18.05/5.39	YES
18.05/5.40	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
18.05/5.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
18.05/5.40	
18.05/5.40	
18.05/5.40	Termination w.r.t. Q of the given QTRS could be proven:
18.05/5.40	
18.05/5.40	(0) QTRS
18.05/5.40	(1) QTRS Reverse [EQUIVALENT, 0 ms]
18.05/5.40	(2) QTRS
18.05/5.40	(3) DependencyPairsProof [EQUIVALENT, 1 ms]
18.05/5.40	(4) QDP
18.05/5.40	(5) QDPOrderProof [EQUIVALENT, 151 ms]
18.05/5.40	(6) QDP
18.05/5.40	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
18.05/5.40	(8) TRUE
18.05/5.40	
18.05/5.40	
18.05/5.40	----------------------------------------
18.05/5.40	
18.05/5.40	(0)
18.05/5.40	Obligation:
18.05/5.40	Q restricted rewrite system:
18.05/5.40	The TRS R consists of the following rules:
18.05/5.40	
18.05/5.40	   a(b(a(x1))) -> b(b(b(a(x1))))
18.05/5.40	   b(b(b(a(x1)))) -> a(a(a(b(x1))))
18.05/5.40	
18.05/5.40	Q is empty.
18.05/5.40	
18.05/5.40	----------------------------------------
18.05/5.40	
18.05/5.40	(1) QTRS Reverse (EQUIVALENT)
18.05/5.40	We applied the QTRS Reverse Processor [REVERSE].
18.05/5.40	----------------------------------------
18.05/5.40	
18.05/5.40	(2)
18.05/5.40	Obligation:
18.05/5.40	Q restricted rewrite system:
18.05/5.40	The TRS R consists of the following rules:
18.05/5.40	
18.05/5.40	   a(b(a(x1))) -> a(b(b(b(x1))))
18.05/5.40	   a(b(b(b(x1)))) -> b(a(a(a(x1))))
18.05/5.40	
18.05/5.40	Q is empty.
18.05/5.40	
18.05/5.40	----------------------------------------
18.05/5.40	
18.05/5.40	(3) DependencyPairsProof (EQUIVALENT)
18.05/5.40	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
18.05/5.40	----------------------------------------
18.05/5.40	
18.05/5.40	(4)
18.05/5.40	Obligation:
18.05/5.40	Q DP problem:
18.05/5.40	The TRS P consists of the following rules:
18.05/5.40	
18.05/5.40	   A(b(a(x1))) -> A(b(b(b(x1))))
18.05/5.40	   A(b(b(b(x1)))) -> A(a(a(x1)))
18.05/5.40	   A(b(b(b(x1)))) -> A(a(x1))
18.05/5.40	   A(b(b(b(x1)))) -> A(x1)
18.05/5.40	
18.05/5.40	The TRS R consists of the following rules:
18.05/5.40	
18.05/5.40	   a(b(a(x1))) -> a(b(b(b(x1))))
18.05/5.40	   a(b(b(b(x1)))) -> b(a(a(a(x1))))
18.05/5.40	
18.05/5.40	Q is empty.
18.05/5.40	We have to consider all minimal (P,Q,R)-chains.
18.05/5.40	----------------------------------------
18.05/5.40	
18.05/5.40	(5) QDPOrderProof (EQUIVALENT)
18.05/5.40	We use the reduction pair processor [LPAR04,JAR06].
18.05/5.40	
18.05/5.40	
18.05/5.40	The following pairs can be oriented strictly and are deleted.
18.05/5.40	
18.05/5.40	   A(b(b(b(x1)))) -> A(a(a(x1)))
18.05/5.40	   A(b(b(b(x1)))) -> A(a(x1))
18.05/5.40	   A(b(b(b(x1)))) -> A(x1)
18.05/5.40	The remaining pairs can at least be oriented weakly.
18.05/5.40	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
18.05/5.40	
18.05/5.40	   <<<
18.05/5.40	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
18.05/5.40	>>>
18.05/5.40	
18.05/5.40	   <<<
18.05/5.40	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, -I, -I], [-I, 1A, -I]] 	* 	x_1
18.05/5.40	>>>
18.05/5.40	
18.05/5.40	   <<<
18.05/5.40	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
18.05/5.40	>>>
18.05/5.40	
18.05/5.40	
18.05/5.40	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
18.05/5.40	
18.05/5.40	   a(b(a(x1))) -> a(b(b(b(x1))))
18.05/5.40	   a(b(b(b(x1)))) -> b(a(a(a(x1))))
18.05/5.40	
18.05/5.40	
18.05/5.40	----------------------------------------
18.05/5.40	
18.05/5.40	(6)
18.05/5.40	Obligation:
18.05/5.40	Q DP problem:
18.05/5.40	The TRS P consists of the following rules:
18.05/5.40	
18.05/5.40	   A(b(a(x1))) -> A(b(b(b(x1))))
18.05/5.40	
18.05/5.40	The TRS R consists of the following rules:
18.05/5.40	
18.05/5.40	   a(b(a(x1))) -> a(b(b(b(x1))))
18.05/5.40	   a(b(b(b(x1)))) -> b(a(a(a(x1))))
18.05/5.40	
18.05/5.40	Q is empty.
18.05/5.40	We have to consider all minimal (P,Q,R)-chains.
18.05/5.40	----------------------------------------
18.05/5.40	
18.05/5.40	(7) DependencyGraphProof (EQUIVALENT)
18.05/5.40	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
18.05/5.40	----------------------------------------
18.05/5.40	
18.05/5.40	(8)
18.05/5.40	TRUE
18.35/5.48	EOF