21.38/6.39	YES
21.38/6.40	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
21.38/6.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.38/6.40	
21.38/6.40	
21.38/6.40	Termination w.r.t. Q of the given QTRS could be proven:
21.38/6.40	
21.38/6.40	(0) QTRS
21.38/6.40	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
21.38/6.40	(2) QDP
21.38/6.40	(3) QDPOrderProof [EQUIVALENT, 27 ms]
21.38/6.40	(4) QDP
21.38/6.40	(5) QDPOrderProof [EQUIVALENT, 127 ms]
21.38/6.40	(6) QDP
21.38/6.40	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
21.38/6.40	(8) TRUE
21.38/6.40	
21.38/6.40	
21.38/6.40	----------------------------------------
21.38/6.40	
21.38/6.40	(0)
21.38/6.40	Obligation:
21.38/6.40	Q restricted rewrite system:
21.38/6.40	The TRS R consists of the following rules:
21.38/6.40	
21.38/6.40	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
21.38/6.40	   a(b(a(a(x1)))) -> a(a(a(a(x1))))
21.38/6.40	   a(a(a(b(x1)))) -> b(b(b(b(x1))))
21.38/6.40	
21.38/6.40	Q is empty.
21.38/6.40	
21.38/6.40	----------------------------------------
21.38/6.40	
21.38/6.40	(1) DependencyPairsProof (EQUIVALENT)
21.38/6.40	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
21.38/6.40	----------------------------------------
21.38/6.40	
21.38/6.40	(2)
21.38/6.40	Obligation:
21.38/6.40	Q DP problem:
21.38/6.40	The TRS P consists of the following rules:
21.38/6.40	
21.38/6.40	   A(a(b(a(x1)))) -> A(b(a(b(x1))))
21.38/6.40	   A(a(b(a(x1)))) -> A(b(x1))
21.38/6.40	   A(b(a(a(x1)))) -> A(a(a(a(x1))))
21.38/6.40	   A(b(a(a(x1)))) -> A(a(a(x1)))
21.38/6.40	
21.38/6.40	The TRS R consists of the following rules:
21.38/6.40	
21.38/6.40	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
21.38/6.40	   a(b(a(a(x1)))) -> a(a(a(a(x1))))
21.38/6.40	   a(a(a(b(x1)))) -> b(b(b(b(x1))))
21.38/6.40	
21.38/6.40	Q is empty.
21.38/6.40	We have to consider all minimal (P,Q,R)-chains.
21.38/6.40	----------------------------------------
21.38/6.40	
21.38/6.40	(3) QDPOrderProof (EQUIVALENT)
21.38/6.40	We use the reduction pair processor [LPAR04,JAR06].
21.38/6.40	
21.38/6.40	
21.38/6.40	The following pairs can be oriented strictly and are deleted.
21.38/6.40	
21.38/6.40	   A(a(b(a(x1)))) -> A(b(x1))
21.38/6.40	   A(b(a(a(x1)))) -> A(a(a(x1)))
21.38/6.40	The remaining pairs can at least be oriented weakly.
21.38/6.40	Used ordering:  Polynomial interpretation [POLO]:
21.38/6.40	
21.38/6.40	   POL(A(x_1)) = x_1
21.38/6.40	   POL(a(x_1)) = 1 + x_1
21.38/6.40	   POL(b(x_1)) = 1 + x_1
21.38/6.40	
21.38/6.40	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.38/6.40	
21.38/6.40	   a(b(a(a(x1)))) -> a(a(a(a(x1))))
21.38/6.40	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
21.38/6.40	   a(a(a(b(x1)))) -> b(b(b(b(x1))))
21.38/6.40	
21.38/6.40	
21.38/6.40	----------------------------------------
21.38/6.40	
21.38/6.40	(4)
21.38/6.40	Obligation:
21.38/6.40	Q DP problem:
21.38/6.40	The TRS P consists of the following rules:
21.38/6.40	
21.38/6.40	   A(a(b(a(x1)))) -> A(b(a(b(x1))))
21.38/6.40	   A(b(a(a(x1)))) -> A(a(a(a(x1))))
21.38/6.40	
21.38/6.40	The TRS R consists of the following rules:
21.38/6.40	
21.38/6.40	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
21.38/6.40	   a(b(a(a(x1)))) -> a(a(a(a(x1))))
21.38/6.40	   a(a(a(b(x1)))) -> b(b(b(b(x1))))
21.38/6.40	
21.38/6.40	Q is empty.
21.38/6.40	We have to consider all minimal (P,Q,R)-chains.
21.38/6.40	----------------------------------------
21.38/6.40	
21.38/6.40	(5) QDPOrderProof (EQUIVALENT)
21.38/6.40	We use the reduction pair processor [LPAR04,JAR06].
21.38/6.40	
21.38/6.40	
21.38/6.40	The following pairs can be oriented strictly and are deleted.
21.38/6.40	
21.38/6.40	   A(b(a(a(x1)))) -> A(a(a(a(x1))))
21.38/6.40	The remaining pairs can at least be oriented weakly.
21.38/6.40	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
21.38/6.40	
21.38/6.40	   <<<
21.38/6.40	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
21.38/6.40	>>>
21.38/6.40	
21.38/6.40	   <<<
21.38/6.40	 POL(a(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
21.38/6.40	>>>
21.38/6.40	
21.38/6.40	   <<<
21.38/6.40	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[-I, 1A, 0A], [-I, -I, -I], [-I, 0A, -I]] 	* 	x_1
21.38/6.40	>>>
21.38/6.40	
21.38/6.40	
21.38/6.40	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.38/6.40	
21.38/6.40	   a(b(a(a(x1)))) -> a(a(a(a(x1))))
21.38/6.40	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
21.38/6.40	   a(a(a(b(x1)))) -> b(b(b(b(x1))))
21.38/6.40	
21.38/6.40	
21.38/6.40	----------------------------------------
21.38/6.40	
21.38/6.40	(6)
21.38/6.40	Obligation:
21.38/6.40	Q DP problem:
21.38/6.40	The TRS P consists of the following rules:
21.38/6.40	
21.38/6.40	   A(a(b(a(x1)))) -> A(b(a(b(x1))))
21.38/6.40	
21.38/6.40	The TRS R consists of the following rules:
21.38/6.40	
21.38/6.40	   a(a(b(a(x1)))) -> a(b(a(b(x1))))
21.38/6.40	   a(b(a(a(x1)))) -> a(a(a(a(x1))))
21.38/6.40	   a(a(a(b(x1)))) -> b(b(b(b(x1))))
21.38/6.40	
21.38/6.40	Q is empty.
21.38/6.40	We have to consider all minimal (P,Q,R)-chains.
21.38/6.40	----------------------------------------
21.38/6.40	
21.38/6.40	(7) DependencyGraphProof (EQUIVALENT)
21.38/6.40	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
21.38/6.40	----------------------------------------
21.38/6.40	
21.38/6.40	(8)
21.38/6.40	TRUE
21.70/6.50	EOF