26.76/7.74	YES
27.28/7.82	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.28/7.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.28/7.82	
27.28/7.82	
27.28/7.82	Termination w.r.t. Q of the given QTRS could be proven:
27.28/7.82	
27.28/7.82	(0) QTRS
27.28/7.82	(1) DependencyPairsProof [EQUIVALENT, 25 ms]
27.28/7.82	(2) QDP
27.28/7.82	(3) MRRProof [EQUIVALENT, 71 ms]
27.28/7.82	(4) QDP
27.28/7.82	(5) QDPOrderProof [EQUIVALENT, 55 ms]
27.28/7.82	(6) QDP
27.28/7.82	(7) QDPOrderProof [EQUIVALENT, 389 ms]
27.28/7.82	(8) QDP
27.28/7.82	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
27.28/7.82	(10) TRUE
27.28/7.82	
27.28/7.82	
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(0)
27.28/7.82	Obligation:
27.28/7.82	Q restricted rewrite system:
27.28/7.82	The TRS R consists of the following rules:
27.28/7.82	
27.28/7.82	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
27.28/7.82	   b(a(b(x1))) -> a(b(b(b(x1))))
27.28/7.82	
27.28/7.82	Q is empty.
27.28/7.82	
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(1) DependencyPairsProof (EQUIVALENT)
27.28/7.82	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(2)
27.28/7.82	Obligation:
27.28/7.82	Q DP problem:
27.28/7.82	The TRS P consists of the following rules:
27.28/7.82	
27.28/7.82	   A(b(b(a(x1)))) -> B(a(a(b(x1))))
27.28/7.82	   A(b(b(a(x1)))) -> A(a(b(x1)))
27.28/7.82	   A(b(b(a(x1)))) -> A(b(x1))
27.28/7.82	   A(b(b(a(x1)))) -> B(x1)
27.28/7.82	   B(a(b(x1))) -> A(b(b(b(x1))))
27.28/7.82	   B(a(b(x1))) -> B(b(b(x1)))
27.28/7.82	   B(a(b(x1))) -> B(b(x1))
27.28/7.82	
27.28/7.82	The TRS R consists of the following rules:
27.28/7.82	
27.28/7.82	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
27.28/7.82	   b(a(b(x1))) -> a(b(b(b(x1))))
27.28/7.82	
27.28/7.82	Q is empty.
27.28/7.82	We have to consider all minimal (P,Q,R)-chains.
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(3) MRRProof (EQUIVALENT)
27.28/7.82	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.
27.28/7.82	
27.28/7.82	Strictly oriented dependency pairs:
27.28/7.82	
27.28/7.82	   A(b(b(a(x1)))) -> A(b(x1))
27.28/7.82	   A(b(b(a(x1)))) -> B(x1)
27.28/7.82	   B(a(b(x1))) -> B(b(b(x1)))
27.28/7.82	   B(a(b(x1))) -> B(b(x1))
27.28/7.82	
27.28/7.82	
27.28/7.82	Used ordering: Polynomial interpretation [POLO]:
27.28/7.82	
27.28/7.82	   POL(A(x_1)) = 2 + 2*x_1
27.28/7.82	   POL(B(x_1)) = 2*x_1
27.28/7.82	   POL(a(x_1)) = 1 + x_1
27.28/7.82	   POL(b(x_1)) = x_1
27.28/7.82	
27.28/7.82	
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(4)
27.28/7.82	Obligation:
27.28/7.82	Q DP problem:
27.28/7.82	The TRS P consists of the following rules:
27.28/7.82	
27.28/7.82	   A(b(b(a(x1)))) -> B(a(a(b(x1))))
27.28/7.82	   A(b(b(a(x1)))) -> A(a(b(x1)))
27.28/7.82	   B(a(b(x1))) -> A(b(b(b(x1))))
27.28/7.82	
27.28/7.82	The TRS R consists of the following rules:
27.28/7.82	
27.28/7.82	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
27.28/7.82	   b(a(b(x1))) -> a(b(b(b(x1))))
27.28/7.82	
27.28/7.82	Q is empty.
27.28/7.82	We have to consider all minimal (P,Q,R)-chains.
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(5) QDPOrderProof (EQUIVALENT)
27.28/7.82	We use the reduction pair processor [LPAR04,JAR06].
27.28/7.82	
27.28/7.82	
27.28/7.82	The following pairs can be oriented strictly and are deleted.
27.28/7.82	
27.28/7.82	   A(b(b(a(x1)))) -> A(a(b(x1)))
27.28/7.82	The remaining pairs can at least be oriented weakly.
27.28/7.82	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.28/7.82	
27.28/7.82	   <<<
27.28/7.82	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
27.28/7.82	>>>
27.28/7.82	
27.28/7.82	   <<<
27.28/7.82	 POL(b(x_1)) =  	[[1A], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, -I, 0A]] 	* 	x_1
27.28/7.82	>>>
27.28/7.82	
27.28/7.82	   <<<
27.28/7.82	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [0A, 0A, -I], [-I, -I, -I]] 	* 	x_1
27.28/7.82	>>>
27.28/7.82	
27.28/7.82	   <<<
27.28/7.82	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
27.28/7.82	>>>
27.28/7.82	
27.28/7.82	
27.28/7.82	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.28/7.82	
27.28/7.82	   b(a(b(x1))) -> a(b(b(b(x1))))
27.28/7.82	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
27.28/7.82	
27.28/7.82	
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(6)
27.28/7.82	Obligation:
27.28/7.82	Q DP problem:
27.28/7.82	The TRS P consists of the following rules:
27.28/7.82	
27.28/7.82	   A(b(b(a(x1)))) -> B(a(a(b(x1))))
27.28/7.82	   B(a(b(x1))) -> A(b(b(b(x1))))
27.28/7.82	
27.28/7.82	The TRS R consists of the following rules:
27.28/7.82	
27.28/7.82	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
27.28/7.82	   b(a(b(x1))) -> a(b(b(b(x1))))
27.28/7.82	
27.28/7.82	Q is empty.
27.28/7.82	We have to consider all minimal (P,Q,R)-chains.
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(7) QDPOrderProof (EQUIVALENT)
27.28/7.82	We use the reduction pair processor [LPAR04,JAR06].
27.28/7.82	
27.28/7.82	
27.28/7.82	The following pairs can be oriented strictly and are deleted.
27.28/7.82	
27.28/7.82	   B(a(b(x1))) -> A(b(b(b(x1))))
27.28/7.82	The remaining pairs can at least be oriented weakly.
27.28/7.82	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.28/7.82	
27.28/7.82	   <<<
27.28/7.82	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
27.28/7.82	>>>
27.28/7.82	
27.28/7.82	   <<<
27.28/7.82	 POL(b(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
27.28/7.82	>>>
27.28/7.82	
27.28/7.82	   <<<
27.28/7.82	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, -I], [-I, -I, -I], [1A, 0A, -I]] 	* 	x_1
27.28/7.82	>>>
27.28/7.82	
27.28/7.82	   <<<
27.28/7.82	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
27.28/7.82	>>>
27.28/7.82	
27.28/7.82	
27.28/7.82	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.28/7.82	
27.28/7.82	   b(a(b(x1))) -> a(b(b(b(x1))))
27.28/7.82	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
27.28/7.82	
27.28/7.82	
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(8)
27.28/7.82	Obligation:
27.28/7.82	Q DP problem:
27.28/7.82	The TRS P consists of the following rules:
27.28/7.82	
27.28/7.82	   A(b(b(a(x1)))) -> B(a(a(b(x1))))
27.28/7.82	
27.28/7.82	The TRS R consists of the following rules:
27.28/7.82	
27.28/7.82	   a(b(b(a(x1)))) -> b(a(a(b(x1))))
27.28/7.82	   b(a(b(x1))) -> a(b(b(b(x1))))
27.28/7.82	
27.28/7.82	Q is empty.
27.28/7.82	We have to consider all minimal (P,Q,R)-chains.
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(9) DependencyGraphProof (EQUIVALENT)
27.28/7.82	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
27.28/7.82	----------------------------------------
27.28/7.82	
27.28/7.82	(10)
27.28/7.82	TRUE
27.45/7.88	EOF