34.89/9.81	YES
35.27/9.90	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
35.27/9.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
35.27/9.90	
35.27/9.90	
35.27/9.90	Termination w.r.t. Q of the given QTRS could be proven:
35.27/9.90	
35.27/9.90	(0) QTRS
35.27/9.90	(1) DependencyPairsProof [EQUIVALENT, 9 ms]
35.27/9.90	(2) QDP
35.27/9.90	(3) QDPOrderProof [EQUIVALENT, 110 ms]
35.27/9.90	(4) QDP
35.27/9.90	(5) QDPOrderProof [EQUIVALENT, 27 ms]
35.27/9.90	(6) QDP
35.27/9.90	(7) QDPOrderProof [EQUIVALENT, 0 ms]
35.27/9.90	(8) QDP
35.27/9.90	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
35.27/9.90	(10) TRUE
35.27/9.90	
35.27/9.90	
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(0)
35.27/9.90	Obligation:
35.27/9.90	Q restricted rewrite system:
35.27/9.90	The TRS R consists of the following rules:
35.27/9.90	
35.27/9.90	   a(x1) -> x1
35.27/9.90	   a(b(b(x1))) -> b(b(a(b(c(x1)))))
35.27/9.90	   b(c(x1)) -> a(x1)
35.27/9.90	
35.27/9.90	Q is empty.
35.27/9.90	
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(1) DependencyPairsProof (EQUIVALENT)
35.27/9.90	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(2)
35.27/9.90	Obligation:
35.27/9.90	Q DP problem:
35.27/9.90	The TRS P consists of the following rules:
35.27/9.90	
35.27/9.90	   A(b(b(x1))) -> B(b(a(b(c(x1)))))
35.27/9.90	   A(b(b(x1))) -> B(a(b(c(x1))))
35.27/9.90	   A(b(b(x1))) -> A(b(c(x1)))
35.27/9.90	   A(b(b(x1))) -> B(c(x1))
35.27/9.90	   B(c(x1)) -> A(x1)
35.27/9.90	
35.27/9.90	The TRS R consists of the following rules:
35.27/9.90	
35.27/9.90	   a(x1) -> x1
35.27/9.90	   a(b(b(x1))) -> b(b(a(b(c(x1)))))
35.27/9.90	   b(c(x1)) -> a(x1)
35.27/9.90	
35.27/9.90	Q is empty.
35.27/9.90	We have to consider all minimal (P,Q,R)-chains.
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(3) QDPOrderProof (EQUIVALENT)
35.27/9.90	We use the reduction pair processor [LPAR04,JAR06].
35.27/9.90	
35.27/9.90	
35.27/9.90	The following pairs can be oriented strictly and are deleted.
35.27/9.90	
35.27/9.90	   A(b(b(x1))) -> B(a(b(c(x1))))
35.27/9.90	The remaining pairs can at least be oriented weakly.
35.27/9.90	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 1A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [0A, 0A, -I]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(c(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, -I], [0A, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	
35.27/9.90	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.27/9.90	
35.27/9.90	   a(b(b(x1))) -> b(b(a(b(c(x1)))))
35.27/9.90	   b(c(x1)) -> a(x1)
35.27/9.90	   a(x1) -> x1
35.27/9.90	
35.27/9.90	
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(4)
35.27/9.90	Obligation:
35.27/9.90	Q DP problem:
35.27/9.90	The TRS P consists of the following rules:
35.27/9.90	
35.27/9.90	   A(b(b(x1))) -> B(b(a(b(c(x1)))))
35.27/9.90	   A(b(b(x1))) -> A(b(c(x1)))
35.27/9.90	   A(b(b(x1))) -> B(c(x1))
35.27/9.90	   B(c(x1)) -> A(x1)
35.27/9.90	
35.27/9.90	The TRS R consists of the following rules:
35.27/9.90	
35.27/9.90	   a(x1) -> x1
35.27/9.90	   a(b(b(x1))) -> b(b(a(b(c(x1)))))
35.27/9.90	   b(c(x1)) -> a(x1)
35.27/9.90	
35.27/9.90	Q is empty.
35.27/9.90	We have to consider all minimal (P,Q,R)-chains.
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(5) QDPOrderProof (EQUIVALENT)
35.27/9.90	We use the reduction pair processor [LPAR04,JAR06].
35.27/9.90	
35.27/9.90	
35.27/9.90	The following pairs can be oriented strictly and are deleted.
35.27/9.90	
35.27/9.90	   A(b(b(x1))) -> A(b(c(x1)))
35.27/9.90	   A(b(b(x1))) -> B(c(x1))
35.27/9.90	The remaining pairs can at least be oriented weakly.
35.27/9.90	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	
35.27/9.90	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.27/9.90	
35.27/9.90	   a(b(b(x1))) -> b(b(a(b(c(x1)))))
35.27/9.90	   b(c(x1)) -> a(x1)
35.27/9.90	   a(x1) -> x1
35.27/9.90	
35.27/9.90	
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(6)
35.27/9.90	Obligation:
35.27/9.90	Q DP problem:
35.27/9.90	The TRS P consists of the following rules:
35.27/9.90	
35.27/9.90	   A(b(b(x1))) -> B(b(a(b(c(x1)))))
35.27/9.90	   B(c(x1)) -> A(x1)
35.27/9.90	
35.27/9.90	The TRS R consists of the following rules:
35.27/9.90	
35.27/9.90	   a(x1) -> x1
35.27/9.90	   a(b(b(x1))) -> b(b(a(b(c(x1)))))
35.27/9.90	   b(c(x1)) -> a(x1)
35.27/9.90	
35.27/9.90	Q is empty.
35.27/9.90	We have to consider all minimal (P,Q,R)-chains.
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(7) QDPOrderProof (EQUIVALENT)
35.27/9.90	We use the reduction pair processor [LPAR04,JAR06].
35.27/9.90	
35.27/9.90	
35.27/9.90	The following pairs can be oriented strictly and are deleted.
35.27/9.90	
35.27/9.90	   A(b(b(x1))) -> B(b(a(b(c(x1)))))
35.27/9.90	The remaining pairs can at least be oriented weakly.
35.27/9.90	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(b(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [1A, 0A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	   <<<
35.27/9.90	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
35.27/9.90	>>>
35.27/9.90	
35.27/9.90	
35.27/9.90	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.27/9.90	
35.27/9.90	   a(b(b(x1))) -> b(b(a(b(c(x1)))))
35.27/9.90	   b(c(x1)) -> a(x1)
35.27/9.90	   a(x1) -> x1
35.27/9.90	
35.27/9.90	
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(8)
35.27/9.90	Obligation:
35.27/9.90	Q DP problem:
35.27/9.90	The TRS P consists of the following rules:
35.27/9.90	
35.27/9.90	   B(c(x1)) -> A(x1)
35.27/9.90	
35.27/9.90	The TRS R consists of the following rules:
35.27/9.90	
35.27/9.90	   a(x1) -> x1
35.27/9.90	   a(b(b(x1))) -> b(b(a(b(c(x1)))))
35.27/9.90	   b(c(x1)) -> a(x1)
35.27/9.90	
35.27/9.90	Q is empty.
35.27/9.90	We have to consider all minimal (P,Q,R)-chains.
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(9) DependencyGraphProof (EQUIVALENT)
35.27/9.90	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
35.27/9.90	----------------------------------------
35.27/9.90	
35.27/9.90	(10)
35.27/9.90	TRUE
35.47/10.02	EOF