26.33/7.64	YES
26.33/7.68	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
26.33/7.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.33/7.68	
26.33/7.68	
26.33/7.68	Termination w.r.t. Q of the given QTRS could be proven:
26.33/7.68	
26.33/7.68	(0) QTRS
26.33/7.68	(1) QTRS Reverse [EQUIVALENT, 0 ms]
26.33/7.68	(2) QTRS
26.33/7.68	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
26.33/7.68	(4) QDP
26.33/7.68	(5) QDPOrderProof [EQUIVALENT, 207 ms]
26.33/7.68	(6) QDP
26.33/7.68	(7) QDPOrderProof [EQUIVALENT, 0 ms]
26.33/7.68	(8) QDP
26.33/7.68	(9) PisEmptyProof [EQUIVALENT, 0 ms]
26.33/7.68	(10) YES
26.33/7.68	
26.33/7.68	
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(0)
26.33/7.68	Obligation:
26.33/7.68	Q restricted rewrite system:
26.33/7.68	The TRS R consists of the following rules:
26.33/7.68	
26.33/7.68	   a(a(a(b(x1)))) -> b(a(b(a(a(a(x1))))))
26.33/7.68	   a(b(x1)) -> x1
26.33/7.68	
26.33/7.68	Q is empty.
26.33/7.68	
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(1) QTRS Reverse (EQUIVALENT)
26.33/7.68	We applied the QTRS Reverse Processor [REVERSE].
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(2)
26.33/7.68	Obligation:
26.33/7.68	Q restricted rewrite system:
26.33/7.68	The TRS R consists of the following rules:
26.33/7.68	
26.33/7.68	   b(a(a(a(x1)))) -> a(a(a(b(a(b(x1))))))
26.33/7.68	   b(a(x1)) -> x1
26.33/7.68	
26.33/7.68	Q is empty.
26.33/7.68	
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(3) DependencyPairsProof (EQUIVALENT)
26.33/7.68	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(4)
26.33/7.68	Obligation:
26.33/7.68	Q DP problem:
26.33/7.68	The TRS P consists of the following rules:
26.33/7.68	
26.33/7.68	   B(a(a(a(x1)))) -> B(a(b(x1)))
26.33/7.68	   B(a(a(a(x1)))) -> B(x1)
26.33/7.68	
26.33/7.68	The TRS R consists of the following rules:
26.33/7.68	
26.33/7.68	   b(a(a(a(x1)))) -> a(a(a(b(a(b(x1))))))
26.33/7.68	   b(a(x1)) -> x1
26.33/7.68	
26.33/7.68	Q is empty.
26.33/7.68	We have to consider all minimal (P,Q,R)-chains.
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(5) QDPOrderProof (EQUIVALENT)
26.33/7.68	We use the reduction pair processor [LPAR04,JAR06].
26.33/7.68	
26.33/7.68	
26.33/7.68	The following pairs can be oriented strictly and are deleted.
26.33/7.68	
26.33/7.68	   B(a(a(a(x1)))) -> B(x1)
26.33/7.68	The remaining pairs can at least be oriented weakly.
26.33/7.68	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.33/7.68	
26.33/7.68	   <<<
26.33/7.68	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
26.33/7.68	>>>
26.33/7.68	
26.33/7.68	   <<<
26.33/7.68	 POL(a(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, 1A, 0A], [-I, -I, 0A], [0A, 0A, -I]] 	* 	x_1
26.33/7.68	>>>
26.33/7.68	
26.33/7.68	   <<<
26.33/7.68	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
26.33/7.68	>>>
26.33/7.68	
26.33/7.68	
26.33/7.68	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.33/7.68	
26.33/7.68	   b(a(a(a(x1)))) -> a(a(a(b(a(b(x1))))))
26.33/7.68	   b(a(x1)) -> x1
26.33/7.68	
26.33/7.68	
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(6)
26.33/7.68	Obligation:
26.33/7.68	Q DP problem:
26.33/7.68	The TRS P consists of the following rules:
26.33/7.68	
26.33/7.68	   B(a(a(a(x1)))) -> B(a(b(x1)))
26.33/7.68	
26.33/7.68	The TRS R consists of the following rules:
26.33/7.68	
26.33/7.68	   b(a(a(a(x1)))) -> a(a(a(b(a(b(x1))))))
26.33/7.68	   b(a(x1)) -> x1
26.33/7.68	
26.33/7.68	Q is empty.
26.33/7.68	We have to consider all minimal (P,Q,R)-chains.
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(7) QDPOrderProof (EQUIVALENT)
26.33/7.68	We use the reduction pair processor [LPAR04,JAR06].
26.33/7.68	
26.33/7.68	
26.33/7.68	The following pairs can be oriented strictly and are deleted.
26.33/7.68	
26.33/7.68	   B(a(a(a(x1)))) -> B(a(b(x1)))
26.33/7.68	The remaining pairs can at least be oriented weakly.
26.33/7.68	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.33/7.68	
26.33/7.68	   <<<
26.33/7.68	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
26.33/7.68	>>>
26.33/7.68	
26.33/7.68	   <<<
26.33/7.68	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, -I, 0A], [1A, 0A, 0A]] 	* 	x_1
26.33/7.68	>>>
26.33/7.68	
26.33/7.68	   <<<
26.33/7.68	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, 0A, -I]] 	* 	x_1
26.33/7.68	>>>
26.33/7.68	
26.33/7.68	
26.33/7.68	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.33/7.68	
26.33/7.68	   b(a(a(a(x1)))) -> a(a(a(b(a(b(x1))))))
26.33/7.68	   b(a(x1)) -> x1
26.33/7.68	
26.33/7.68	
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(8)
26.33/7.68	Obligation:
26.33/7.68	Q DP problem:
26.33/7.68	P is empty.
26.33/7.68	The TRS R consists of the following rules:
26.33/7.68	
26.33/7.68	   b(a(a(a(x1)))) -> a(a(a(b(a(b(x1))))))
26.33/7.68	   b(a(x1)) -> x1
26.33/7.68	
26.33/7.68	Q is empty.
26.33/7.68	We have to consider all minimal (P,Q,R)-chains.
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(9) PisEmptyProof (EQUIVALENT)
26.33/7.68	The TRS P is empty. Hence, there is no (P,Q,R) chain.
26.33/7.68	----------------------------------------
26.33/7.68	
26.33/7.68	(10)
26.33/7.68	YES
26.56/7.75	EOF