61.07/16.58	YES
61.47/16.64	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
61.47/16.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
61.47/16.64	
61.47/16.64	
61.47/16.64	Termination w.r.t. Q of the given QTRS could be proven:
61.47/16.64	
61.47/16.64	(0) QTRS
61.47/16.64	(1) QTRS Reverse [EQUIVALENT, 0 ms]
61.47/16.64	(2) QTRS
61.47/16.64	(3) DependencyPairsProof [EQUIVALENT, 23 ms]
61.47/16.64	(4) QDP
61.47/16.64	(5) QDPOrderProof [EQUIVALENT, 73 ms]
61.47/16.64	(6) QDP
61.47/16.64	(7) QDPOrderProof [EQUIVALENT, 227 ms]
61.47/16.64	(8) QDP
61.47/16.64	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
61.47/16.64	(10) QDP
61.47/16.64	(11) QDPOrderProof [EQUIVALENT, 351 ms]
61.47/16.64	(12) QDP
61.47/16.64	(13) PisEmptyProof [EQUIVALENT, 0 ms]
61.47/16.64	(14) YES
61.47/16.64	
61.47/16.64	
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(0)
61.47/16.64	Obligation:
61.47/16.64	Q restricted rewrite system:
61.47/16.64	The TRS R consists of the following rules:
61.47/16.64	
61.47/16.64	   b(a(b(b(x1)))) -> a(a(b(a(x1))))
61.47/16.64	   a(a(a(a(x1)))) -> b(a(b(b(x1))))
61.47/16.64	   b(b(b(a(x1)))) -> b(b(a(a(x1))))
61.47/16.64	
61.47/16.64	Q is empty.
61.47/16.64	
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(1) QTRS Reverse (EQUIVALENT)
61.47/16.64	We applied the QTRS Reverse Processor [REVERSE].
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(2)
61.47/16.64	Obligation:
61.47/16.64	Q restricted rewrite system:
61.47/16.64	The TRS R consists of the following rules:
61.47/16.64	
61.47/16.64	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
61.47/16.64	   a(a(a(a(x1)))) -> b(b(a(b(x1))))
61.47/16.64	   a(b(b(b(x1)))) -> a(a(b(b(x1))))
61.47/16.64	
61.47/16.64	Q is empty.
61.47/16.64	
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(3) DependencyPairsProof (EQUIVALENT)
61.47/16.64	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(4)
61.47/16.64	Obligation:
61.47/16.64	Q DP problem:
61.47/16.64	The TRS P consists of the following rules:
61.47/16.64	
61.47/16.64	   B(b(a(b(x1)))) -> A(b(a(a(x1))))
61.47/16.64	   B(b(a(b(x1)))) -> B(a(a(x1)))
61.47/16.64	   B(b(a(b(x1)))) -> A(a(x1))
61.47/16.64	   B(b(a(b(x1)))) -> A(x1)
61.47/16.64	   A(a(a(a(x1)))) -> B(b(a(b(x1))))
61.47/16.64	   A(a(a(a(x1)))) -> B(a(b(x1)))
61.47/16.64	   A(a(a(a(x1)))) -> A(b(x1))
61.47/16.64	   A(a(a(a(x1)))) -> B(x1)
61.47/16.64	   A(b(b(b(x1)))) -> A(a(b(b(x1))))
61.47/16.64	   A(b(b(b(x1)))) -> A(b(b(x1)))
61.47/16.64	
61.47/16.64	The TRS R consists of the following rules:
61.47/16.64	
61.47/16.64	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
61.47/16.64	   a(a(a(a(x1)))) -> b(b(a(b(x1))))
61.47/16.64	   a(b(b(b(x1)))) -> a(a(b(b(x1))))
61.47/16.64	
61.47/16.64	Q is empty.
61.47/16.64	We have to consider all minimal (P,Q,R)-chains.
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(5) QDPOrderProof (EQUIVALENT)
61.47/16.64	We use the reduction pair processor [LPAR04,JAR06].
61.47/16.64	
61.47/16.64	
61.47/16.64	The following pairs can be oriented strictly and are deleted.
61.47/16.64	
61.47/16.64	   B(b(a(b(x1)))) -> B(a(a(x1)))
61.47/16.64	   B(b(a(b(x1)))) -> A(a(x1))
61.47/16.64	   B(b(a(b(x1)))) -> A(x1)
61.47/16.64	   A(a(a(a(x1)))) -> B(a(b(x1)))
61.47/16.64	   A(a(a(a(x1)))) -> A(b(x1))
61.47/16.64	   A(a(a(a(x1)))) -> B(x1)
61.47/16.64	   A(b(b(b(x1)))) -> A(b(b(x1)))
61.47/16.64	The remaining pairs can at least be oriented weakly.
61.47/16.64	Used ordering:  Polynomial interpretation [POLO]:
61.47/16.64	
61.47/16.64	   POL(A(x_1)) = 1 + x_1
61.47/16.64	   POL(B(x_1)) = 1 + x_1
61.47/16.64	   POL(a(x_1)) = 1 + x_1
61.47/16.64	   POL(b(x_1)) = 1 + x_1
61.47/16.64	
61.47/16.64	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
61.47/16.64	
61.47/16.64	   a(a(a(a(x1)))) -> b(b(a(b(x1))))
61.47/16.64	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
61.47/16.64	   a(b(b(b(x1)))) -> a(a(b(b(x1))))
61.47/16.64	
61.47/16.64	
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(6)
61.47/16.64	Obligation:
61.47/16.64	Q DP problem:
61.47/16.64	The TRS P consists of the following rules:
61.47/16.64	
61.47/16.64	   B(b(a(b(x1)))) -> A(b(a(a(x1))))
61.47/16.64	   A(a(a(a(x1)))) -> B(b(a(b(x1))))
61.47/16.64	   A(b(b(b(x1)))) -> A(a(b(b(x1))))
61.47/16.64	
61.47/16.64	The TRS R consists of the following rules:
61.47/16.64	
61.47/16.64	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
61.47/16.64	   a(a(a(a(x1)))) -> b(b(a(b(x1))))
61.47/16.64	   a(b(b(b(x1)))) -> a(a(b(b(x1))))
61.47/16.64	
61.47/16.64	Q is empty.
61.47/16.64	We have to consider all minimal (P,Q,R)-chains.
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(7) QDPOrderProof (EQUIVALENT)
61.47/16.64	We use the reduction pair processor [LPAR04,JAR06].
61.47/16.64	
61.47/16.64	
61.47/16.64	The following pairs can be oriented strictly and are deleted.
61.47/16.64	
61.47/16.64	   B(b(a(b(x1)))) -> A(b(a(a(x1))))
61.47/16.64	The remaining pairs can at least be oriented weakly.
61.47/16.64	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
61.47/16.64	
61.47/16.64	   <<<
61.47/16.64	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
61.47/16.64	>>>
61.47/16.64	
61.47/16.64	   <<<
61.47/16.64	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 1A, 0A], [-I, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
61.47/16.64	>>>
61.47/16.64	
61.47/16.64	   <<<
61.47/16.64	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, 1A, 0A], [0A, 0A, -I]] 	* 	x_1
61.47/16.64	>>>
61.47/16.64	
61.47/16.64	   <<<
61.47/16.64	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
61.47/16.64	>>>
61.47/16.64	
61.47/16.64	
61.47/16.64	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
61.47/16.64	
61.47/16.64	   a(a(a(a(x1)))) -> b(b(a(b(x1))))
61.47/16.64	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
61.47/16.64	   a(b(b(b(x1)))) -> a(a(b(b(x1))))
61.47/16.64	
61.47/16.64	
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(8)
61.47/16.64	Obligation:
61.47/16.64	Q DP problem:
61.47/16.64	The TRS P consists of the following rules:
61.47/16.64	
61.47/16.64	   A(a(a(a(x1)))) -> B(b(a(b(x1))))
61.47/16.64	   A(b(b(b(x1)))) -> A(a(b(b(x1))))
61.47/16.64	
61.47/16.64	The TRS R consists of the following rules:
61.47/16.64	
61.47/16.64	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
61.47/16.64	   a(a(a(a(x1)))) -> b(b(a(b(x1))))
61.47/16.64	   a(b(b(b(x1)))) -> a(a(b(b(x1))))
61.47/16.64	
61.47/16.64	Q is empty.
61.47/16.64	We have to consider all minimal (P,Q,R)-chains.
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(9) DependencyGraphProof (EQUIVALENT)
61.47/16.64	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(10)
61.47/16.64	Obligation:
61.47/16.64	Q DP problem:
61.47/16.64	The TRS P consists of the following rules:
61.47/16.64	
61.47/16.64	   A(b(b(b(x1)))) -> A(a(b(b(x1))))
61.47/16.64	
61.47/16.64	The TRS R consists of the following rules:
61.47/16.64	
61.47/16.64	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
61.47/16.64	   a(a(a(a(x1)))) -> b(b(a(b(x1))))
61.47/16.64	   a(b(b(b(x1)))) -> a(a(b(b(x1))))
61.47/16.64	
61.47/16.64	Q is empty.
61.47/16.64	We have to consider all minimal (P,Q,R)-chains.
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(11) QDPOrderProof (EQUIVALENT)
61.47/16.64	We use the reduction pair processor [LPAR04,JAR06].
61.47/16.64	
61.47/16.64	
61.47/16.64	The following pairs can be oriented strictly and are deleted.
61.47/16.64	
61.47/16.64	   A(b(b(b(x1)))) -> A(a(b(b(x1))))
61.47/16.64	The remaining pairs can at least be oriented weakly.
61.47/16.64	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
61.47/16.64	
61.47/16.64	   <<<
61.47/16.64	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
61.47/16.64	>>>
61.47/16.64	
61.47/16.64	   <<<
61.47/16.64	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 1A, 1A], [1A, -I, -I]] 	* 	x_1
61.47/16.64	>>>
61.47/16.64	
61.47/16.64	   <<<
61.47/16.64	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, -I], [-I, 0A, 1A], [1A, 0A, 0A]] 	* 	x_1
61.47/16.64	>>>
61.47/16.64	
61.47/16.64	
61.47/16.64	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
61.47/16.64	
61.47/16.64	   a(a(a(a(x1)))) -> b(b(a(b(x1))))
61.47/16.64	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
61.47/16.64	   a(b(b(b(x1)))) -> a(a(b(b(x1))))
61.47/16.64	
61.47/16.64	
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(12)
61.47/16.64	Obligation:
61.47/16.64	Q DP problem:
61.47/16.64	P is empty.
61.47/16.64	The TRS R consists of the following rules:
61.47/16.64	
61.47/16.64	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
61.47/16.64	   a(a(a(a(x1)))) -> b(b(a(b(x1))))
61.47/16.64	   a(b(b(b(x1)))) -> a(a(b(b(x1))))
61.47/16.64	
61.47/16.64	Q is empty.
61.47/16.64	We have to consider all minimal (P,Q,R)-chains.
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(13) PisEmptyProof (EQUIVALENT)
61.47/16.64	The TRS P is empty. Hence, there is no (P,Q,R) chain.
61.47/16.64	----------------------------------------
61.47/16.64	
61.47/16.64	(14)
61.47/16.64	YES
61.57/16.71	EOF