26.18/7.69	YES
26.18/7.71	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
26.18/7.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.18/7.71	
26.18/7.71	
26.18/7.71	Termination w.r.t. Q of the given QTRS could be proven:
26.18/7.71	
26.18/7.71	(0) QTRS
26.18/7.71	(1) DependencyPairsProof [EQUIVALENT, 23 ms]
26.18/7.71	(2) QDP
26.18/7.71	(3) DependencyGraphProof [EQUIVALENT, 7 ms]
26.18/7.71	(4) QDP
26.18/7.71	(5) QDPOrderProof [EQUIVALENT, 120 ms]
26.18/7.71	(6) QDP
26.18/7.71	(7) QDPOrderProof [EQUIVALENT, 0 ms]
26.18/7.71	(8) QDP
26.18/7.71	(9) PisEmptyProof [EQUIVALENT, 0 ms]
26.18/7.71	(10) YES
26.18/7.71	
26.18/7.71	
26.18/7.71	----------------------------------------
26.18/7.71	
26.18/7.71	(0)
26.18/7.71	Obligation:
26.18/7.71	Q restricted rewrite system:
26.18/7.71	The TRS R consists of the following rules:
26.18/7.71	
26.18/7.71	   a(b(x1)) -> x1
26.18/7.71	   a(c(x1)) -> b(b(x1))
26.18/7.71	   c(b(x1)) -> a(a(c(c(x1))))
26.18/7.71	
26.18/7.71	Q is empty.
26.18/7.71	
26.18/7.71	----------------------------------------
26.18/7.71	
26.18/7.71	(1) DependencyPairsProof (EQUIVALENT)
26.18/7.71	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.18/7.71	----------------------------------------
26.18/7.71	
26.18/7.71	(2)
26.18/7.71	Obligation:
26.18/7.71	Q DP problem:
26.18/7.71	The TRS P consists of the following rules:
26.18/7.71	
26.18/7.71	   C(b(x1)) -> A(a(c(c(x1))))
26.18/7.71	   C(b(x1)) -> A(c(c(x1)))
26.18/7.72	   C(b(x1)) -> C(c(x1))
26.18/7.72	   C(b(x1)) -> C(x1)
26.18/7.72	
26.18/7.72	The TRS R consists of the following rules:
26.18/7.72	
26.18/7.72	   a(b(x1)) -> x1
26.18/7.72	   a(c(x1)) -> b(b(x1))
26.18/7.72	   c(b(x1)) -> a(a(c(c(x1))))
26.18/7.72	
26.18/7.72	Q is empty.
26.18/7.72	We have to consider all minimal (P,Q,R)-chains.
26.18/7.72	----------------------------------------
26.18/7.72	
26.18/7.72	(3) DependencyGraphProof (EQUIVALENT)
26.18/7.72	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
26.18/7.72	----------------------------------------
26.18/7.72	
26.18/7.72	(4)
26.18/7.72	Obligation:
26.18/7.72	Q DP problem:
26.18/7.72	The TRS P consists of the following rules:
26.18/7.72	
26.18/7.72	   C(b(x1)) -> C(x1)
26.18/7.72	   C(b(x1)) -> C(c(x1))
26.18/7.72	
26.18/7.72	The TRS R consists of the following rules:
26.18/7.72	
26.18/7.72	   a(b(x1)) -> x1
26.18/7.72	   a(c(x1)) -> b(b(x1))
26.18/7.72	   c(b(x1)) -> a(a(c(c(x1))))
26.18/7.72	
26.18/7.72	Q is empty.
26.18/7.72	We have to consider all minimal (P,Q,R)-chains.
26.18/7.72	----------------------------------------
26.18/7.72	
26.18/7.72	(5) QDPOrderProof (EQUIVALENT)
26.18/7.72	We use the reduction pair processor [LPAR04,JAR06].
26.18/7.72	
26.18/7.72	
26.18/7.72	The following pairs can be oriented strictly and are deleted.
26.18/7.72	
26.18/7.72	   C(b(x1)) -> C(x1)
26.18/7.72	The remaining pairs can at least be oriented weakly.
26.18/7.72	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.55/7.72	
26.55/7.72	   <<<
26.55/7.72	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
26.55/7.72	>>>
26.55/7.72	
26.55/7.72	   <<<
26.55/7.72	 POL(b(x_1)) =  	[[1A], [-I], [1A]] 	 +  	[[-I, 0A, 0A], [1A, 1A, 1A], [0A, -I, 0A]] 	* 	x_1
26.55/7.72	>>>
26.55/7.72	
26.55/7.72	   <<<
26.55/7.72	 POL(c(x_1)) =  	[[-I], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [1A, 1A, 0A], [1A, 1A, 1A]] 	* 	x_1
26.55/7.72	>>>
26.55/7.72	
26.55/7.72	   <<<
26.55/7.72	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [0A, -I, 1A], [0A, -I, -I]] 	* 	x_1
26.55/7.72	>>>
26.55/7.72	
26.55/7.72	
26.55/7.72	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.55/7.72	
26.55/7.72	   c(b(x1)) -> a(a(c(c(x1))))
26.55/7.72	   a(b(x1)) -> x1
26.55/7.72	   a(c(x1)) -> b(b(x1))
26.55/7.72	
26.55/7.72	
26.55/7.72	----------------------------------------
26.55/7.72	
26.55/7.72	(6)
26.55/7.72	Obligation:
26.55/7.72	Q DP problem:
26.55/7.72	The TRS P consists of the following rules:
26.55/7.72	
26.55/7.72	   C(b(x1)) -> C(c(x1))
26.55/7.72	
26.55/7.72	The TRS R consists of the following rules:
26.55/7.72	
26.55/7.72	   a(b(x1)) -> x1
26.55/7.72	   a(c(x1)) -> b(b(x1))
26.55/7.72	   c(b(x1)) -> a(a(c(c(x1))))
26.55/7.72	
26.55/7.72	Q is empty.
26.55/7.72	We have to consider all minimal (P,Q,R)-chains.
26.55/7.72	----------------------------------------
26.55/7.72	
26.55/7.72	(7) QDPOrderProof (EQUIVALENT)
26.55/7.72	We use the reduction pair processor [LPAR04,JAR06].
26.55/7.72	
26.55/7.72	
26.55/7.72	The following pairs can be oriented strictly and are deleted.
26.55/7.72	
26.55/7.72	   C(b(x1)) -> C(c(x1))
26.55/7.72	The remaining pairs can at least be oriented weakly.
26.55/7.72	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.55/7.72	
26.55/7.72	   <<<
26.55/7.72	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
26.55/7.72	>>>
26.55/7.72	
26.55/7.72	   <<<
26.55/7.72	 POL(b(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [1A, 0A, 1A]] 	* 	x_1
26.55/7.72	>>>
26.55/7.72	
26.55/7.72	   <<<
26.55/7.72	 POL(c(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, -I, 0A], [1A, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
26.55/7.72	>>>
26.55/7.72	
26.55/7.72	   <<<
26.55/7.72	 POL(a(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, -I], [-I, -I, 0A], [0A, 1A, -I]] 	* 	x_1
26.55/7.72	>>>
26.55/7.72	
26.55/7.72	
26.55/7.72	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.55/7.72	
26.55/7.72	   c(b(x1)) -> a(a(c(c(x1))))
26.55/7.72	   a(b(x1)) -> x1
26.55/7.72	   a(c(x1)) -> b(b(x1))
26.55/7.72	
26.55/7.72	
26.55/7.72	----------------------------------------
26.55/7.72	
26.55/7.72	(8)
26.55/7.72	Obligation:
26.55/7.72	Q DP problem:
26.55/7.72	P is empty.
26.55/7.72	The TRS R consists of the following rules:
26.55/7.72	
26.55/7.72	   a(b(x1)) -> x1
26.55/7.72	   a(c(x1)) -> b(b(x1))
26.55/7.72	   c(b(x1)) -> a(a(c(c(x1))))
26.55/7.72	
26.55/7.72	Q is empty.
26.55/7.72	We have to consider all minimal (P,Q,R)-chains.
26.55/7.72	----------------------------------------
26.55/7.72	
26.55/7.72	(9) PisEmptyProof (EQUIVALENT)
26.55/7.72	The TRS P is empty. Hence, there is no (P,Q,R) chain.
26.55/7.72	----------------------------------------
26.55/7.72	
26.55/7.72	(10)
26.55/7.72	YES
26.76/7.80	EOF