24.19/7.23	YES
24.60/7.33	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
24.60/7.33	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
24.60/7.33	
24.60/7.33	
24.60/7.33	Termination w.r.t. Q of the given QTRS could be proven:
24.60/7.33	
24.60/7.33	(0) QTRS
24.60/7.33	(1) DependencyPairsProof [EQUIVALENT, 3 ms]
24.60/7.33	(2) QDP
24.60/7.33	(3) QDPOrderProof [EQUIVALENT, 110 ms]
24.60/7.33	(4) QDP
24.60/7.33	(5) QDPOrderProof [EQUIVALENT, 0 ms]
24.60/7.33	(6) QDP
24.60/7.33	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
24.60/7.33	(8) TRUE
24.60/7.33	
24.60/7.33	
24.60/7.33	----------------------------------------
24.60/7.33	
24.60/7.33	(0)
24.60/7.33	Obligation:
24.60/7.33	Q restricted rewrite system:
24.60/7.33	The TRS R consists of the following rules:
24.60/7.33	
24.60/7.33	   a(x1) -> x1
24.60/7.33	   a(b(b(x1))) -> b(b(b(c(x1))))
24.60/7.33	   b(c(x1)) -> a(a(x1))
24.60/7.33	
24.60/7.33	Q is empty.
24.60/7.33	
24.60/7.33	----------------------------------------
24.60/7.33	
24.60/7.33	(1) DependencyPairsProof (EQUIVALENT)
24.60/7.33	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
24.60/7.33	----------------------------------------
24.60/7.33	
24.60/7.33	(2)
24.60/7.33	Obligation:
24.60/7.33	Q DP problem:
24.60/7.33	The TRS P consists of the following rules:
24.60/7.33	
24.60/7.33	   A(b(b(x1))) -> B(b(b(c(x1))))
24.60/7.33	   A(b(b(x1))) -> B(b(c(x1)))
24.60/7.33	   A(b(b(x1))) -> B(c(x1))
24.60/7.33	   B(c(x1)) -> A(a(x1))
24.60/7.33	   B(c(x1)) -> A(x1)
24.60/7.33	
24.60/7.33	The TRS R consists of the following rules:
24.60/7.33	
24.60/7.33	   a(x1) -> x1
24.60/7.33	   a(b(b(x1))) -> b(b(b(c(x1))))
24.60/7.33	   b(c(x1)) -> a(a(x1))
24.60/7.33	
24.60/7.33	Q is empty.
24.60/7.33	We have to consider all minimal (P,Q,R)-chains.
24.60/7.33	----------------------------------------
24.60/7.33	
24.60/7.33	(3) QDPOrderProof (EQUIVALENT)
24.60/7.33	We use the reduction pair processor [LPAR04,JAR06].
24.60/7.33	
24.60/7.33	
24.60/7.33	The following pairs can be oriented strictly and are deleted.
24.60/7.33	
24.60/7.33	   A(b(b(x1))) -> B(b(c(x1)))
24.60/7.33	The remaining pairs can at least be oriented weakly.
24.60/7.33	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, 0A], [0A, -I, 0A], [0A, 1A, -I]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [0A, -I, 0A]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	
24.60/7.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.60/7.33	
24.60/7.33	   a(b(b(x1))) -> b(b(b(c(x1))))
24.60/7.33	   b(c(x1)) -> a(a(x1))
24.60/7.33	   a(x1) -> x1
24.60/7.33	
24.60/7.33	
24.60/7.33	----------------------------------------
24.60/7.33	
24.60/7.33	(4)
24.60/7.33	Obligation:
24.60/7.33	Q DP problem:
24.60/7.33	The TRS P consists of the following rules:
24.60/7.33	
24.60/7.33	   A(b(b(x1))) -> B(b(b(c(x1))))
24.60/7.33	   A(b(b(x1))) -> B(c(x1))
24.60/7.33	   B(c(x1)) -> A(a(x1))
24.60/7.33	   B(c(x1)) -> A(x1)
24.60/7.33	
24.60/7.33	The TRS R consists of the following rules:
24.60/7.33	
24.60/7.33	   a(x1) -> x1
24.60/7.33	   a(b(b(x1))) -> b(b(b(c(x1))))
24.60/7.33	   b(c(x1)) -> a(a(x1))
24.60/7.33	
24.60/7.33	Q is empty.
24.60/7.33	We have to consider all minimal (P,Q,R)-chains.
24.60/7.33	----------------------------------------
24.60/7.33	
24.60/7.33	(5) QDPOrderProof (EQUIVALENT)
24.60/7.33	We use the reduction pair processor [LPAR04,JAR06].
24.60/7.33	
24.60/7.33	
24.60/7.33	The following pairs can be oriented strictly and are deleted.
24.60/7.33	
24.60/7.33	   A(b(b(x1))) -> B(b(b(c(x1))))
24.60/7.33	   A(b(b(x1))) -> B(c(x1))
24.60/7.33	The remaining pairs can at least be oriented weakly.
24.60/7.33	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(A(x_1)) =  	[[-I]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [1A, 0A, 0A]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	   <<<
24.60/7.33	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
24.60/7.33	>>>
24.60/7.33	
24.60/7.33	
24.60/7.33	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
24.60/7.33	
24.60/7.33	   a(b(b(x1))) -> b(b(b(c(x1))))
24.60/7.33	   b(c(x1)) -> a(a(x1))
24.60/7.33	   a(x1) -> x1
24.60/7.33	
24.60/7.33	
24.60/7.33	----------------------------------------
24.60/7.33	
24.60/7.33	(6)
24.60/7.33	Obligation:
24.60/7.33	Q DP problem:
24.60/7.33	The TRS P consists of the following rules:
24.60/7.33	
24.60/7.33	   B(c(x1)) -> A(a(x1))
24.60/7.33	   B(c(x1)) -> A(x1)
24.60/7.33	
24.60/7.33	The TRS R consists of the following rules:
24.60/7.33	
24.60/7.33	   a(x1) -> x1
24.60/7.33	   a(b(b(x1))) -> b(b(b(c(x1))))
24.60/7.33	   b(c(x1)) -> a(a(x1))
24.60/7.33	
24.60/7.33	Q is empty.
24.60/7.33	We have to consider all minimal (P,Q,R)-chains.
24.60/7.33	----------------------------------------
24.60/7.33	
24.60/7.33	(7) DependencyGraphProof (EQUIVALENT)
24.60/7.33	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
24.60/7.33	----------------------------------------
24.60/7.33	
24.60/7.33	(8)
24.60/7.33	TRUE
24.60/7.38	EOF