27.19/7.81	YES
27.19/7.82	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.19/7.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.19/7.82	
27.19/7.82	
27.19/7.82	Termination w.r.t. Q of the given QTRS could be proven:
27.19/7.82	
27.19/7.82	(0) QTRS
27.19/7.82	(1) DependencyPairsProof [EQUIVALENT, 21 ms]
27.19/7.82	(2) QDP
27.19/7.82	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
27.19/7.82	(4) QDP
27.19/7.82	(5) QDPOrderProof [EQUIVALENT, 92 ms]
27.19/7.82	(6) QDP
27.19/7.82	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
27.19/7.82	(8) QDP
27.19/7.82	(9) QDPOrderProof [EQUIVALENT, 0 ms]
27.19/7.82	(10) QDP
27.19/7.82	(11) PisEmptyProof [EQUIVALENT, 0 ms]
27.19/7.82	(12) YES
27.19/7.82	
27.19/7.82	
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(0)
27.19/7.82	Obligation:
27.19/7.82	Q restricted rewrite system:
27.19/7.82	The TRS R consists of the following rules:
27.19/7.82	
27.19/7.82	   a(a(x1)) -> x1
27.19/7.82	   b(b(x1)) -> c(c(c(c(x1))))
27.19/7.82	   c(c(x1)) -> a(c(b(x1)))
27.19/7.82	
27.19/7.82	Q is empty.
27.19/7.82	
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(1) DependencyPairsProof (EQUIVALENT)
27.19/7.82	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(2)
27.19/7.82	Obligation:
27.19/7.82	Q DP problem:
27.19/7.82	The TRS P consists of the following rules:
27.19/7.82	
27.19/7.82	   B(b(x1)) -> C(c(c(c(x1))))
27.19/7.82	   B(b(x1)) -> C(c(c(x1)))
27.19/7.82	   B(b(x1)) -> C(c(x1))
27.19/7.82	   B(b(x1)) -> C(x1)
27.19/7.82	   C(c(x1)) -> A(c(b(x1)))
27.19/7.82	   C(c(x1)) -> C(b(x1))
27.19/7.82	   C(c(x1)) -> B(x1)
27.19/7.82	
27.19/7.82	The TRS R consists of the following rules:
27.19/7.82	
27.19/7.82	   a(a(x1)) -> x1
27.19/7.82	   b(b(x1)) -> c(c(c(c(x1))))
27.19/7.82	   c(c(x1)) -> a(c(b(x1)))
27.19/7.82	
27.19/7.82	Q is empty.
27.19/7.82	We have to consider all minimal (P,Q,R)-chains.
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(3) DependencyGraphProof (EQUIVALENT)
27.19/7.82	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(4)
27.19/7.82	Obligation:
27.19/7.82	Q DP problem:
27.19/7.82	The TRS P consists of the following rules:
27.19/7.82	
27.19/7.82	   C(c(x1)) -> C(b(x1))
27.19/7.82	   C(c(x1)) -> B(x1)
27.19/7.82	   B(b(x1)) -> C(c(c(c(x1))))
27.19/7.82	   B(b(x1)) -> C(c(c(x1)))
27.19/7.82	   B(b(x1)) -> C(c(x1))
27.19/7.82	   B(b(x1)) -> C(x1)
27.19/7.82	
27.19/7.82	The TRS R consists of the following rules:
27.19/7.82	
27.19/7.82	   a(a(x1)) -> x1
27.19/7.82	   b(b(x1)) -> c(c(c(c(x1))))
27.19/7.82	   c(c(x1)) -> a(c(b(x1)))
27.19/7.82	
27.19/7.82	Q is empty.
27.19/7.82	We have to consider all minimal (P,Q,R)-chains.
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(5) QDPOrderProof (EQUIVALENT)
27.19/7.82	We use the reduction pair processor [LPAR04,JAR06].
27.19/7.82	
27.19/7.82	
27.19/7.82	The following pairs can be oriented strictly and are deleted.
27.19/7.82	
27.19/7.82	   B(b(x1)) -> C(c(c(c(x1))))
27.19/7.82	   B(b(x1)) -> C(c(c(x1)))
27.19/7.82	   B(b(x1)) -> C(c(x1))
27.19/7.82	   B(b(x1)) -> C(x1)
27.19/7.82	The remaining pairs can at least be oriented weakly.
27.19/7.82	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.19/7.82	
27.19/7.82	   <<<
27.19/7.82	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
27.19/7.82	>>>
27.19/7.82	
27.19/7.82	   <<<
27.19/7.82	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, -I, 1A], [1A, -I, 0A], [-I, -I, -I]] 	* 	x_1
27.19/7.82	>>>
27.19/7.82	
27.19/7.82	   <<<
27.19/7.82	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, 0A], [0A, 0A, 1A], [0A, -I, 1A]] 	* 	x_1
27.19/7.82	>>>
27.19/7.82	
27.19/7.82	   <<<
27.19/7.82	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, -I, 1A]] 	* 	x_1
27.19/7.82	>>>
27.19/7.82	
27.19/7.82	   <<<
27.19/7.82	 POL(a(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
27.19/7.82	>>>
27.19/7.82	
27.19/7.82	
27.19/7.82	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.19/7.82	
27.19/7.82	   b(b(x1)) -> c(c(c(c(x1))))
27.19/7.82	   c(c(x1)) -> a(c(b(x1)))
27.19/7.82	   a(a(x1)) -> x1
27.19/7.82	
27.19/7.82	
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(6)
27.19/7.82	Obligation:
27.19/7.82	Q DP problem:
27.19/7.82	The TRS P consists of the following rules:
27.19/7.82	
27.19/7.82	   C(c(x1)) -> C(b(x1))
27.19/7.82	   C(c(x1)) -> B(x1)
27.19/7.82	
27.19/7.82	The TRS R consists of the following rules:
27.19/7.82	
27.19/7.82	   a(a(x1)) -> x1
27.19/7.82	   b(b(x1)) -> c(c(c(c(x1))))
27.19/7.82	   c(c(x1)) -> a(c(b(x1)))
27.19/7.82	
27.19/7.82	Q is empty.
27.19/7.82	We have to consider all minimal (P,Q,R)-chains.
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(7) DependencyGraphProof (EQUIVALENT)
27.19/7.82	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(8)
27.19/7.82	Obligation:
27.19/7.82	Q DP problem:
27.19/7.82	The TRS P consists of the following rules:
27.19/7.82	
27.19/7.82	   C(c(x1)) -> C(b(x1))
27.19/7.82	
27.19/7.82	The TRS R consists of the following rules:
27.19/7.82	
27.19/7.82	   a(a(x1)) -> x1
27.19/7.82	   b(b(x1)) -> c(c(c(c(x1))))
27.19/7.82	   c(c(x1)) -> a(c(b(x1)))
27.19/7.82	
27.19/7.82	Q is empty.
27.19/7.82	We have to consider all minimal (P,Q,R)-chains.
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(9) QDPOrderProof (EQUIVALENT)
27.19/7.82	We use the reduction pair processor [LPAR04,JAR06].
27.19/7.82	
27.19/7.82	
27.19/7.82	The following pairs can be oriented strictly and are deleted.
27.19/7.82	
27.19/7.82	   C(c(x1)) -> C(b(x1))
27.19/7.82	The remaining pairs can at least be oriented weakly.
27.19/7.82	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.19/7.82	
27.19/7.82	   <<<
27.19/7.82	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
27.19/7.82	>>>
27.19/7.82	
27.19/7.82	   <<<
27.19/7.82	 POL(c(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-I, -I, -I], [-I, -I, 1A], [1A, -I, 0A]] 	* 	x_1
27.19/7.82	>>>
27.19/7.82	
27.19/7.82	   <<<
27.19/7.82	 POL(b(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[1A, -I, 0A], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
27.19/7.82	>>>
27.19/7.82	
27.19/7.82	   <<<
27.19/7.82	 POL(a(x_1)) =  	[[-I], [-I], [1A]] 	 +  	[[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] 	* 	x_1
27.19/7.82	>>>
27.19/7.82	
27.19/7.82	
27.19/7.82	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.19/7.82	
27.19/7.82	   b(b(x1)) -> c(c(c(c(x1))))
27.19/7.82	   c(c(x1)) -> a(c(b(x1)))
27.19/7.82	   a(a(x1)) -> x1
27.19/7.82	
27.19/7.82	
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(10)
27.19/7.82	Obligation:
27.19/7.82	Q DP problem:
27.19/7.82	P is empty.
27.19/7.82	The TRS R consists of the following rules:
27.19/7.82	
27.19/7.82	   a(a(x1)) -> x1
27.19/7.82	   b(b(x1)) -> c(c(c(c(x1))))
27.19/7.82	   c(c(x1)) -> a(c(b(x1)))
27.19/7.82	
27.19/7.82	Q is empty.
27.19/7.82	We have to consider all minimal (P,Q,R)-chains.
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(11) PisEmptyProof (EQUIVALENT)
27.19/7.82	The TRS P is empty. Hence, there is no (P,Q,R) chain.
27.19/7.82	----------------------------------------
27.19/7.82	
27.19/7.82	(12)
27.19/7.82	YES
27.47/7.96	EOF