25.24/7.32	YES
25.29/7.32	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
25.29/7.32	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
25.29/7.32	
25.29/7.32	
25.29/7.32	Termination w.r.t. Q of the given QTRS could be proven:
25.29/7.32	
25.29/7.32	(0) QTRS
25.29/7.32	(1) DependencyPairsProof [EQUIVALENT, 6 ms]
25.29/7.32	(2) QDP
25.29/7.32	(3) QDPOrderProof [EQUIVALENT, 152 ms]
25.29/7.32	(4) QDP
25.29/7.32	(5) QDPOrderProof [EQUIVALENT, 0 ms]
25.29/7.32	(6) QDP
25.29/7.32	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
25.29/7.32	(8) TRUE
25.29/7.32	
25.29/7.32	
25.29/7.32	----------------------------------------
25.29/7.32	
25.29/7.32	(0)
25.29/7.32	Obligation:
25.29/7.32	Q restricted rewrite system:
25.29/7.32	The TRS R consists of the following rules:
25.29/7.32	
25.29/7.32	   a(a(b(x1))) -> a(x1)
25.29/7.32	   a(b(a(x1))) -> b(b(a(x1)))
25.29/7.32	   b(b(x1)) -> a(a(a(x1)))
25.29/7.32	
25.29/7.32	Q is empty.
25.29/7.32	
25.29/7.32	----------------------------------------
25.29/7.32	
25.29/7.32	(1) DependencyPairsProof (EQUIVALENT)
25.29/7.32	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
25.29/7.32	----------------------------------------
25.29/7.32	
25.29/7.32	(2)
25.29/7.32	Obligation:
25.29/7.32	Q DP problem:
25.29/7.32	The TRS P consists of the following rules:
25.29/7.32	
25.29/7.32	   A(a(b(x1))) -> A(x1)
25.29/7.32	   A(b(a(x1))) -> B(b(a(x1)))
25.29/7.32	   B(b(x1)) -> A(a(a(x1)))
25.29/7.32	   B(b(x1)) -> A(a(x1))
25.29/7.32	   B(b(x1)) -> A(x1)
25.29/7.32	
25.29/7.32	The TRS R consists of the following rules:
25.29/7.32	
25.29/7.32	   a(a(b(x1))) -> a(x1)
25.29/7.32	   a(b(a(x1))) -> b(b(a(x1)))
25.29/7.32	   b(b(x1)) -> a(a(a(x1)))
25.29/7.32	
25.29/7.32	Q is empty.
25.29/7.32	We have to consider all minimal (P,Q,R)-chains.
25.29/7.32	----------------------------------------
25.29/7.32	
25.29/7.32	(3) QDPOrderProof (EQUIVALENT)
25.29/7.32	We use the reduction pair processor [LPAR04,JAR06].
25.29/7.32	
25.29/7.32	
25.29/7.32	The following pairs can be oriented strictly and are deleted.
25.29/7.32	
25.29/7.32	   A(a(b(x1))) -> A(x1)
25.29/7.32	The remaining pairs can at least be oriented weakly.
25.29/7.32	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
25.29/7.32	
25.29/7.32	   <<<
25.29/7.32	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
25.29/7.32	>>>
25.29/7.32	
25.29/7.32	   <<<
25.29/7.32	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
25.29/7.32	>>>
25.29/7.32	
25.29/7.32	   <<<
25.29/7.32	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, -I, 0A], [1A, -I, 0A]] 	* 	x_1
25.29/7.32	>>>
25.29/7.32	
25.29/7.32	   <<<
25.29/7.32	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
25.29/7.32	>>>
25.29/7.32	
25.29/7.32	
25.29/7.32	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.29/7.32	
25.29/7.32	   a(b(a(x1))) -> b(b(a(x1)))
25.29/7.32	   b(b(x1)) -> a(a(a(x1)))
25.29/7.32	   a(a(b(x1))) -> a(x1)
25.29/7.32	
25.29/7.32	
25.29/7.32	----------------------------------------
25.29/7.32	
25.29/7.32	(4)
25.29/7.32	Obligation:
25.29/7.32	Q DP problem:
25.29/7.32	The TRS P consists of the following rules:
25.29/7.32	
25.29/7.32	   A(b(a(x1))) -> B(b(a(x1)))
25.29/7.32	   B(b(x1)) -> A(a(a(x1)))
25.29/7.32	   B(b(x1)) -> A(a(x1))
25.29/7.32	   B(b(x1)) -> A(x1)
25.29/7.32	
25.29/7.32	The TRS R consists of the following rules:
25.29/7.32	
25.29/7.32	   a(a(b(x1))) -> a(x1)
25.29/7.32	   a(b(a(x1))) -> b(b(a(x1)))
25.29/7.32	   b(b(x1)) -> a(a(a(x1)))
25.29/7.32	
25.29/7.32	Q is empty.
25.29/7.32	We have to consider all minimal (P,Q,R)-chains.
25.29/7.32	----------------------------------------
25.29/7.32	
25.29/7.32	(5) QDPOrderProof (EQUIVALENT)
25.29/7.32	We use the reduction pair processor [LPAR04,JAR06].
25.29/7.32	
25.29/7.32	
25.29/7.32	The following pairs can be oriented strictly and are deleted.
25.29/7.32	
25.29/7.32	   A(b(a(x1))) -> B(b(a(x1)))
25.29/7.32	The remaining pairs can at least be oriented weakly.
25.29/7.32	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
25.29/7.32	
25.29/7.32	   <<<
25.29/7.32	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
25.29/7.32	>>>
25.29/7.32	
25.29/7.32	   <<<
25.29/7.32	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
25.29/7.32	>>>
25.29/7.32	
25.29/7.32	   <<<
25.29/7.32	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
25.29/7.32	>>>
25.29/7.32	
25.29/7.32	   <<<
25.29/7.32	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
25.29/7.32	>>>
25.29/7.32	
25.29/7.32	
25.29/7.32	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.29/7.32	
25.29/7.32	   a(b(a(x1))) -> b(b(a(x1)))
25.29/7.32	   b(b(x1)) -> a(a(a(x1)))
25.29/7.32	   a(a(b(x1))) -> a(x1)
25.29/7.32	
25.29/7.32	
25.29/7.32	----------------------------------------
25.29/7.32	
25.29/7.32	(6)
25.29/7.32	Obligation:
25.29/7.32	Q DP problem:
25.29/7.32	The TRS P consists of the following rules:
25.29/7.32	
25.29/7.32	   B(b(x1)) -> A(a(a(x1)))
25.29/7.32	   B(b(x1)) -> A(a(x1))
25.29/7.32	   B(b(x1)) -> A(x1)
25.29/7.32	
25.29/7.32	The TRS R consists of the following rules:
25.29/7.32	
25.29/7.32	   a(a(b(x1))) -> a(x1)
25.29/7.32	   a(b(a(x1))) -> b(b(a(x1)))
25.29/7.32	   b(b(x1)) -> a(a(a(x1)))
25.29/7.32	
25.29/7.32	Q is empty.
25.29/7.32	We have to consider all minimal (P,Q,R)-chains.
25.29/7.32	----------------------------------------
25.29/7.32	
25.29/7.32	(7) DependencyGraphProof (EQUIVALENT)
25.29/7.32	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
25.29/7.32	----------------------------------------
25.29/7.32	
25.29/7.32	(8)
25.29/7.32	TRUE
25.44/10.86	EOF