32.65/9.30	YES
32.65/9.31	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
32.65/9.31	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.65/9.31	
32.65/9.31	
32.65/9.31	Termination w.r.t. Q of the given QTRS could be proven:
32.65/9.31	
32.65/9.31	(0) QTRS
32.65/9.31	(1) DependencyPairsProof [EQUIVALENT, 10 ms]
32.65/9.31	(2) QDP
32.65/9.31	(3) QDPOrderProof [EQUIVALENT, 77 ms]
32.65/9.31	(4) QDP
32.65/9.31	(5) QDPOrderProof [EQUIVALENT, 0 ms]
32.65/9.31	(6) QDP
32.65/9.31	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
32.65/9.31	(8) AND
32.65/9.31	    (9) QDP
32.65/9.31	        (10) QDPOrderProof [EQUIVALENT, 0 ms]
32.65/9.31	        (11) QDP
32.65/9.31	        (12) PisEmptyProof [EQUIVALENT, 0 ms]
32.65/9.31	        (13) YES
32.65/9.31	    (14) QDP
32.65/9.31	        (15) QDPOrderProof [EQUIVALENT, 0 ms]
32.65/9.31	        (16) QDP
32.65/9.31	        (17) PisEmptyProof [EQUIVALENT, 0 ms]
32.65/9.31	        (18) YES
32.65/9.31	
32.65/9.31	
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(0)
32.65/9.31	Obligation:
32.65/9.31	Q restricted rewrite system:
32.65/9.31	The TRS R consists of the following rules:
32.65/9.31	
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	
32.65/9.31	Q is empty.
32.65/9.31	
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(1) DependencyPairsProof (EQUIVALENT)
32.65/9.31	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(2)
32.65/9.31	Obligation:
32.65/9.31	Q DP problem:
32.65/9.31	The TRS P consists of the following rules:
32.65/9.31	
32.65/9.31	   A(a(b(a(x1)))) -> A(b(b(x1)))
32.65/9.31	   A(a(b(a(x1)))) -> B(b(x1))
32.65/9.31	   A(a(b(a(x1)))) -> B(x1)
32.65/9.31	   B(b(x1)) -> B(a(a(a(x1))))
32.65/9.31	   B(b(x1)) -> A(a(a(x1)))
32.65/9.31	   B(b(x1)) -> A(a(x1))
32.65/9.31	   B(b(x1)) -> A(x1)
32.65/9.31	
32.65/9.31	The TRS R consists of the following rules:
32.65/9.31	
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	
32.65/9.31	Q is empty.
32.65/9.31	We have to consider all minimal (P,Q,R)-chains.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(3) QDPOrderProof (EQUIVALENT)
32.65/9.31	We use the reduction pair processor [LPAR04,JAR06].
32.65/9.31	
32.65/9.31	
32.65/9.31	The following pairs can be oriented strictly and are deleted.
32.65/9.31	
32.65/9.31	   A(a(b(a(x1)))) -> B(x1)
32.65/9.31	The remaining pairs can at least be oriented weakly.
32.65/9.31	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.65/9.31	
32.65/9.31	   <<<
32.65/9.31	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
32.65/9.31	>>>
32.65/9.31	
32.65/9.31	   <<<
32.65/9.31	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
32.65/9.31	>>>
32.65/9.31	
32.65/9.31	   <<<
32.65/9.31	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 1A, 0A], [0A, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
32.65/9.31	>>>
32.65/9.31	
32.65/9.31	   <<<
32.65/9.31	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
32.65/9.31	>>>
32.65/9.31	
32.65/9.31	
32.65/9.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.65/9.31	
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	
32.65/9.31	
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(4)
32.65/9.31	Obligation:
32.65/9.31	Q DP problem:
32.65/9.31	The TRS P consists of the following rules:
32.65/9.31	
32.65/9.31	   A(a(b(a(x1)))) -> A(b(b(x1)))
32.65/9.31	   A(a(b(a(x1)))) -> B(b(x1))
32.65/9.31	   B(b(x1)) -> B(a(a(a(x1))))
32.65/9.31	   B(b(x1)) -> A(a(a(x1)))
32.65/9.31	   B(b(x1)) -> A(a(x1))
32.65/9.31	   B(b(x1)) -> A(x1)
32.65/9.31	
32.65/9.31	The TRS R consists of the following rules:
32.65/9.31	
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	
32.65/9.31	Q is empty.
32.65/9.31	We have to consider all minimal (P,Q,R)-chains.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(5) QDPOrderProof (EQUIVALENT)
32.65/9.31	We use the reduction pair processor [LPAR04,JAR06].
32.65/9.31	
32.65/9.31	
32.65/9.31	The following pairs can be oriented strictly and are deleted.
32.65/9.31	
32.65/9.31	   A(a(b(a(x1)))) -> B(b(x1))
32.65/9.31	The remaining pairs can at least be oriented weakly.
32.65/9.31	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.65/9.31	
32.65/9.31	   <<<
32.65/9.31	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
32.65/9.31	>>>
32.65/9.31	
32.65/9.31	   <<<
32.65/9.31	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-I, -I, -I], [0A, 0A, -I], [0A, -I, 0A]] 	* 	x_1
32.65/9.31	>>>
32.65/9.31	
32.65/9.31	   <<<
32.65/9.31	 POL(b(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, 1A, 0A], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
32.65/9.31	>>>
32.65/9.31	
32.65/9.31	   <<<
32.65/9.31	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
32.65/9.31	>>>
32.65/9.31	
32.65/9.31	
32.65/9.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.65/9.31	
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	
32.65/9.31	
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(6)
32.65/9.31	Obligation:
32.65/9.31	Q DP problem:
32.65/9.31	The TRS P consists of the following rules:
32.65/9.31	
32.65/9.31	   A(a(b(a(x1)))) -> A(b(b(x1)))
32.65/9.31	   B(b(x1)) -> B(a(a(a(x1))))
32.65/9.31	   B(b(x1)) -> A(a(a(x1)))
32.65/9.31	   B(b(x1)) -> A(a(x1))
32.65/9.31	   B(b(x1)) -> A(x1)
32.65/9.31	
32.65/9.31	The TRS R consists of the following rules:
32.65/9.31	
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	
32.65/9.31	Q is empty.
32.65/9.31	We have to consider all minimal (P,Q,R)-chains.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(7) DependencyGraphProof (EQUIVALENT)
32.65/9.31	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(8)
32.65/9.31	Complex Obligation (AND)
32.65/9.31	
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(9)
32.65/9.31	Obligation:
32.65/9.31	Q DP problem:
32.65/9.31	The TRS P consists of the following rules:
32.65/9.31	
32.65/9.31	   A(a(b(a(x1)))) -> A(b(b(x1)))
32.65/9.31	
32.65/9.31	The TRS R consists of the following rules:
32.65/9.31	
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	
32.65/9.31	Q is empty.
32.65/9.31	We have to consider all minimal (P,Q,R)-chains.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(10) QDPOrderProof (EQUIVALENT)
32.65/9.31	We use the reduction pair processor [LPAR04,JAR06].
32.65/9.31	
32.65/9.31	
32.65/9.31	The following pairs can be oriented strictly and are deleted.
32.65/9.31	
32.65/9.31	   A(a(b(a(x1)))) -> A(b(b(x1)))
32.65/9.31	The remaining pairs can at least be oriented weakly.
32.65/9.31	Used ordering:  Polynomial interpretation [POLO]:
32.65/9.31	
32.65/9.31	   POL(A(x_1)) = x_1
32.65/9.31	   POL(a(x_1)) = 1 + x_1
32.65/9.31	   POL(b(x_1)) = 1
32.65/9.31	
32.65/9.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.65/9.31	
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	
32.65/9.31	
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(11)
32.65/9.31	Obligation:
32.65/9.31	Q DP problem:
32.65/9.31	P is empty.
32.65/9.31	The TRS R consists of the following rules:
32.65/9.31	
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	
32.65/9.31	Q is empty.
32.65/9.31	We have to consider all minimal (P,Q,R)-chains.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(12) PisEmptyProof (EQUIVALENT)
32.65/9.31	The TRS P is empty. Hence, there is no (P,Q,R) chain.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(13)
32.65/9.31	YES
32.65/9.31	
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(14)
32.65/9.31	Obligation:
32.65/9.31	Q DP problem:
32.65/9.31	The TRS P consists of the following rules:
32.65/9.31	
32.65/9.31	   B(b(x1)) -> B(a(a(a(x1))))
32.65/9.31	
32.65/9.31	The TRS R consists of the following rules:
32.65/9.31	
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	
32.65/9.31	Q is empty.
32.65/9.31	We have to consider all minimal (P,Q,R)-chains.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(15) QDPOrderProof (EQUIVALENT)
32.65/9.31	We use the reduction pair processor [LPAR04,JAR06].
32.65/9.31	
32.65/9.31	
32.65/9.31	The following pairs can be oriented strictly and are deleted.
32.65/9.31	
32.65/9.31	   B(b(x1)) -> B(a(a(a(x1))))
32.65/9.31	The remaining pairs can at least be oriented weakly.
32.65/9.31	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
32.65/9.31	
32.65/9.31	POL( B_1(x_1) ) = x_1 + 1
32.65/9.31	POL( a_1(x_1) ) = max{0, -2}
32.65/9.31	POL( b_1(x_1) ) = 2
32.65/9.31	
32.65/9.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.65/9.31	
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	
32.65/9.31	
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(16)
32.65/9.31	Obligation:
32.65/9.31	Q DP problem:
32.65/9.31	P is empty.
32.65/9.31	The TRS R consists of the following rules:
32.65/9.31	
32.65/9.31	   a(a(b(a(x1)))) -> a(b(b(x1)))
32.65/9.31	   b(b(x1)) -> b(a(a(a(x1))))
32.65/9.31	
32.65/9.31	Q is empty.
32.65/9.31	We have to consider all minimal (P,Q,R)-chains.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(17) PisEmptyProof (EQUIVALENT)
32.65/9.31	The TRS P is empty. Hence, there is no (P,Q,R) chain.
32.65/9.31	----------------------------------------
32.65/9.31	
32.65/9.31	(18)
32.65/9.31	YES
32.76/9.35	EOF