37.82/10.57	YES
37.82/10.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
37.82/10.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
37.82/10.59	
37.82/10.59	
37.82/10.59	Termination w.r.t. Q of the given QTRS could be proven:
37.82/10.59	
37.82/10.59	(0) QTRS
37.82/10.59	(1) DependencyPairsProof [EQUIVALENT, 21 ms]
37.82/10.59	(2) QDP
37.82/10.59	(3) DependencyGraphProof [EQUIVALENT, 8 ms]
37.82/10.59	(4) QDP
37.82/10.59	(5) MRRProof [EQUIVALENT, 59 ms]
37.82/10.59	(6) QDP
37.82/10.59	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
37.82/10.59	(8) AND
37.82/10.59	    (9) QDP
37.82/10.59	        (10) QDPOrderProof [EQUIVALENT, 118 ms]
37.82/10.59	        (11) QDP
37.82/10.59	        (12) PisEmptyProof [EQUIVALENT, 0 ms]
37.82/10.59	        (13) YES
37.82/10.59	    (14) QDP
37.82/10.59	        (15) QDPOrderProof [EQUIVALENT, 416 ms]
37.82/10.59	        (16) QDP
37.82/10.59	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
37.82/10.59	        (18) YES
37.82/10.59	
37.82/10.59	
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(0)
37.82/10.59	Obligation:
37.82/10.59	Q restricted rewrite system:
37.82/10.59	The TRS R consists of the following rules:
37.82/10.59	
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	
37.82/10.59	Q is empty.
37.82/10.59	
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(1) DependencyPairsProof (EQUIVALENT)
37.82/10.59	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(2)
37.82/10.59	Obligation:
37.82/10.59	Q DP problem:
37.82/10.59	The TRS P consists of the following rules:
37.82/10.59	
37.82/10.59	   B(b(a(a(x1)))) -> B(b(a(b(x1))))
37.82/10.59	   B(b(a(a(x1)))) -> B(a(b(x1)))
37.82/10.59	   B(b(a(a(x1)))) -> A(b(x1))
37.82/10.59	   B(b(a(a(x1)))) -> B(x1)
37.82/10.59	   A(a(b(b(x1)))) -> B(b(a(a(x1))))
37.82/10.59	   A(a(b(b(x1)))) -> B(a(a(x1)))
37.82/10.59	   A(a(b(b(x1)))) -> A(a(x1))
37.82/10.59	   A(a(b(b(x1)))) -> A(x1)
37.82/10.59	   B(a(a(b(x1)))) -> B(a(a(a(x1))))
37.82/10.59	   B(a(a(b(x1)))) -> A(a(a(x1)))
37.82/10.59	   B(a(a(b(x1)))) -> A(a(x1))
37.82/10.59	   B(a(a(b(x1)))) -> A(x1)
37.82/10.59	
37.82/10.59	The TRS R consists of the following rules:
37.82/10.59	
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	
37.82/10.59	Q is empty.
37.82/10.59	We have to consider all minimal (P,Q,R)-chains.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(3) DependencyGraphProof (EQUIVALENT)
37.82/10.59	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(4)
37.82/10.59	Obligation:
37.82/10.59	Q DP problem:
37.82/10.59	The TRS P consists of the following rules:
37.82/10.59	
37.82/10.59	   B(b(a(a(x1)))) -> B(a(b(x1)))
37.82/10.59	   B(b(a(a(x1)))) -> B(b(a(b(x1))))
37.82/10.59	   B(b(a(a(x1)))) -> B(x1)
37.82/10.59	   B(a(a(b(x1)))) -> B(a(a(a(x1))))
37.82/10.59	   B(a(a(b(x1)))) -> A(a(a(x1)))
37.82/10.59	   A(a(b(b(x1)))) -> B(b(a(a(x1))))
37.82/10.59	   A(a(b(b(x1)))) -> B(a(a(x1)))
37.82/10.59	   B(a(a(b(x1)))) -> A(a(x1))
37.82/10.59	   A(a(b(b(x1)))) -> A(a(x1))
37.82/10.59	   A(a(b(b(x1)))) -> A(x1)
37.82/10.59	   B(a(a(b(x1)))) -> A(x1)
37.82/10.59	
37.82/10.59	The TRS R consists of the following rules:
37.82/10.59	
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	
37.82/10.59	Q is empty.
37.82/10.59	We have to consider all minimal (P,Q,R)-chains.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(5) MRRProof (EQUIVALENT)
37.82/10.59	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
37.82/10.59	
37.82/10.59	Strictly oriented dependency pairs:
37.82/10.59	
37.82/10.59	   B(b(a(a(x1)))) -> B(a(b(x1)))
37.82/10.59	   B(b(a(a(x1)))) -> B(x1)
37.82/10.59	   B(a(a(b(x1)))) -> A(a(a(x1)))
37.82/10.59	   A(a(b(b(x1)))) -> B(b(a(a(x1))))
37.82/10.59	   A(a(b(b(x1)))) -> B(a(a(x1)))
37.82/10.59	   B(a(a(b(x1)))) -> A(a(x1))
37.82/10.59	   A(a(b(b(x1)))) -> A(a(x1))
37.82/10.59	   A(a(b(b(x1)))) -> A(x1)
37.82/10.59	   B(a(a(b(x1)))) -> A(x1)
37.82/10.59	
37.82/10.59	
37.82/10.59	Used ordering: Polynomial interpretation [POLO]:
37.82/10.59	
37.82/10.59	   POL(A(x_1)) = 1 + 2*x_1
37.82/10.59	   POL(B(x_1)) = 2*x_1
37.82/10.59	   POL(a(x_1)) = 1 + x_1
37.82/10.59	   POL(b(x_1)) = 1 + x_1
37.82/10.59	
37.82/10.59	
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(6)
37.82/10.59	Obligation:
37.82/10.59	Q DP problem:
37.82/10.59	The TRS P consists of the following rules:
37.82/10.59	
37.82/10.59	   B(b(a(a(x1)))) -> B(b(a(b(x1))))
37.82/10.59	   B(a(a(b(x1)))) -> B(a(a(a(x1))))
37.82/10.59	
37.82/10.59	The TRS R consists of the following rules:
37.82/10.59	
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	
37.82/10.59	Q is empty.
37.82/10.59	We have to consider all minimal (P,Q,R)-chains.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(7) DependencyGraphProof (EQUIVALENT)
37.82/10.59	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(8)
37.82/10.59	Complex Obligation (AND)
37.82/10.59	
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(9)
37.82/10.59	Obligation:
37.82/10.59	Q DP problem:
37.82/10.59	The TRS P consists of the following rules:
37.82/10.59	
37.82/10.59	   B(b(a(a(x1)))) -> B(b(a(b(x1))))
37.82/10.59	
37.82/10.59	The TRS R consists of the following rules:
37.82/10.59	
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	
37.82/10.59	Q is empty.
37.82/10.59	We have to consider all minimal (P,Q,R)-chains.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(10) QDPOrderProof (EQUIVALENT)
37.82/10.59	We use the reduction pair processor [LPAR04,JAR06].
37.82/10.59	
37.82/10.59	
37.82/10.59	The following pairs can be oriented strictly and are deleted.
37.82/10.59	
37.82/10.59	   B(b(a(a(x1)))) -> B(b(a(b(x1))))
37.82/10.59	The remaining pairs can at least be oriented weakly.
37.82/10.59	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.82/10.59	
37.82/10.59	   <<<
37.82/10.59	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
37.82/10.59	>>>
37.82/10.59	
37.82/10.59	   <<<
37.82/10.59	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, -I], [-I, 0A, 0A], [-I, -I, -I]] 	* 	x_1
37.82/10.59	>>>
37.82/10.59	
37.82/10.59	   <<<
37.82/10.59	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
37.82/10.59	>>>
37.82/10.59	
37.82/10.59	
37.82/10.59	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.82/10.59	
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	
37.82/10.59	
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(11)
37.82/10.59	Obligation:
37.82/10.59	Q DP problem:
37.82/10.59	P is empty.
37.82/10.59	The TRS R consists of the following rules:
37.82/10.59	
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	
37.82/10.59	Q is empty.
37.82/10.59	We have to consider all minimal (P,Q,R)-chains.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(12) PisEmptyProof (EQUIVALENT)
37.82/10.59	The TRS P is empty. Hence, there is no (P,Q,R) chain.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(13)
37.82/10.59	YES
37.82/10.59	
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(14)
37.82/10.59	Obligation:
37.82/10.59	Q DP problem:
37.82/10.59	The TRS P consists of the following rules:
37.82/10.59	
37.82/10.59	   B(a(a(b(x1)))) -> B(a(a(a(x1))))
37.82/10.59	
37.82/10.59	The TRS R consists of the following rules:
37.82/10.59	
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	
37.82/10.59	Q is empty.
37.82/10.59	We have to consider all minimal (P,Q,R)-chains.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(15) QDPOrderProof (EQUIVALENT)
37.82/10.59	We use the reduction pair processor [LPAR04,JAR06].
37.82/10.59	
37.82/10.59	
37.82/10.59	The following pairs can be oriented strictly and are deleted.
37.82/10.59	
37.82/10.59	   B(a(a(b(x1)))) -> B(a(a(a(x1))))
37.82/10.59	The remaining pairs can at least be oriented weakly.
37.82/10.59	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
37.82/10.59	
37.82/10.59	   <<<
37.82/10.59	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
37.82/10.59	>>>
37.82/10.59	
37.82/10.59	   <<<
37.82/10.59	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [1A, -I, -I], [-I, 0A, -I]] 	* 	x_1
37.82/10.59	>>>
37.82/10.59	
37.82/10.59	   <<<
37.82/10.59	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [1A, 1A, -I]] 	* 	x_1
37.82/10.59	>>>
37.82/10.59	
37.82/10.59	
37.82/10.59	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
37.82/10.59	
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	
37.82/10.59	
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(16)
37.82/10.59	Obligation:
37.82/10.59	Q DP problem:
37.82/10.59	P is empty.
37.82/10.59	The TRS R consists of the following rules:
37.82/10.59	
37.82/10.59	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
37.82/10.59	   a(a(b(b(x1)))) -> b(b(a(a(x1))))
37.82/10.59	   b(a(a(b(x1)))) -> b(a(a(a(x1))))
37.82/10.59	
37.82/10.59	Q is empty.
37.82/10.59	We have to consider all minimal (P,Q,R)-chains.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(17) PisEmptyProof (EQUIVALENT)
37.82/10.59	The TRS P is empty. Hence, there is no (P,Q,R) chain.
37.82/10.59	----------------------------------------
37.82/10.59	
37.82/10.59	(18)
37.82/10.59	YES
37.82/10.64	EOF