36.71/10.18	YES
36.71/10.19	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
36.71/10.19	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
36.71/10.19	
36.71/10.19	
36.71/10.19	Termination w.r.t. Q of the given QTRS could be proven:
36.71/10.19	
36.71/10.19	(0) QTRS
36.71/10.19	(1) DependencyPairsProof [EQUIVALENT, 9 ms]
36.71/10.19	(2) QDP
36.71/10.19	(3) QDPOrderProof [EQUIVALENT, 217 ms]
36.71/10.19	(4) QDP
36.71/10.19	(5) QDPOrderProof [EQUIVALENT, 27 ms]
36.71/10.19	(6) QDP
36.71/10.19	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
36.71/10.19	(8) QDP
36.71/10.19	(9) QDPOrderProof [EQUIVALENT, 14 ms]
36.71/10.19	(10) QDP
36.71/10.19	(11) PisEmptyProof [EQUIVALENT, 0 ms]
36.71/10.19	(12) YES
36.71/10.19	
36.71/10.19	
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(0)
36.71/10.19	Obligation:
36.71/10.19	Q restricted rewrite system:
36.71/10.19	The TRS R consists of the following rules:
36.71/10.19	
36.71/10.19	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
36.71/10.19	   b(b(b(x1))) -> b(a(a(a(x1))))
36.71/10.19	
36.71/10.19	Q is empty.
36.71/10.19	
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(1) DependencyPairsProof (EQUIVALENT)
36.71/10.19	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(2)
36.71/10.19	Obligation:
36.71/10.19	Q DP problem:
36.71/10.19	The TRS P consists of the following rules:
36.71/10.19	
36.71/10.19	   A(a(b(a(x1)))) -> A(b(b(b(x1))))
36.71/10.19	   A(a(b(a(x1)))) -> B(b(b(x1)))
36.71/10.19	   A(a(b(a(x1)))) -> B(b(x1))
36.71/10.19	   A(a(b(a(x1)))) -> B(x1)
36.71/10.19	   B(b(b(x1))) -> B(a(a(a(x1))))
36.71/10.19	   B(b(b(x1))) -> A(a(a(x1)))
36.71/10.19	   B(b(b(x1))) -> A(a(x1))
36.71/10.19	   B(b(b(x1))) -> A(x1)
36.71/10.19	
36.71/10.19	The TRS R consists of the following rules:
36.71/10.19	
36.71/10.19	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
36.71/10.19	   b(b(b(x1))) -> b(a(a(a(x1))))
36.71/10.19	
36.71/10.19	Q is empty.
36.71/10.19	We have to consider all minimal (P,Q,R)-chains.
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(3) QDPOrderProof (EQUIVALENT)
36.71/10.19	We use the reduction pair processor [LPAR04,JAR06].
36.71/10.19	
36.71/10.19	
36.71/10.19	The following pairs can be oriented strictly and are deleted.
36.71/10.19	
36.71/10.19	   A(a(b(a(x1)))) -> A(b(b(b(x1))))
36.71/10.19	The remaining pairs can at least be oriented weakly.
36.71/10.19	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.71/10.19	
36.71/10.19	   <<<
36.71/10.19	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
36.71/10.19	>>>
36.71/10.19	
36.71/10.19	   <<<
36.71/10.19	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
36.71/10.19	>>>
36.71/10.19	
36.71/10.19	   <<<
36.71/10.19	 POL(b(x_1)) =  	[[-I], [1A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [-I, -I, -I]] 	* 	x_1
36.71/10.19	>>>
36.71/10.19	
36.71/10.19	   <<<
36.71/10.19	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
36.71/10.19	>>>
36.71/10.19	
36.71/10.19	
36.71/10.19	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.71/10.19	
36.71/10.19	   b(b(b(x1))) -> b(a(a(a(x1))))
36.71/10.19	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
36.71/10.19	
36.71/10.19	
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(4)
36.71/10.19	Obligation:
36.71/10.19	Q DP problem:
36.71/10.19	The TRS P consists of the following rules:
36.71/10.19	
36.71/10.19	   A(a(b(a(x1)))) -> B(b(b(x1)))
36.71/10.19	   A(a(b(a(x1)))) -> B(b(x1))
36.71/10.19	   A(a(b(a(x1)))) -> B(x1)
36.71/10.19	   B(b(b(x1))) -> B(a(a(a(x1))))
36.71/10.19	   B(b(b(x1))) -> A(a(a(x1)))
36.71/10.19	   B(b(b(x1))) -> A(a(x1))
36.71/10.19	   B(b(b(x1))) -> A(x1)
36.71/10.19	
36.71/10.19	The TRS R consists of the following rules:
36.71/10.19	
36.71/10.19	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
36.71/10.19	   b(b(b(x1))) -> b(a(a(a(x1))))
36.71/10.19	
36.71/10.19	Q is empty.
36.71/10.19	We have to consider all minimal (P,Q,R)-chains.
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(5) QDPOrderProof (EQUIVALENT)
36.71/10.19	We use the reduction pair processor [LPAR04,JAR06].
36.71/10.19	
36.71/10.19	
36.71/10.19	The following pairs can be oriented strictly and are deleted.
36.71/10.19	
36.71/10.19	   A(a(b(a(x1)))) -> B(b(b(x1)))
36.71/10.19	   A(a(b(a(x1)))) -> B(b(x1))
36.71/10.19	   A(a(b(a(x1)))) -> B(x1)
36.71/10.19	The remaining pairs can at least be oriented weakly.
36.71/10.19	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.71/10.19	
36.71/10.19	   <<<
36.71/10.19	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
36.71/10.19	>>>
36.71/10.19	
36.71/10.19	   <<<
36.71/10.19	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
36.71/10.19	>>>
36.71/10.19	
36.71/10.19	   <<<
36.71/10.19	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
36.71/10.19	>>>
36.71/10.19	
36.71/10.19	   <<<
36.71/10.19	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
36.71/10.19	>>>
36.71/10.19	
36.71/10.19	
36.71/10.19	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.71/10.19	
36.71/10.19	   b(b(b(x1))) -> b(a(a(a(x1))))
36.71/10.19	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
36.71/10.19	
36.71/10.19	
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(6)
36.71/10.19	Obligation:
36.71/10.19	Q DP problem:
36.71/10.19	The TRS P consists of the following rules:
36.71/10.19	
36.71/10.19	   B(b(b(x1))) -> B(a(a(a(x1))))
36.71/10.19	   B(b(b(x1))) -> A(a(a(x1)))
36.71/10.19	   B(b(b(x1))) -> A(a(x1))
36.71/10.19	   B(b(b(x1))) -> A(x1)
36.71/10.19	
36.71/10.19	The TRS R consists of the following rules:
36.71/10.19	
36.71/10.19	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
36.71/10.19	   b(b(b(x1))) -> b(a(a(a(x1))))
36.71/10.19	
36.71/10.19	Q is empty.
36.71/10.19	We have to consider all minimal (P,Q,R)-chains.
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(7) DependencyGraphProof (EQUIVALENT)
36.71/10.19	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(8)
36.71/10.19	Obligation:
36.71/10.19	Q DP problem:
36.71/10.19	The TRS P consists of the following rules:
36.71/10.19	
36.71/10.19	   B(b(b(x1))) -> B(a(a(a(x1))))
36.71/10.19	
36.71/10.19	The TRS R consists of the following rules:
36.71/10.19	
36.71/10.19	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
36.71/10.19	   b(b(b(x1))) -> b(a(a(a(x1))))
36.71/10.19	
36.71/10.19	Q is empty.
36.71/10.19	We have to consider all minimal (P,Q,R)-chains.
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(9) QDPOrderProof (EQUIVALENT)
36.71/10.19	We use the reduction pair processor [LPAR04,JAR06].
36.71/10.19	
36.71/10.19	
36.71/10.19	The following pairs can be oriented strictly and are deleted.
36.71/10.19	
36.71/10.19	   B(b(b(x1))) -> B(a(a(a(x1))))
36.71/10.19	The remaining pairs can at least be oriented weakly.
36.71/10.19	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
36.71/10.19	
36.71/10.19	POL( B_1(x_1) ) = x_1 + 2
36.71/10.19	POL( a_1(x_1) ) = 2
36.71/10.19	POL( b_1(x_1) ) = x_1 + 2
36.71/10.19	
36.71/10.19	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.71/10.19	
36.71/10.19	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
36.71/10.19	
36.71/10.19	
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(10)
36.71/10.19	Obligation:
36.71/10.19	Q DP problem:
36.71/10.19	P is empty.
36.71/10.19	The TRS R consists of the following rules:
36.71/10.19	
36.71/10.19	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
36.71/10.19	   b(b(b(x1))) -> b(a(a(a(x1))))
36.71/10.19	
36.71/10.19	Q is empty.
36.71/10.19	We have to consider all minimal (P,Q,R)-chains.
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(11) PisEmptyProof (EQUIVALENT)
36.71/10.19	The TRS P is empty. Hence, there is no (P,Q,R) chain.
36.71/10.19	----------------------------------------
36.71/10.19	
36.71/10.19	(12)
36.71/10.19	YES
37.16/10.34	EOF