28.45/8.22	YES
30.45/8.81	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
30.45/8.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.45/8.81	
30.45/8.81	
30.45/8.81	Termination w.r.t. Q of the given QTRS could be proven:
30.45/8.81	
30.45/8.81	(0) QTRS
30.45/8.81	(1) DependencyPairsProof [EQUIVALENT, 14 ms]
30.45/8.81	(2) QDP
30.45/8.81	(3) QDPOrderProof [EQUIVALENT, 552 ms]
30.45/8.81	(4) QDP
30.45/8.81	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
30.45/8.81	(6) TRUE
30.45/8.81	
30.45/8.81	
30.45/8.81	----------------------------------------
30.45/8.81	
30.45/8.81	(0)
30.45/8.81	Obligation:
30.45/8.81	Q restricted rewrite system:
30.45/8.81	The TRS R consists of the following rules:
30.45/8.81	
30.45/8.81	   b(b(x1)) -> b(a(b(x1)))
30.45/8.81	   b(b(a(b(x1)))) -> b(a(b(a(a(b(b(x1)))))))
30.45/8.81	   b(a(b(x1))) -> b(a(a(b(x1))))
30.45/8.81	   b(a(a(b(a(b(x1)))))) -> b(b(x1))
30.45/8.81	
30.45/8.81	Q is empty.
30.45/8.81	
30.45/8.81	----------------------------------------
30.45/8.81	
30.45/8.81	(1) DependencyPairsProof (EQUIVALENT)
30.45/8.81	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
30.45/8.81	----------------------------------------
30.45/8.81	
30.45/8.81	(2)
30.45/8.81	Obligation:
30.45/8.81	Q DP problem:
30.45/8.81	The TRS P consists of the following rules:
30.45/8.81	
30.45/8.81	   B(b(x1)) -> B(a(b(x1)))
30.45/8.81	   B(b(a(b(x1)))) -> B(a(b(a(a(b(b(x1)))))))
30.45/8.81	   B(b(a(b(x1)))) -> B(a(a(b(b(x1)))))
30.45/8.81	   B(b(a(b(x1)))) -> B(b(x1))
30.45/8.81	   B(a(b(x1))) -> B(a(a(b(x1))))
30.45/8.81	   B(a(a(b(a(b(x1)))))) -> B(b(x1))
30.45/8.81	
30.45/8.81	The TRS R consists of the following rules:
30.45/8.81	
30.45/8.81	   b(b(x1)) -> b(a(b(x1)))
30.45/8.81	   b(b(a(b(x1)))) -> b(a(b(a(a(b(b(x1)))))))
30.45/8.81	   b(a(b(x1))) -> b(a(a(b(x1))))
30.45/8.81	   b(a(a(b(a(b(x1)))))) -> b(b(x1))
30.45/8.81	
30.45/8.81	Q is empty.
30.45/8.81	We have to consider all minimal (P,Q,R)-chains.
30.45/8.81	----------------------------------------
30.45/8.81	
30.45/8.81	(3) QDPOrderProof (EQUIVALENT)
30.45/8.81	We use the reduction pair processor [LPAR04,JAR06].
30.45/8.81	
30.45/8.81	
30.45/8.81	The following pairs can be oriented strictly and are deleted.
30.45/8.81	
30.45/8.81	   B(b(a(b(x1)))) -> B(b(x1))
30.45/8.81	   B(a(a(b(a(b(x1)))))) -> B(b(x1))
30.45/8.81	The remaining pairs can at least be oriented weakly.
30.45/8.81	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.45/8.81	
30.45/8.81	   <<<
30.45/8.81	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
30.45/8.81	>>>
30.45/8.81	
30.45/8.81	   <<<
30.45/8.81	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 1A, 0A], [0A, 1A, 0A]] 	* 	x_1
30.45/8.81	>>>
30.45/8.81	
30.45/8.81	   <<<
30.45/8.81	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
30.45/8.81	>>>
30.45/8.81	
30.45/8.81	
30.45/8.81	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.45/8.81	
30.45/8.81	   b(a(b(x1))) -> b(a(a(b(x1))))
30.45/8.81	   b(a(a(b(a(b(x1)))))) -> b(b(x1))
30.45/8.81	   b(b(x1)) -> b(a(b(x1)))
30.45/8.81	   b(b(a(b(x1)))) -> b(a(b(a(a(b(b(x1)))))))
30.45/8.81	
30.45/8.81	
30.45/8.81	----------------------------------------
30.45/8.81	
30.45/8.81	(4)
30.45/8.81	Obligation:
30.45/8.81	Q DP problem:
30.45/8.81	The TRS P consists of the following rules:
30.45/8.81	
30.45/8.81	   B(b(x1)) -> B(a(b(x1)))
30.45/8.81	   B(b(a(b(x1)))) -> B(a(b(a(a(b(b(x1)))))))
30.45/8.81	   B(b(a(b(x1)))) -> B(a(a(b(b(x1)))))
30.45/8.81	   B(a(b(x1))) -> B(a(a(b(x1))))
30.45/8.81	
30.45/8.81	The TRS R consists of the following rules:
30.45/8.81	
30.45/8.81	   b(b(x1)) -> b(a(b(x1)))
30.45/8.81	   b(b(a(b(x1)))) -> b(a(b(a(a(b(b(x1)))))))
30.45/8.81	   b(a(b(x1))) -> b(a(a(b(x1))))
30.45/8.81	   b(a(a(b(a(b(x1)))))) -> b(b(x1))
30.45/8.81	
30.45/8.81	Q is empty.
30.45/8.81	We have to consider all minimal (P,Q,R)-chains.
30.45/8.81	----------------------------------------
30.45/8.81	
30.45/8.81	(5) DependencyGraphProof (EQUIVALENT)
30.45/8.81	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
30.45/8.81	----------------------------------------
30.45/8.81	
30.45/8.81	(6)
30.45/8.81	TRUE
30.70/8.90	EOF