24.09/6.99	YES
24.28/7.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
24.28/7.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
24.28/7.06	
24.28/7.06	
24.28/7.06	Termination w.r.t. Q of the given QTRS could be proven:
24.28/7.06	
24.28/7.06	(0) QTRS
24.28/7.06	(1) QTRS Reverse [EQUIVALENT, 0 ms]
24.28/7.06	(2) QTRS
24.28/7.06	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
24.28/7.06	(4) QDP
24.28/7.06	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
24.28/7.06	(6) QDP
24.28/7.06	(7) QDPOrderProof [EQUIVALENT, 181 ms]
24.28/7.06	(8) QDP
24.28/7.06	(9) QDPOrderProof [EQUIVALENT, 0 ms]
24.28/7.06	(10) QDP
24.28/7.06	(11) PisEmptyProof [EQUIVALENT, 0 ms]
24.28/7.06	(12) YES
24.28/7.06	
24.28/7.06	
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(0)
24.28/7.06	Obligation:
24.28/7.06	Q restricted rewrite system:
24.28/7.06	The TRS R consists of the following rules:
24.28/7.06	
24.28/7.06	   a(x1) -> x1
24.28/7.06	   a(a(b(x1))) -> b(a(b(a(a(x1)))))
24.28/7.06	   b(b(x1)) -> b(x1)
24.28/7.06	
24.28/7.06	Q is empty.
24.28/7.06	
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(1) QTRS Reverse (EQUIVALENT)
24.28/7.06	We applied the QTRS Reverse Processor [REVERSE].
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(2)
24.28/7.06	Obligation:
24.28/7.06	Q restricted rewrite system:
24.28/7.06	The TRS R consists of the following rules:
24.28/7.06	
24.28/7.06	   a(x1) -> x1
24.28/7.06	   b(a(a(x1))) -> a(a(b(a(b(x1)))))
24.28/7.06	   b(b(x1)) -> b(x1)
24.28/7.06	
24.28/7.06	Q is empty.
24.28/7.06	
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(3) DependencyPairsProof (EQUIVALENT)
24.28/7.06	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(4)
24.28/7.06	Obligation:
24.28/7.06	Q DP problem:
24.28/7.06	The TRS P consists of the following rules:
24.28/7.06	
24.28/7.06	   B(a(a(x1))) -> A(a(b(a(b(x1)))))
24.28/7.06	   B(a(a(x1))) -> A(b(a(b(x1))))
24.28/7.06	   B(a(a(x1))) -> B(a(b(x1)))
24.28/7.06	   B(a(a(x1))) -> A(b(x1))
24.28/7.06	   B(a(a(x1))) -> B(x1)
24.28/7.06	
24.28/7.06	The TRS R consists of the following rules:
24.28/7.06	
24.28/7.06	   a(x1) -> x1
24.28/7.06	   b(a(a(x1))) -> a(a(b(a(b(x1)))))
24.28/7.06	   b(b(x1)) -> b(x1)
24.28/7.06	
24.28/7.06	Q is empty.
24.28/7.06	We have to consider all minimal (P,Q,R)-chains.
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(5) DependencyGraphProof (EQUIVALENT)
24.28/7.06	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(6)
24.28/7.06	Obligation:
24.28/7.06	Q DP problem:
24.28/7.06	The TRS P consists of the following rules:
24.28/7.06	
24.28/7.06	   B(a(a(x1))) -> B(x1)
24.28/7.06	   B(a(a(x1))) -> B(a(b(x1)))
24.28/7.06	
24.28/7.06	The TRS R consists of the following rules:
24.28/7.06	
24.28/7.06	   a(x1) -> x1
24.28/7.06	   b(a(a(x1))) -> a(a(b(a(b(x1)))))
24.28/7.06	   b(b(x1)) -> b(x1)
24.28/7.06	
24.28/7.06	Q is empty.
24.28/7.06	We have to consider all minimal (P,Q,R)-chains.
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(7) QDPOrderProof (EQUIVALENT)
24.28/7.06	We use the reduction pair processor [LPAR04,JAR06].
24.28/7.06	
24.28/7.06	
24.28/7.06	The following pairs can be oriented strictly and are deleted.
24.28/7.06	
24.28/7.06	   B(a(a(x1))) -> B(x1)
24.28/7.06	The remaining pairs can at least be oriented weakly.
24.28/7.06	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.28/7.06	
24.28/7.06	   <<<
24.28/7.06	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
24.28/7.06	>>>
24.28/7.06	
24.28/7.06	   <<<
24.28/7.06	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [1A, -I, 0A]] 	* 	x_1
24.28/7.06	>>>
24.28/7.06	
24.28/7.06	   <<<
24.28/7.06	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
24.28/7.06	>>>
24.28/7.06	
24.28/7.06	
24.28/7.06	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.28/7.06	
24.28/7.06	   b(a(a(x1))) -> a(a(b(a(b(x1)))))
24.28/7.06	   b(b(x1)) -> b(x1)
24.28/7.06	   a(x1) -> x1
24.28/7.06	
24.28/7.06	
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(8)
24.28/7.06	Obligation:
24.28/7.06	Q DP problem:
24.28/7.06	The TRS P consists of the following rules:
24.28/7.06	
24.28/7.06	   B(a(a(x1))) -> B(a(b(x1)))
24.28/7.06	
24.28/7.06	The TRS R consists of the following rules:
24.28/7.06	
24.28/7.06	   a(x1) -> x1
24.28/7.06	   b(a(a(x1))) -> a(a(b(a(b(x1)))))
24.28/7.06	   b(b(x1)) -> b(x1)
24.28/7.06	
24.28/7.06	Q is empty.
24.28/7.06	We have to consider all minimal (P,Q,R)-chains.
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(9) QDPOrderProof (EQUIVALENT)
24.28/7.06	We use the reduction pair processor [LPAR04,JAR06].
24.28/7.06	
24.28/7.06	
24.28/7.06	The following pairs can be oriented strictly and are deleted.
24.28/7.06	
24.28/7.06	   B(a(a(x1))) -> B(a(b(x1)))
24.28/7.06	The remaining pairs can at least be oriented weakly.
24.28/7.06	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.28/7.06	
24.28/7.06	   <<<
24.28/7.06	 POL(B(x_1)) =  	[[-I]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
24.28/7.06	>>>
24.28/7.06	
24.28/7.06	   <<<
24.28/7.06	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
24.28/7.06	>>>
24.28/7.06	
24.28/7.06	   <<<
24.28/7.06	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
24.28/7.06	>>>
24.28/7.06	
24.28/7.06	
24.28/7.06	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.28/7.06	
24.28/7.06	   b(a(a(x1))) -> a(a(b(a(b(x1)))))
24.28/7.06	   b(b(x1)) -> b(x1)
24.28/7.06	   a(x1) -> x1
24.28/7.06	
24.28/7.06	
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(10)
24.28/7.06	Obligation:
24.28/7.06	Q DP problem:
24.28/7.06	P is empty.
24.28/7.06	The TRS R consists of the following rules:
24.28/7.06	
24.28/7.06	   a(x1) -> x1
24.28/7.06	   b(a(a(x1))) -> a(a(b(a(b(x1)))))
24.28/7.06	   b(b(x1)) -> b(x1)
24.28/7.06	
24.28/7.06	Q is empty.
24.28/7.06	We have to consider all minimal (P,Q,R)-chains.
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(11) PisEmptyProof (EQUIVALENT)
24.28/7.06	The TRS P is empty. Hence, there is no (P,Q,R) chain.
24.28/7.06	----------------------------------------
24.28/7.06	
24.28/7.06	(12)
24.28/7.06	YES
24.51/7.14	EOF