35.74/10.19	YES
36.14/10.26	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
36.14/10.26	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
36.14/10.26	
36.14/10.26	
36.14/10.26	Termination w.r.t. Q of the given QTRS could be proven:
36.14/10.26	
36.14/10.26	(0) QTRS
36.14/10.26	(1) QTRS Reverse [EQUIVALENT, 0 ms]
36.14/10.26	(2) QTRS
36.14/10.26	(3) DependencyPairsProof [EQUIVALENT, 2 ms]
36.14/10.26	(4) QDP
36.14/10.26	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
36.14/10.26	(6) QDP
36.14/10.26	(7) QDPOrderProof [EQUIVALENT, 61 ms]
36.14/10.26	(8) QDP
36.14/10.26	(9) QDPOrderProof [EQUIVALENT, 20 ms]
36.14/10.26	(10) QDP
36.14/10.26	(11) PisEmptyProof [EQUIVALENT, 0 ms]
36.14/10.26	(12) YES
36.14/10.26	
36.14/10.26	
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(0)
36.14/10.26	Obligation:
36.14/10.26	Q restricted rewrite system:
36.14/10.26	The TRS R consists of the following rules:
36.14/10.26	
36.14/10.26	   a(x1) -> x1
36.14/10.26	   a(a(b(x1))) -> a(b(b(a(x1))))
36.14/10.26	   b(b(b(x1))) -> a(x1)
36.14/10.26	
36.14/10.26	Q is empty.
36.14/10.26	
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(1) QTRS Reverse (EQUIVALENT)
36.14/10.26	We applied the QTRS Reverse Processor [REVERSE].
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(2)
36.14/10.26	Obligation:
36.14/10.26	Q restricted rewrite system:
36.14/10.26	The TRS R consists of the following rules:
36.14/10.26	
36.14/10.26	   a(x1) -> x1
36.14/10.26	   b(a(a(x1))) -> a(b(b(a(x1))))
36.14/10.26	   b(b(b(x1))) -> a(x1)
36.14/10.26	
36.14/10.26	Q is empty.
36.14/10.26	
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(3) DependencyPairsProof (EQUIVALENT)
36.14/10.26	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(4)
36.14/10.26	Obligation:
36.14/10.26	Q DP problem:
36.14/10.26	The TRS P consists of the following rules:
36.14/10.26	
36.14/10.26	   B(a(a(x1))) -> A(b(b(a(x1))))
36.14/10.26	   B(a(a(x1))) -> B(b(a(x1)))
36.14/10.26	   B(a(a(x1))) -> B(a(x1))
36.14/10.26	   B(b(b(x1))) -> A(x1)
36.14/10.26	
36.14/10.26	The TRS R consists of the following rules:
36.14/10.26	
36.14/10.26	   a(x1) -> x1
36.14/10.26	   b(a(a(x1))) -> a(b(b(a(x1))))
36.14/10.26	   b(b(b(x1))) -> a(x1)
36.14/10.26	
36.14/10.26	Q is empty.
36.14/10.26	We have to consider all minimal (P,Q,R)-chains.
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(5) DependencyGraphProof (EQUIVALENT)
36.14/10.26	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(6)
36.14/10.26	Obligation:
36.14/10.26	Q DP problem:
36.14/10.26	The TRS P consists of the following rules:
36.14/10.26	
36.14/10.26	   B(a(a(x1))) -> B(a(x1))
36.14/10.26	   B(a(a(x1))) -> B(b(a(x1)))
36.14/10.26	
36.14/10.26	The TRS R consists of the following rules:
36.14/10.26	
36.14/10.26	   a(x1) -> x1
36.14/10.26	   b(a(a(x1))) -> a(b(b(a(x1))))
36.14/10.26	   b(b(b(x1))) -> a(x1)
36.14/10.26	
36.14/10.26	Q is empty.
36.14/10.26	We have to consider all minimal (P,Q,R)-chains.
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(7) QDPOrderProof (EQUIVALENT)
36.14/10.26	We use the reduction pair processor [LPAR04,JAR06].
36.14/10.26	
36.14/10.26	
36.14/10.26	The following pairs can be oriented strictly and are deleted.
36.14/10.26	
36.14/10.26	   B(a(a(x1))) -> B(a(x1))
36.14/10.26	The remaining pairs can at least be oriented weakly.
36.14/10.26	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.14/10.26	
36.14/10.26	   <<<
36.14/10.26	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
36.14/10.26	>>>
36.14/10.26	
36.14/10.26	   <<<
36.14/10.26	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
36.14/10.26	>>>
36.14/10.26	
36.14/10.26	   <<<
36.14/10.26	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
36.14/10.26	>>>
36.14/10.26	
36.14/10.26	
36.14/10.26	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.14/10.26	
36.14/10.26	   a(x1) -> x1
36.14/10.26	   b(a(a(x1))) -> a(b(b(a(x1))))
36.14/10.26	   b(b(b(x1))) -> a(x1)
36.14/10.26	
36.14/10.26	
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(8)
36.14/10.26	Obligation:
36.14/10.26	Q DP problem:
36.14/10.26	The TRS P consists of the following rules:
36.14/10.26	
36.14/10.26	   B(a(a(x1))) -> B(b(a(x1)))
36.14/10.26	
36.14/10.26	The TRS R consists of the following rules:
36.14/10.26	
36.14/10.26	   a(x1) -> x1
36.14/10.26	   b(a(a(x1))) -> a(b(b(a(x1))))
36.14/10.26	   b(b(b(x1))) -> a(x1)
36.14/10.26	
36.14/10.26	Q is empty.
36.14/10.26	We have to consider all minimal (P,Q,R)-chains.
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(9) QDPOrderProof (EQUIVALENT)
36.14/10.26	We use the reduction pair processor [LPAR04,JAR06].
36.14/10.26	
36.14/10.26	
36.14/10.26	The following pairs can be oriented strictly and are deleted.
36.14/10.26	
36.14/10.26	   B(a(a(x1))) -> B(b(a(x1)))
36.14/10.26	The remaining pairs can at least be oriented weakly.
36.14/10.26	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.14/10.26	
36.14/10.26	   <<<
36.14/10.26	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
36.14/10.26	>>>
36.14/10.26	
36.14/10.26	   <<<
36.14/10.26	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 1A, 0A], [0A, 0A, 0A]] 	* 	x_1
36.14/10.26	>>>
36.14/10.26	
36.14/10.26	   <<<
36.14/10.26	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
36.14/10.26	>>>
36.14/10.26	
36.14/10.26	
36.14/10.26	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.14/10.26	
36.14/10.26	   a(x1) -> x1
36.14/10.26	   b(a(a(x1))) -> a(b(b(a(x1))))
36.14/10.26	   b(b(b(x1))) -> a(x1)
36.14/10.26	
36.14/10.26	
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(10)
36.14/10.26	Obligation:
36.14/10.26	Q DP problem:
36.14/10.26	P is empty.
36.14/10.26	The TRS R consists of the following rules:
36.14/10.26	
36.14/10.26	   a(x1) -> x1
36.14/10.26	   b(a(a(x1))) -> a(b(b(a(x1))))
36.14/10.26	   b(b(b(x1))) -> a(x1)
36.14/10.26	
36.14/10.26	Q is empty.
36.14/10.26	We have to consider all minimal (P,Q,R)-chains.
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(11) PisEmptyProof (EQUIVALENT)
36.14/10.26	The TRS P is empty. Hence, there is no (P,Q,R) chain.
36.14/10.26	----------------------------------------
36.14/10.26	
36.14/10.26	(12)
36.14/10.26	YES
36.52/10.39	EOF