26.81/7.79	YES
27.73/8.03	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
27.73/8.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.73/8.03	
27.73/8.03	
27.73/8.03	Termination w.r.t. Q of the given QTRS could be proven:
27.73/8.03	
27.73/8.03	(0) QTRS
27.73/8.03	(1) QTRS Reverse [EQUIVALENT, 0 ms]
27.73/8.03	(2) QTRS
27.73/8.03	(3) DependencyPairsProof [EQUIVALENT, 17 ms]
27.73/8.03	(4) QDP
27.73/8.03	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
27.73/8.03	(6) QDP
27.73/8.03	(7) QDPOrderProof [EQUIVALENT, 55 ms]
27.73/8.03	(8) QDP
27.73/8.03	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
27.73/8.03	(10) QDP
27.73/8.03	(11) QDPOrderProof [EQUIVALENT, 37 ms]
27.73/8.03	(12) QDP
27.73/8.03	(13) PisEmptyProof [EQUIVALENT, 0 ms]
27.73/8.03	(14) YES
27.73/8.03	
27.73/8.03	
27.73/8.03	----------------------------------------
27.73/8.03	
27.73/8.03	(0)
27.73/8.03	Obligation:
27.73/8.03	Q restricted rewrite system:
27.73/8.03	The TRS R consists of the following rules:
27.73/8.03	
27.73/8.03	   a(x1) -> x1
27.73/8.03	   a(b(x1)) -> c(a(x1))
27.73/8.03	   c(c(x1)) -> b(c(b(a(x1))))
27.73/8.03	
27.73/8.03	Q is empty.
27.73/8.03	
27.73/8.03	----------------------------------------
27.73/8.03	
27.73/8.03	(1) QTRS Reverse (EQUIVALENT)
27.73/8.03	We applied the QTRS Reverse Processor [REVERSE].
27.73/8.03	----------------------------------------
27.73/8.03	
27.73/8.03	(2)
27.73/8.03	Obligation:
27.73/8.03	Q restricted rewrite system:
27.73/8.03	The TRS R consists of the following rules:
27.73/8.03	
27.73/8.03	   a(x1) -> x1
27.73/8.03	   b(a(x1)) -> a(c(x1))
27.73/8.03	   c(c(x1)) -> a(b(c(b(x1))))
27.73/8.03	
27.73/8.03	Q is empty.
27.73/8.03	
27.73/8.03	----------------------------------------
27.73/8.03	
27.73/8.03	(3) DependencyPairsProof (EQUIVALENT)
27.73/8.03	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.73/8.03	----------------------------------------
27.73/8.03	
27.73/8.03	(4)
27.73/8.03	Obligation:
27.73/8.03	Q DP problem:
27.73/8.03	The TRS P consists of the following rules:
27.73/8.03	
27.73/8.03	   B(a(x1)) -> A(c(x1))
27.73/8.03	   B(a(x1)) -> C(x1)
27.73/8.03	   C(c(x1)) -> A(b(c(b(x1))))
27.73/8.03	   C(c(x1)) -> B(c(b(x1)))
27.73/8.03	   C(c(x1)) -> C(b(x1))
27.73/8.03	   C(c(x1)) -> B(x1)
27.73/8.03	
27.73/8.03	The TRS R consists of the following rules:
27.73/8.03	
27.73/8.03	   a(x1) -> x1
27.73/8.03	   b(a(x1)) -> a(c(x1))
27.73/8.03	   c(c(x1)) -> a(b(c(b(x1))))
27.73/8.03	
27.73/8.03	Q is empty.
27.73/8.03	We have to consider all minimal (P,Q,R)-chains.
27.73/8.03	----------------------------------------
27.73/8.03	
27.73/8.03	(5) DependencyGraphProof (EQUIVALENT)
27.73/8.03	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
27.73/8.03	----------------------------------------
27.73/8.03	
27.73/8.03	(6)
27.73/8.03	Obligation:
27.73/8.03	Q DP problem:
27.73/8.03	The TRS P consists of the following rules:
27.73/8.03	
27.73/8.03	   B(a(x1)) -> C(x1)
27.73/8.03	   C(c(x1)) -> B(c(b(x1)))
27.73/8.04	   C(c(x1)) -> C(b(x1))
27.73/8.04	   C(c(x1)) -> B(x1)
27.73/8.04	
27.73/8.04	The TRS R consists of the following rules:
27.73/8.04	
27.73/8.04	   a(x1) -> x1
27.73/8.04	   b(a(x1)) -> a(c(x1))
27.73/8.04	   c(c(x1)) -> a(b(c(b(x1))))
27.73/8.04	
27.73/8.04	Q is empty.
27.73/8.04	We have to consider all minimal (P,Q,R)-chains.
27.73/8.04	----------------------------------------
27.73/8.04	
27.73/8.04	(7) QDPOrderProof (EQUIVALENT)
27.73/8.04	We use the reduction pair processor [LPAR04,JAR06].
27.73/8.04	
27.73/8.04	
27.73/8.04	The following pairs can be oriented strictly and are deleted.
27.73/8.04	
27.73/8.04	   C(c(x1)) -> B(c(b(x1)))
27.73/8.04	   C(c(x1)) -> B(x1)
27.73/8.04	The remaining pairs can at least be oriented weakly.
27.73/8.04	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.73/8.04	
27.73/8.04	   <<<
27.73/8.04	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
27.73/8.04	>>>
27.73/8.04	
27.73/8.04	   <<<
27.73/8.04	 POL(a(x_1)) =  	[[0A], [1A], [1A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 1A], [0A, 0A, 1A]] 	* 	x_1
27.73/8.04	>>>
27.73/8.04	
27.73/8.04	   <<<
27.73/8.04	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
27.73/8.04	>>>
27.73/8.04	
27.73/8.04	   <<<
27.73/8.04	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
27.73/8.04	>>>
27.73/8.04	
27.73/8.04	   <<<
27.73/8.04	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, 1A, 0A], [-I, 0A, -I]] 	* 	x_1
27.73/8.04	>>>
27.73/8.04	
27.73/8.04	
27.73/8.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.73/8.04	
27.73/8.04	   b(a(x1)) -> a(c(x1))
27.73/8.04	   c(c(x1)) -> a(b(c(b(x1))))
27.73/8.04	   a(x1) -> x1
27.73/8.04	
27.73/8.04	
27.73/8.04	----------------------------------------
27.73/8.04	
27.73/8.04	(8)
27.73/8.04	Obligation:
27.73/8.04	Q DP problem:
27.73/8.04	The TRS P consists of the following rules:
27.73/8.04	
27.73/8.04	   B(a(x1)) -> C(x1)
27.73/8.04	   C(c(x1)) -> C(b(x1))
27.73/8.04	
27.73/8.04	The TRS R consists of the following rules:
27.73/8.04	
27.73/8.04	   a(x1) -> x1
27.73/8.04	   b(a(x1)) -> a(c(x1))
27.73/8.04	   c(c(x1)) -> a(b(c(b(x1))))
27.73/8.04	
27.73/8.04	Q is empty.
27.73/8.04	We have to consider all minimal (P,Q,R)-chains.
27.73/8.04	----------------------------------------
27.73/8.04	
27.73/8.04	(9) DependencyGraphProof (EQUIVALENT)
27.73/8.04	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
27.73/8.04	----------------------------------------
27.73/8.04	
27.73/8.04	(10)
27.73/8.04	Obligation:
27.73/8.04	Q DP problem:
27.73/8.04	The TRS P consists of the following rules:
27.73/8.04	
27.73/8.04	   C(c(x1)) -> C(b(x1))
27.73/8.04	
27.73/8.04	The TRS R consists of the following rules:
27.73/8.04	
27.73/8.04	   a(x1) -> x1
27.73/8.04	   b(a(x1)) -> a(c(x1))
27.73/8.04	   c(c(x1)) -> a(b(c(b(x1))))
27.73/8.04	
27.73/8.04	Q is empty.
27.73/8.04	We have to consider all minimal (P,Q,R)-chains.
27.73/8.04	----------------------------------------
27.73/8.04	
27.73/8.04	(11) QDPOrderProof (EQUIVALENT)
27.73/8.04	We use the reduction pair processor [LPAR04,JAR06].
27.73/8.04	
27.73/8.04	
27.73/8.04	The following pairs can be oriented strictly and are deleted.
27.73/8.04	
27.73/8.04	   C(c(x1)) -> C(b(x1))
27.73/8.04	The remaining pairs can at least be oriented weakly.
27.73/8.04	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.73/8.04	
27.73/8.04	   <<<
27.73/8.04	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
27.73/8.04	>>>
27.73/8.04	
27.73/8.04	   <<<
27.73/8.04	 POL(c(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 0A, 1A], [-I, 0A, -I]] 	* 	x_1
27.73/8.04	>>>
27.73/8.04	
27.73/8.04	   <<<
27.73/8.04	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 1A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
27.73/8.04	>>>
27.73/8.04	
27.73/8.04	   <<<
27.73/8.04	 POL(a(x_1)) =  	[[1A], [0A], [1A]] 	 +  	[[0A, 1A, 1A], [0A, 0A, 0A], [0A, 0A, 1A]] 	* 	x_1
27.73/8.04	>>>
27.73/8.04	
27.73/8.04	
27.73/8.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.73/8.04	
27.73/8.04	   b(a(x1)) -> a(c(x1))
27.73/8.04	   c(c(x1)) -> a(b(c(b(x1))))
27.73/8.04	   a(x1) -> x1
27.73/8.04	
27.73/8.04	
27.73/8.04	----------------------------------------
27.73/8.04	
27.73/8.04	(12)
27.73/8.04	Obligation:
27.73/8.04	Q DP problem:
27.73/8.04	P is empty.
27.73/8.04	The TRS R consists of the following rules:
27.73/8.04	
27.73/8.04	   a(x1) -> x1
27.73/8.04	   b(a(x1)) -> a(c(x1))
27.73/8.04	   c(c(x1)) -> a(b(c(b(x1))))
27.73/8.04	
27.73/8.04	Q is empty.
27.73/8.04	We have to consider all minimal (P,Q,R)-chains.
27.73/8.04	----------------------------------------
27.73/8.04	
27.73/8.04	(13) PisEmptyProof (EQUIVALENT)
27.73/8.04	The TRS P is empty. Hence, there is no (P,Q,R) chain.
27.73/8.04	----------------------------------------
27.73/8.04	
27.73/8.04	(14)
27.73/8.04	YES
28.05/8.12	EOF