23.58/7.09	YES
23.58/7.10	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
23.58/7.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
23.58/7.10	
23.58/7.10	
23.58/7.10	Termination w.r.t. Q of the given QTRS could be proven:
23.58/7.10	
23.58/7.10	(0) QTRS
23.58/7.10	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
23.58/7.10	(2) QDP
23.58/7.10	(3) QDPOrderProof [EQUIVALENT, 120 ms]
23.58/7.10	(4) QDP
23.58/7.10	(5) QDPOrderProof [EQUIVALENT, 57 ms]
23.58/7.10	(6) QDP
23.58/7.10	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
23.58/7.10	(8) TRUE
23.58/7.10	
23.58/7.10	
23.58/7.10	----------------------------------------
23.58/7.10	
23.58/7.10	(0)
23.58/7.10	Obligation:
23.58/7.10	Q restricted rewrite system:
23.58/7.10	The TRS R consists of the following rules:
23.58/7.10	
23.58/7.10	   a(b(b(x1))) -> a(x1)
23.58/7.10	   a(a(x1)) -> b(b(b(x1)))
23.58/7.10	   b(b(a(x1))) -> a(b(a(x1)))
23.58/7.10	
23.58/7.10	Q is empty.
23.58/7.10	
23.58/7.10	----------------------------------------
23.58/7.10	
23.58/7.10	(1) DependencyPairsProof (EQUIVALENT)
23.58/7.10	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
23.58/7.10	----------------------------------------
23.58/7.10	
23.58/7.10	(2)
23.58/7.10	Obligation:
23.58/7.10	Q DP problem:
23.58/7.10	The TRS P consists of the following rules:
23.58/7.10	
23.58/7.10	   A(b(b(x1))) -> A(x1)
23.58/7.10	   A(a(x1)) -> B(b(b(x1)))
23.58/7.10	   A(a(x1)) -> B(b(x1))
23.58/7.10	   A(a(x1)) -> B(x1)
23.58/7.10	   B(b(a(x1))) -> A(b(a(x1)))
23.58/7.10	
23.58/7.10	The TRS R consists of the following rules:
23.58/7.10	
23.58/7.10	   a(b(b(x1))) -> a(x1)
23.58/7.10	   a(a(x1)) -> b(b(b(x1)))
23.58/7.10	   b(b(a(x1))) -> a(b(a(x1)))
23.58/7.10	
23.58/7.10	Q is empty.
23.58/7.10	We have to consider all minimal (P,Q,R)-chains.
23.58/7.10	----------------------------------------
23.58/7.10	
23.58/7.10	(3) QDPOrderProof (EQUIVALENT)
23.58/7.10	We use the reduction pair processor [LPAR04,JAR06].
23.58/7.10	
23.58/7.10	
23.58/7.10	The following pairs can be oriented strictly and are deleted.
23.58/7.10	
23.58/7.10	   A(b(b(x1))) -> A(x1)
23.58/7.10	The remaining pairs can at least be oriented weakly.
23.58/7.10	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.58/7.10	
23.58/7.10	   <<<
23.58/7.10	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
23.58/7.10	>>>
23.58/7.10	
23.58/7.10	   <<<
23.58/7.10	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [-I, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
23.58/7.10	>>>
23.58/7.10	
23.58/7.10	   <<<
23.58/7.10	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[1A, 0A, 1A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
23.58/7.10	>>>
23.58/7.10	
23.58/7.10	   <<<
23.58/7.10	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
23.58/7.10	>>>
23.58/7.10	
23.58/7.10	
23.58/7.10	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.58/7.10	
23.58/7.10	   a(a(x1)) -> b(b(b(x1)))
23.58/7.10	   b(b(a(x1))) -> a(b(a(x1)))
23.58/7.10	   a(b(b(x1))) -> a(x1)
23.58/7.10	
23.58/7.10	
23.58/7.10	----------------------------------------
23.58/7.10	
23.58/7.10	(4)
23.58/7.10	Obligation:
23.58/7.10	Q DP problem:
23.58/7.10	The TRS P consists of the following rules:
23.58/7.10	
23.58/7.10	   A(a(x1)) -> B(b(b(x1)))
23.58/7.10	   A(a(x1)) -> B(b(x1))
23.58/7.10	   A(a(x1)) -> B(x1)
23.58/7.10	   B(b(a(x1))) -> A(b(a(x1)))
23.58/7.10	
23.58/7.10	The TRS R consists of the following rules:
23.58/7.10	
23.58/7.10	   a(b(b(x1))) -> a(x1)
23.58/7.10	   a(a(x1)) -> b(b(b(x1)))
23.58/7.10	   b(b(a(x1))) -> a(b(a(x1)))
23.58/7.10	
23.58/7.10	Q is empty.
23.58/7.10	We have to consider all minimal (P,Q,R)-chains.
23.58/7.10	----------------------------------------
23.58/7.10	
23.58/7.10	(5) QDPOrderProof (EQUIVALENT)
23.58/7.10	We use the reduction pair processor [LPAR04,JAR06].
23.58/7.10	
23.58/7.10	
23.58/7.10	The following pairs can be oriented strictly and are deleted.
23.58/7.10	
23.58/7.10	   B(b(a(x1))) -> A(b(a(x1)))
23.58/7.10	The remaining pairs can at least be oriented weakly.
23.58/7.10	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
23.58/7.10	
23.58/7.10	   <<<
23.58/7.10	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
23.58/7.10	>>>
23.58/7.10	
23.58/7.10	   <<<
23.58/7.10	 POL(a(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, -I], [1A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
23.58/7.10	>>>
23.58/7.10	
23.58/7.10	   <<<
23.58/7.10	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
23.58/7.10	>>>
23.58/7.10	
23.58/7.10	   <<<
23.58/7.10	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
23.58/7.10	>>>
23.58/7.10	
23.58/7.10	
23.58/7.10	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
23.58/7.10	
23.58/7.10	   a(a(x1)) -> b(b(b(x1)))
23.58/7.10	   b(b(a(x1))) -> a(b(a(x1)))
23.58/7.10	   a(b(b(x1))) -> a(x1)
23.58/7.10	
23.58/7.10	
23.58/7.10	----------------------------------------
23.58/7.10	
23.58/7.10	(6)
23.58/7.10	Obligation:
23.58/7.10	Q DP problem:
23.58/7.10	The TRS P consists of the following rules:
23.58/7.10	
23.58/7.10	   A(a(x1)) -> B(b(b(x1)))
23.58/7.10	   A(a(x1)) -> B(b(x1))
23.58/7.10	   A(a(x1)) -> B(x1)
23.58/7.10	
23.58/7.10	The TRS R consists of the following rules:
23.58/7.10	
23.58/7.10	   a(b(b(x1))) -> a(x1)
23.58/7.10	   a(a(x1)) -> b(b(b(x1)))
23.58/7.10	   b(b(a(x1))) -> a(b(a(x1)))
23.58/7.10	
23.58/7.10	Q is empty.
23.58/7.10	We have to consider all minimal (P,Q,R)-chains.
23.58/7.10	----------------------------------------
23.58/7.10	
23.58/7.10	(7) DependencyGraphProof (EQUIVALENT)
23.58/7.10	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
23.58/7.10	----------------------------------------
23.58/7.10	
23.58/7.10	(8)
23.58/7.10	TRUE
24.04/7.18	EOF