1)))) -> a(a(a(b(x1))))
29.55/8.58	   a(a(a(b(x1)))) -> a(b(b(a(x1))))
29.55/8.58	
29.55/8.58	
29.55/8.58	----------------------------------------
29.55/8.58	
29.55/8.58	(4)
29.55/8.58	Obligation:
29.55/8.58	Q DP problem:
29.55/8.58	The TRS P consists of the following rules:
29.55/8.58	
29.55/8.58	   A(a(a(b(x1)))) -> A(b(b(a(x1))))
29.55/8.58	
29.55/8.58	The TRS R consists of the following rules:
29.55/8.58	
29.55/8.58	   b(b(a(b(x1)))) -> a(b(b(b(x1))))
29.55/8.58	   a(a(a(b(x1)))) -> a(b(b(a(x1))))
29.55/8.58	   b(a(a(b(x1)))) -> a(a(a(b(x1))))
29.55/8.58	
29.55/8.58	Q is empty.
29.55/8.58	We have to consider all minimal (P,Q,R)-chains.
29.55/8.58	----------------------------------------
29.55/8.58	
29.55/8.58	(5) QDPOrderProof (EQUIVALENT)
29.55/8.58	We use the reduction pair processor [LPAR04,JAR06].
29.55/8.58	
29.55/8.58	
29.55/8.58	The following pairs can be oriented strictly and are deleted.
29.55/8.58	
29.55/8.58	   A(a(a(b(x1)))) -> A(b(b(a(x1))))
29.55/8.58	The remaining pairs can at least be oriented weakly.
29.55/8.58	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
29.55/8.58	
29.55/8.58	   <<<
29.55/8.58	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
29.55/8.58	>>>
29.55/8.58	
29.55/8.58	   <<<
29.55/8.58	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [0A, 0A, -I]] 	* 	x_1
29.55/8.58	>>>
29.55/8.58	
29.55/8.58	   <<<
29.55/8.58	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
29.55/8.58	>>>
29.55/8.58	
29.55/8.58	
29.55/8.58	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
29.55/8.58	
29.55/8.58	   a(a(a(b(x1)))) -> a(b(b(a(x1))))
29.55/8.58	   b(b(a(b(x1)))) -> a(b(b(b(x1))))
29.55/8.58	   b(a(a(b(x1)))) -> a(a(a(b(x1))))
29.55/8.58	
29.55/8.58	
29.55/8.58	----------------------------------------
29.55/8.58	
29.55/8.58	(6)
29.55/8.58	Obligation:
29.55/8.58	Q DP problem:
29.55/8.58	P is empty.
29.55/8.58	The TRS R consists of the following rules:
29.55/8.58	
29.55/8.58	   b(b(a(b(x1)))) -> a(b(b(b(x1))))
29.55/8.58	   a(a(a(b(x1)))) -> a(b(b(a(x1))))
29.55/8.58	   b(a(a(b(x1)))) -> a(a(a(b(x1))))
29.55/8.58	
29.55/8.58	Q is empty.
29.55/8.58	We have to consider all minimal (P,Q,R)-chains.
29.55/8.58	----------------------------------------
29.55/8.58	
29.55/8.58	(7) PisEmptyProof (EQUIVALENT)
29.55/8.58	The TRS P is empty. Hence, there is no (P,Q,R) chain.
29.55/8.58	----------------------------------------
29.55/8.58	
29.55/8.58	(8)
29.55/8.58	YES
30.07/8.72	EOF