25.52/7.42	YES
25.52/7.45	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
25.52/7.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
25.52/7.45	
25.52/7.45	
25.52/7.45	Termination w.r.t. Q of the given QTRS could be proven:
25.52/7.45	
25.52/7.45	(0) QTRS
25.52/7.45	(1) DependencyPairsProof [EQUIVALENT, 1 ms]
25.52/7.45	(2) QDP
25.52/7.45	(3) DependencyGraphProof [EQUIVALENT, 9 ms]
25.52/7.45	(4) QDP
25.52/7.45	(5) QDPOrderProof [EQUIVALENT, 128 ms]
25.52/7.45	(6) QDP
25.52/7.45	(7) QDPOrderProof [EQUIVALENT, 48 ms]
25.52/7.45	(8) QDP
25.52/7.45	(9) PisEmptyProof [EQUIVALENT, 0 ms]
25.52/7.45	(10) YES
25.52/7.45	
25.52/7.45	
25.52/7.45	----------------------------------------
25.52/7.45	
25.52/7.45	(0)
25.52/7.45	Obligation:
25.52/7.45	Q restricted rewrite system:
25.52/7.45	The TRS R consists of the following rules:
25.52/7.45	
25.52/7.45	   a(b(x1)) -> x1
25.52/7.45	   a(c(x1)) -> b(b(x1))
25.52/7.45	   c(b(x1)) -> a(c(c(a(x1))))
25.52/7.45	
25.52/7.45	Q is empty.
25.52/7.45	
25.52/7.45	----------------------------------------
25.52/7.45	
25.52/7.45	(1) DependencyPairsProof (EQUIVALENT)
25.52/7.45	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
25.52/7.45	----------------------------------------
25.52/7.45	
25.52/7.45	(2)
25.52/7.45	Obligation:
25.52/7.45	Q DP problem:
25.52/7.45	The TRS P consists of the following rules:
25.52/7.45	
25.52/7.45	   C(b(x1)) -> A(c(c(a(x1))))
25.52/7.45	   C(b(x1)) -> C(c(a(x1)))
25.52/7.45	   C(b(x1)) -> C(a(x1))
25.52/7.45	   C(b(x1)) -> A(x1)
25.52/7.45	
25.52/7.45	The TRS R consists of the following rules:
25.52/7.45	
25.52/7.45	   a(b(x1)) -> x1
25.52/7.45	   a(c(x1)) -> b(b(x1))
25.52/7.45	   c(b(x1)) -> a(c(c(a(x1))))
25.52/7.45	
25.52/7.45	Q is empty.
25.52/7.45	We have to consider all minimal (P,Q,R)-chains.
25.52/7.45	----------------------------------------
25.52/7.45	
25.52/7.45	(3) DependencyGraphProof (EQUIVALENT)
25.52/7.45	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
25.52/7.45	----------------------------------------
25.52/7.45	
25.52/7.45	(4)
25.52/7.45	Obligation:
25.52/7.45	Q DP problem:
25.52/7.45	The TRS P consists of the following rules:
25.52/7.45	
25.52/7.45	   C(b(x1)) -> C(a(x1))
25.52/7.45	   C(b(x1)) -> C(c(a(x1)))
25.52/7.45	
25.52/7.45	The TRS R consists of the following rules:
25.52/7.45	
25.52/7.45	   a(b(x1)) -> x1
25.52/7.45	   a(c(x1)) -> b(b(x1))
25.52/7.45	   c(b(x1)) -> a(c(c(a(x1))))
25.52/7.45	
25.52/7.45	Q is empty.
25.52/7.45	We have to consider all minimal (P,Q,R)-chains.
25.52/7.45	----------------------------------------
25.52/7.45	
25.52/7.45	(5) QDPOrderProof (EQUIVALENT)
25.52/7.45	We use the reduction pair processor [LPAR04,JAR06].
25.52/7.45	
25.52/7.45	
25.52/7.45	The following pairs can be oriented strictly and are deleted.
25.52/7.45	
25.52/7.45	   C(b(x1)) -> C(a(x1))
25.52/7.45	The remaining pairs can at least be oriented weakly.
25.52/7.45	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
25.52/7.45	
25.52/7.45	   <<<
25.52/7.45	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
25.52/7.45	>>>
25.52/7.45	
25.52/7.45	   <<<
25.52/7.45	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [1A, 0A, 0A]] 	* 	x_1
25.52/7.45	>>>
25.52/7.45	
25.52/7.45	   <<<
25.52/7.45	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
25.52/7.45	>>>
25.52/7.45	
25.52/7.45	   <<<
25.52/7.45	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[1A, 1A, 1A], [0A, 0A, 1A], [0A, 0A, 1A]] 	* 	x_1
25.52/7.45	>>>
25.52/7.45	
25.52/7.45	
25.52/7.45	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.52/7.45	
25.52/7.45	   a(b(x1)) -> x1
25.52/7.45	   a(c(x1)) -> b(b(x1))
25.52/7.45	   c(b(x1)) -> a(c(c(a(x1))))
25.52/7.45	
25.52/7.45	
25.52/7.45	----------------------------------------
25.52/7.45	
25.52/7.45	(6)
25.52/7.45	Obligation:
25.52/7.45	Q DP problem:
25.52/7.45	The TRS P consists of the following rules:
25.52/7.45	
25.52/7.45	   C(b(x1)) -> C(c(a(x1)))
25.52/7.45	
25.52/7.45	The TRS R consists of the following rules:
25.52/7.45	
25.52/7.45	   a(b(x1)) -> x1
25.52/7.45	   a(c(x1)) -> b(b(x1))
25.52/7.45	   c(b(x1)) -> a(c(c(a(x1))))
25.52/7.45	
25.52/7.45	Q is empty.
25.52/7.45	We have to consider all minimal (P,Q,R)-chains.
25.52/7.45	----------------------------------------
25.52/7.45	
25.52/7.45	(7) QDPOrderProof (EQUIVALENT)
25.52/7.45	We use the reduction pair processor [LPAR04,JAR06].
25.52/7.45	
25.52/7.45	
25.52/7.45	The following pairs can be oriented strictly and are deleted.
25.52/7.45	
25.52/7.45	   C(b(x1)) -> C(c(a(x1)))
25.52/7.45	The remaining pairs can at least be oriented weakly.
25.52/7.45	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
25.52/7.45	
25.52/7.45	   <<<
25.52/7.45	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
25.52/7.45	>>>
25.52/7.45	
25.52/7.45	   <<<
25.52/7.45	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [-I, -I, 0A], [0A, 1A, -I]] 	* 	x_1
25.52/7.45	>>>
25.52/7.45	
25.52/7.45	   <<<
25.52/7.45	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
25.52/7.45	>>>
25.52/7.45	
25.52/7.45	   <<<
25.52/7.45	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, 1A, -I], [-I, 0A, -I]] 	* 	x_1
25.52/7.45	>>>
25.52/7.45	
25.52/7.45	
25.52/7.45	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.52/7.45	
25.52/7.45	   a(b(x1)) -> x1
25.52/7.45	   a(c(x1)) -> b(b(x1))
25.52/7.45	   c(b(x1)) -> a(c(c(a(x1))))
25.52/7.45	
25.52/7.45	
25.52/7.45	----------------------------------------
25.52/7.45	
25.52/7.45	(8)
25.52/7.45	Obligation:
25.52/7.45	Q DP problem:
25.52/7.45	P is empty.
25.52/7.45	The TRS R consists of the following rules:
25.52/7.45	
25.52/7.45	   a(b(x1)) -> x1
25.52/7.45	   a(c(x1)) -> b(b(x1))
25.67/7.45	   c(b(x1)) -> a(c(c(a(x1))))
25.67/7.45	
25.67/7.45	Q is empty.
25.67/7.45	We have to consider all minimal (P,Q,R)-chains.
25.67/7.45	----------------------------------------
25.67/7.45	
25.67/7.45	(9) PisEmptyProof (EQUIVALENT)
25.67/7.45	The TRS P is empty. Hence, there is no (P,Q,R) chain.
25.67/7.45	----------------------------------------
25.67/7.45	
25.67/7.45	(10)
25.67/7.45	YES
25.67/7.52	EOF