38.92/10.92	YES
38.92/10.93	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
38.92/10.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
38.92/10.93	
38.92/10.93	
38.92/10.93	Termination w.r.t. Q of the given QTRS could be proven:
38.92/10.93	
38.92/10.93	(0) QTRS
38.92/10.93	(1) QTRS Reverse [EQUIVALENT, 0 ms]
38.92/10.93	(2) QTRS
38.92/10.93	(3) DependencyPairsProof [EQUIVALENT, 12 ms]
38.92/10.93	(4) QDP
38.92/10.93	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
38.92/10.93	(6) QDP
38.92/10.93	(7) QDPOrderProof [EQUIVALENT, 126 ms]
38.92/10.93	(8) QDP
38.92/10.93	(9) QDPOrderProof [EQUIVALENT, 45 ms]
38.92/10.93	(10) QDP
38.92/10.93	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
38.92/10.93	(12) QDP
38.92/10.93	(13) QDPOrderProof [EQUIVALENT, 115 ms]
38.92/10.93	(14) QDP
38.92/10.93	(15) PisEmptyProof [EQUIVALENT, 0 ms]
38.92/10.93	(16) YES
38.92/10.93	
38.92/10.93	
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(0)
38.92/10.93	Obligation:
38.92/10.93	Q restricted rewrite system:
38.92/10.93	The TRS R consists of the following rules:
38.92/10.93	
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	   a(b(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> b(c(b(a(a(x1)))))
38.92/10.93	
38.92/10.93	Q is empty.
38.92/10.93	
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(1) QTRS Reverse (EQUIVALENT)
38.92/10.93	We applied the QTRS Reverse Processor [REVERSE].
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(2)
38.92/10.93	Obligation:
38.92/10.93	Q restricted rewrite system:
38.92/10.93	The TRS R consists of the following rules:
38.92/10.93	
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	
38.92/10.93	Q is empty.
38.92/10.93	
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(3) DependencyPairsProof (EQUIVALENT)
38.92/10.93	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(4)
38.92/10.93	Obligation:
38.92/10.93	Q DP problem:
38.92/10.93	The TRS P consists of the following rules:
38.92/10.93	
38.92/10.93	   B(a(x1)) -> C(x1)
38.92/10.93	   C(c(x1)) -> A(a(b(c(b(x1)))))
38.92/10.93	   C(c(x1)) -> A(b(c(b(x1))))
38.92/10.93	   C(c(x1)) -> B(c(b(x1)))
38.92/10.93	   C(c(x1)) -> C(b(x1))
38.92/10.93	   C(c(x1)) -> B(x1)
38.92/10.93	
38.92/10.93	The TRS R consists of the following rules:
38.92/10.93	
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	
38.92/10.93	Q is empty.
38.92/10.93	We have to consider all minimal (P,Q,R)-chains.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(5) DependencyGraphProof (EQUIVALENT)
38.92/10.93	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(6)
38.92/10.93	Obligation:
38.92/10.93	Q DP problem:
38.92/10.93	The TRS P consists of the following rules:
38.92/10.93	
38.92/10.93	   C(c(x1)) -> B(c(b(x1)))
38.92/10.93	   B(a(x1)) -> C(x1)
38.92/10.93	   C(c(x1)) -> C(b(x1))
38.92/10.93	   C(c(x1)) -> B(x1)
38.92/10.93	
38.92/10.93	The TRS R consists of the following rules:
38.92/10.93	
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	
38.92/10.93	Q is empty.
38.92/10.93	We have to consider all minimal (P,Q,R)-chains.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(7) QDPOrderProof (EQUIVALENT)
38.92/10.93	We use the reduction pair processor [LPAR04,JAR06].
38.92/10.93	
38.92/10.93	
38.92/10.93	The following pairs can be oriented strictly and are deleted.
38.92/10.93	
38.92/10.93	   C(c(x1)) -> B(c(b(x1)))
38.92/10.93	The remaining pairs can at least be oriented weakly.
38.92/10.93	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(c(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [0A, 0A, 0A]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(b(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, -I], [-I, -I, 0A], [0A, 0A, -I]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	
38.92/10.93	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.92/10.93	
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	
38.92/10.93	
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(8)
38.92/10.93	Obligation:
38.92/10.93	Q DP problem:
38.92/10.93	The TRS P consists of the following rules:
38.92/10.93	
38.92/10.93	   B(a(x1)) -> C(x1)
38.92/10.93	   C(c(x1)) -> C(b(x1))
38.92/10.93	   C(c(x1)) -> B(x1)
38.92/10.93	
38.92/10.93	The TRS R consists of the following rules:
38.92/10.93	
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	
38.92/10.93	Q is empty.
38.92/10.93	We have to consider all minimal (P,Q,R)-chains.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(9) QDPOrderProof (EQUIVALENT)
38.92/10.93	We use the reduction pair processor [LPAR04,JAR06].
38.92/10.93	
38.92/10.93	
38.92/10.93	The following pairs can be oriented strictly and are deleted.
38.92/10.93	
38.92/10.93	   B(a(x1)) -> C(x1)
38.92/10.93	The remaining pairs can at least be oriented weakly.
38.92/10.93	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 1A, 0A], [-I, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(C(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(c(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, 1A], [-I, 1A, 0A]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	
38.92/10.93	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.92/10.93	
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	
38.92/10.93	
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(10)
38.92/10.93	Obligation:
38.92/10.93	Q DP problem:
38.92/10.93	The TRS P consists of the following rules:
38.92/10.93	
38.92/10.93	   C(c(x1)) -> C(b(x1))
38.92/10.93	   C(c(x1)) -> B(x1)
38.92/10.93	
38.92/10.93	The TRS R consists of the following rules:
38.92/10.93	
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	
38.92/10.93	Q is empty.
38.92/10.93	We have to consider all minimal (P,Q,R)-chains.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(11) DependencyGraphProof (EQUIVALENT)
38.92/10.93	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(12)
38.92/10.93	Obligation:
38.92/10.93	Q DP problem:
38.92/10.93	The TRS P consists of the following rules:
38.92/10.93	
38.92/10.93	   C(c(x1)) -> C(b(x1))
38.92/10.93	
38.92/10.93	The TRS R consists of the following rules:
38.92/10.93	
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	
38.92/10.93	Q is empty.
38.92/10.93	We have to consider all minimal (P,Q,R)-chains.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(13) QDPOrderProof (EQUIVALENT)
38.92/10.93	We use the reduction pair processor [LPAR04,JAR06].
38.92/10.93	
38.92/10.93	
38.92/10.93	The following pairs can be oriented strictly and are deleted.
38.92/10.93	
38.92/10.93	   C(c(x1)) -> C(b(x1))
38.92/10.93	The remaining pairs can at least be oriented weakly.
38.92/10.93	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [1A, 0A, -I], [0A, -I, 0A]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, -I, 0A], [-I, 0A, -I]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	   <<<
38.92/10.93	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [1A, 0A, -I]] 	* 	x_1
38.92/10.93	>>>
38.92/10.93	
38.92/10.93	
38.92/10.93	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
38.92/10.93	
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	
38.92/10.93	
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(14)
38.92/10.93	Obligation:
38.92/10.93	Q DP problem:
38.92/10.93	P is empty.
38.92/10.93	The TRS R consists of the following rules:
38.92/10.93	
38.92/10.93	   a(a(x1)) -> x1
38.92/10.93	   b(a(x1)) -> c(x1)
38.92/10.93	   c(c(x1)) -> a(a(b(c(b(x1)))))
38.92/10.93	
38.92/10.93	Q is empty.
38.92/10.93	We have to consider all minimal (P,Q,R)-chains.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(15) PisEmptyProof (EQUIVALENT)
38.92/10.93	The TRS P is empty. Hence, there is no (P,Q,R) chain.
38.92/10.93	----------------------------------------
38.92/10.93	
38.92/10.93	(16)
38.92/10.93	YES
39.17/10.99	EOF