36.34/10.21	YES
36.34/10.22	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
36.34/10.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
36.34/10.22	
36.34/10.22	
36.34/10.22	Termination w.r.t. Q of the given QTRS could be proven:
36.34/10.22	
36.34/10.22	(0) QTRS
36.34/10.22	(1) DependencyPairsProof [EQUIVALENT, 20 ms]
36.34/10.22	(2) QDP
36.34/10.22	(3) QDPOrderProof [EQUIVALENT, 131 ms]
36.34/10.22	(4) QDP
36.34/10.22	(5) QDPOrderProof [EQUIVALENT, 49 ms]
36.34/10.22	(6) QDP
36.34/10.22	(7) QDPOrderProof [EQUIVALENT, 0 ms]
36.34/10.22	(8) QDP
36.34/10.22	(9) QDPOrderProof [EQUIVALENT, 0 ms]
36.34/10.22	(10) QDP
36.34/10.22	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
36.34/10.22	(12) TRUE
36.34/10.22	
36.34/10.22	
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(0)
36.34/10.22	Obligation:
36.34/10.22	Q restricted rewrite system:
36.34/10.22	The TRS R consists of the following rules:
36.34/10.22	
36.34/10.22	   a(x1) -> x1
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	
36.34/10.22	Q is empty.
36.34/10.22	
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(1) DependencyPairsProof (EQUIVALENT)
36.34/10.22	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(2)
36.34/10.22	Obligation:
36.34/10.22	Q DP problem:
36.34/10.22	The TRS P consists of the following rules:
36.34/10.22	
36.34/10.22	   A(a(x1)) -> B(x1)
36.34/10.22	   B(a(b(x1))) -> A(a(b(b(b(x1)))))
36.34/10.22	   B(a(b(x1))) -> A(b(b(b(x1))))
36.34/10.22	   B(a(b(x1))) -> B(b(b(x1)))
36.34/10.22	   B(a(b(x1))) -> B(b(x1))
36.34/10.22	
36.34/10.22	The TRS R consists of the following rules:
36.34/10.22	
36.34/10.22	   a(x1) -> x1
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	
36.34/10.22	Q is empty.
36.34/10.22	We have to consider all minimal (P,Q,R)-chains.
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(3) QDPOrderProof (EQUIVALENT)
36.34/10.22	We use the reduction pair processor [LPAR04,JAR06].
36.34/10.22	
36.34/10.22	
36.34/10.22	The following pairs can be oriented strictly and are deleted.
36.34/10.22	
36.34/10.22	   B(a(b(x1))) -> B(b(x1))
36.34/10.22	The remaining pairs can at least be oriented weakly.
36.34/10.22	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 1A], [-I, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	
36.34/10.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.34/10.22	
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	   a(x1) -> x1
36.34/10.22	
36.34/10.22	
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(4)
36.34/10.22	Obligation:
36.34/10.22	Q DP problem:
36.34/10.22	The TRS P consists of the following rules:
36.34/10.22	
36.34/10.22	   A(a(x1)) -> B(x1)
36.34/10.22	   B(a(b(x1))) -> A(a(b(b(b(x1)))))
36.34/10.22	   B(a(b(x1))) -> A(b(b(b(x1))))
36.34/10.22	   B(a(b(x1))) -> B(b(b(x1)))
36.34/10.22	
36.34/10.22	The TRS R consists of the following rules:
36.34/10.22	
36.34/10.22	   a(x1) -> x1
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	
36.34/10.22	Q is empty.
36.34/10.22	We have to consider all minimal (P,Q,R)-chains.
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(5) QDPOrderProof (EQUIVALENT)
36.34/10.22	We use the reduction pair processor [LPAR04,JAR06].
36.34/10.22	
36.34/10.22	
36.34/10.22	The following pairs can be oriented strictly and are deleted.
36.34/10.22	
36.34/10.22	   B(a(b(x1))) -> A(b(b(b(x1))))
36.34/10.22	The remaining pairs can at least be oriented weakly.
36.34/10.22	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 1A, -I], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	
36.34/10.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.34/10.22	
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	   a(x1) -> x1
36.34/10.22	
36.34/10.22	
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(6)
36.34/10.22	Obligation:
36.34/10.22	Q DP problem:
36.34/10.22	The TRS P consists of the following rules:
36.34/10.22	
36.34/10.22	   A(a(x1)) -> B(x1)
36.34/10.22	   B(a(b(x1))) -> A(a(b(b(b(x1)))))
36.34/10.22	   B(a(b(x1))) -> B(b(b(x1)))
36.34/10.22	
36.34/10.22	The TRS R consists of the following rules:
36.34/10.22	
36.34/10.22	   a(x1) -> x1
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	
36.34/10.22	Q is empty.
36.34/10.22	We have to consider all minimal (P,Q,R)-chains.
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(7) QDPOrderProof (EQUIVALENT)
36.34/10.22	We use the reduction pair processor [LPAR04,JAR06].
36.34/10.22	
36.34/10.22	
36.34/10.22	The following pairs can be oriented strictly and are deleted.
36.34/10.22	
36.34/10.22	   B(a(b(x1))) -> B(b(b(x1)))
36.34/10.22	The remaining pairs can at least be oriented weakly.
36.34/10.22	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 1A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(a(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, -I, 0A], [0A, 0A, -I], [1A, -I, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(B(x_1)) =  	[[0A]] 	 +  	[[1A, -I, -I]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [1A, -I, -I]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	
36.34/10.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.34/10.22	
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	   a(x1) -> x1
36.34/10.22	
36.34/10.22	
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(8)
36.34/10.22	Obligation:
36.34/10.22	Q DP problem:
36.34/10.22	The TRS P consists of the following rules:
36.34/10.22	
36.34/10.22	   A(a(x1)) -> B(x1)
36.34/10.22	   B(a(b(x1))) -> A(a(b(b(b(x1)))))
36.34/10.22	
36.34/10.22	The TRS R consists of the following rules:
36.34/10.22	
36.34/10.22	   a(x1) -> x1
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	
36.34/10.22	Q is empty.
36.34/10.22	We have to consider all minimal (P,Q,R)-chains.
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(9) QDPOrderProof (EQUIVALENT)
36.34/10.22	We use the reduction pair processor [LPAR04,JAR06].
36.34/10.22	
36.34/10.22	
36.34/10.22	The following pairs can be oriented strictly and are deleted.
36.34/10.22	
36.34/10.22	   A(a(x1)) -> B(x1)
36.34/10.22	The remaining pairs can at least be oriented weakly.
36.34/10.22	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(a(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[1A, 1A, 0A], [0A, 0A, 1A], [0A, 1A, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	   <<<
36.34/10.22	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
36.34/10.22	>>>
36.34/10.22	
36.34/10.22	
36.34/10.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
36.34/10.22	
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	   a(x1) -> x1
36.34/10.22	
36.34/10.22	
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(10)
36.34/10.22	Obligation:
36.34/10.22	Q DP problem:
36.34/10.22	The TRS P consists of the following rules:
36.34/10.22	
36.34/10.22	   B(a(b(x1))) -> A(a(b(b(b(x1)))))
36.34/10.22	
36.34/10.22	The TRS R consists of the following rules:
36.34/10.22	
36.34/10.22	   a(x1) -> x1
36.34/10.22	   a(a(x1)) -> b(x1)
36.34/10.22	   b(a(b(x1))) -> a(a(b(b(b(x1)))))
36.34/10.22	
36.34/10.22	Q is empty.
36.34/10.22	We have to consider all minimal (P,Q,R)-chains.
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(11) DependencyGraphProof (EQUIVALENT)
36.34/10.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
36.34/10.22	----------------------------------------
36.34/10.22	
36.34/10.22	(12)
36.34/10.22	TRUE
36.99/10.39	EOF