63.22/17.01	YES
63.77/17.14	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
63.77/17.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
63.77/17.14	
63.77/17.14	
63.77/17.14	Termination w.r.t. Q of the given QTRS could be proven:
63.77/17.14	
63.77/17.14	(0) QTRS
63.77/17.14	(1) DependencyPairsProof [EQUIVALENT, 21 ms]
63.77/17.14	(2) QDP
63.77/17.14	(3) DependencyGraphProof [EQUIVALENT, 6 ms]
63.77/17.14	(4) QDP
63.77/17.14	(5) QDPOrderProof [EQUIVALENT, 2246 ms]
63.77/17.14	(6) QDP
63.77/17.14	(7) QDPOrderProof [EQUIVALENT, 85 ms]
63.77/17.14	(8) QDP
63.77/17.14	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
63.77/17.14	(10) TRUE
63.77/17.14	
63.77/17.14	
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(0)
63.77/17.14	Obligation:
63.77/17.14	Q restricted rewrite system:
63.77/17.14	The TRS R consists of the following rules:
63.77/17.14	
63.77/17.14	   b(b(b(a(b(x1))))) -> b(a(b(b(a(b(x1))))))
63.77/17.14	   b(a(b(b(x1)))) -> b(b(a(b(a(b(x1))))))
63.77/17.14	   b(a(b(a(a(b(b(x1))))))) -> b(a(a(b(a(a(b(b(b(a(b(x1)))))))))))
63.77/17.14	
63.77/17.14	Q is empty.
63.77/17.14	
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(1) DependencyPairsProof (EQUIVALENT)
63.77/17.14	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(2)
63.77/17.14	Obligation:
63.77/17.14	Q DP problem:
63.77/17.14	The TRS P consists of the following rules:
63.77/17.14	
63.77/17.14	   B(b(b(a(b(x1))))) -> B(a(b(b(a(b(x1))))))
63.77/17.14	   B(a(b(b(x1)))) -> B(b(a(b(a(b(x1))))))
63.77/17.14	   B(a(b(b(x1)))) -> B(a(b(a(b(x1)))))
63.77/17.14	   B(a(b(b(x1)))) -> B(a(b(x1)))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(a(a(b(a(a(b(b(b(a(b(x1)))))))))))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(a(a(b(b(b(a(b(x1))))))))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(b(b(a(b(x1)))))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(b(a(b(x1))))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(a(b(x1)))
63.77/17.14	
63.77/17.14	The TRS R consists of the following rules:
63.77/17.14	
63.77/17.14	   b(b(b(a(b(x1))))) -> b(a(b(b(a(b(x1))))))
63.77/17.14	   b(a(b(b(x1)))) -> b(b(a(b(a(b(x1))))))
63.77/17.14	   b(a(b(a(a(b(b(x1))))))) -> b(a(a(b(a(a(b(b(b(a(b(x1)))))))))))
63.77/17.14	
63.77/17.14	Q is empty.
63.77/17.14	We have to consider all minimal (P,Q,R)-chains.
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(3) DependencyGraphProof (EQUIVALENT)
63.77/17.14	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(4)
63.77/17.14	Obligation:
63.77/17.14	Q DP problem:
63.77/17.14	The TRS P consists of the following rules:
63.77/17.14	
63.77/17.14	   B(a(b(b(x1)))) -> B(b(a(b(a(b(x1))))))
63.77/17.14	   B(b(b(a(b(x1))))) -> B(a(b(b(a(b(x1))))))
63.77/17.14	   B(a(b(b(x1)))) -> B(a(b(a(b(x1)))))
63.77/17.14	   B(a(b(b(x1)))) -> B(a(b(x1)))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(b(b(a(b(x1)))))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(b(a(b(x1))))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(a(b(x1)))
63.77/17.14	
63.77/17.14	The TRS R consists of the following rules:
63.77/17.14	
63.77/17.14	   b(b(b(a(b(x1))))) -> b(a(b(b(a(b(x1))))))
63.77/17.14	   b(a(b(b(x1)))) -> b(b(a(b(a(b(x1))))))
63.77/17.14	   b(a(b(a(a(b(b(x1))))))) -> b(a(a(b(a(a(b(b(b(a(b(x1)))))))))))
63.77/17.14	
63.77/17.14	Q is empty.
63.77/17.14	We have to consider all minimal (P,Q,R)-chains.
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(5) QDPOrderProof (EQUIVALENT)
63.77/17.14	We use the reduction pair processor [LPAR04,JAR06].
63.77/17.14	
63.77/17.14	
63.77/17.14	The following pairs can be oriented strictly and are deleted.
63.77/17.14	
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(b(b(a(b(x1)))))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(b(a(b(x1))))
63.77/17.14	   B(a(b(a(a(b(b(x1))))))) -> B(a(b(x1)))
63.77/17.14	The remaining pairs can at least be oriented weakly.
63.77/17.14	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
63.77/17.14	
63.77/17.14	   <<<
63.77/17.14	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
63.77/17.14	>>>
63.77/17.14	
63.77/17.14	   <<<
63.77/17.14	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
63.77/17.14	>>>
63.77/17.14	
63.77/17.14	   <<<
63.77/17.14	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] 	* 	x_1
63.77/17.14	>>>
63.77/17.14	
63.77/17.14	
63.77/17.14	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
63.77/17.14	
63.77/17.14	   b(a(b(b(x1)))) -> b(b(a(b(a(b(x1))))))
63.77/17.14	   b(b(b(a(b(x1))))) -> b(a(b(b(a(b(x1))))))
63.77/17.14	   b(a(b(a(a(b(b(x1))))))) -> b(a(a(b(a(a(b(b(b(a(b(x1)))))))))))
63.77/17.14	
63.77/17.14	
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(6)
63.77/17.14	Obligation:
63.77/17.14	Q DP problem:
63.77/17.14	The TRS P consists of the following rules:
63.77/17.14	
63.77/17.14	   B(a(b(b(x1)))) -> B(b(a(b(a(b(x1))))))
63.77/17.14	   B(b(b(a(b(x1))))) -> B(a(b(b(a(b(x1))))))
63.77/17.14	   B(a(b(b(x1)))) -> B(a(b(a(b(x1)))))
63.77/17.14	   B(a(b(b(x1)))) -> B(a(b(x1)))
63.77/17.14	
63.77/17.14	The TRS R consists of the following rules:
63.77/17.14	
63.77/17.14	   b(b(b(a(b(x1))))) -> b(a(b(b(a(b(x1))))))
63.77/17.14	   b(a(b(b(x1)))) -> b(b(a(b(a(b(x1))))))
63.77/17.14	   b(a(b(a(a(b(b(x1))))))) -> b(a(a(b(a(a(b(b(b(a(b(x1)))))))))))
63.77/17.14	
63.77/17.14	Q is empty.
63.77/17.14	We have to consider all minimal (P,Q,R)-chains.
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(7) QDPOrderProof (EQUIVALENT)
63.77/17.14	We use the reduction pair processor [LPAR04,JAR06].
63.77/17.14	
63.77/17.14	
63.77/17.14	The following pairs can be oriented strictly and are deleted.
63.77/17.14	
63.77/17.14	   B(b(b(a(b(x1))))) -> B(a(b(b(a(b(x1))))))
63.77/17.14	   B(a(b(b(x1)))) -> B(a(b(a(b(x1)))))
63.77/17.14	   B(a(b(b(x1)))) -> B(a(b(x1)))
63.77/17.14	The remaining pairs can at least be oriented weakly.
63.77/17.14	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
63.77/17.14	
63.77/17.14	POL( B_1(x_1) ) = max{0, 2x_1 - 2}
63.77/17.14	POL( b_1(x_1) ) = x_1 + 2
63.77/17.14	POL( a_1(x_1) ) = max{0, x_1 - 2}
63.77/17.14	
63.77/17.14	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
63.77/17.14	
63.77/17.14	   b(a(b(b(x1)))) -> b(b(a(b(a(b(x1))))))
63.77/17.14	   b(b(b(a(b(x1))))) -> b(a(b(b(a(b(x1))))))
63.77/17.14	   b(a(b(a(a(b(b(x1))))))) -> b(a(a(b(a(a(b(b(b(a(b(x1)))))))))))
63.77/17.14	
63.77/17.14	
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(8)
63.77/17.14	Obligation:
63.77/17.14	Q DP problem:
63.77/17.14	The TRS P consists of the following rules:
63.77/17.14	
63.77/17.14	   B(a(b(b(x1)))) -> B(b(a(b(a(b(x1))))))
63.77/17.14	
63.77/17.14	The TRS R consists of the following rules:
63.77/17.14	
63.77/17.14	   b(b(b(a(b(x1))))) -> b(a(b(b(a(b(x1))))))
63.77/17.14	   b(a(b(b(x1)))) -> b(b(a(b(a(b(x1))))))
63.77/17.14	   b(a(b(a(a(b(b(x1))))))) -> b(a(a(b(a(a(b(b(b(a(b(x1)))))))))))
63.77/17.14	
63.77/17.14	Q is empty.
63.77/17.14	We have to consider all minimal (P,Q,R)-chains.
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(9) DependencyGraphProof (EQUIVALENT)
63.77/17.14	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
63.77/17.14	----------------------------------------
63.77/17.14	
63.77/17.14	(10)
63.77/17.14	TRUE
64.25/17.28	EOF