33.03/9.27	YES
33.57/9.46	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
33.57/9.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.57/9.46	
33.57/9.46	
33.57/9.46	Termination w.r.t. Q of the given QTRS could be proven:
33.57/9.46	
33.57/9.46	(0) QTRS
33.57/9.46	(1) QTRS Reverse [EQUIVALENT, 0 ms]
33.57/9.46	(2) QTRS
33.57/9.46	(3) DependencyPairsProof [EQUIVALENT, 20 ms]
33.57/9.46	(4) QDP
33.57/9.46	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
33.57/9.46	(6) QDP
33.57/9.46	(7) QDPOrderProof [EQUIVALENT, 187 ms]
33.57/9.46	(8) QDP
33.57/9.46	(9) QDPOrderProof [EQUIVALENT, 0 ms]
33.57/9.46	(10) QDP
33.57/9.46	(11) QDPOrderProof [EQUIVALENT, 31 ms]
33.57/9.46	(12) QDP
33.57/9.46	(13) PisEmptyProof [EQUIVALENT, 0 ms]
33.57/9.46	(14) YES
33.57/9.46	
33.57/9.46	
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(0)
33.57/9.46	Obligation:
33.57/9.46	Q restricted rewrite system:
33.57/9.46	The TRS R consists of the following rules:
33.57/9.46	
33.57/9.46	   a(x1) -> x1
33.57/9.46	   a(b(x1)) -> x1
33.57/9.46	   a(c(c(x1))) -> c(c(b(a(c(a(x1))))))
33.57/9.46	
33.57/9.46	Q is empty.
33.57/9.46	
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(1) QTRS Reverse (EQUIVALENT)
33.57/9.46	We applied the QTRS Reverse Processor [REVERSE].
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(2)
33.57/9.46	Obligation:
33.57/9.46	Q restricted rewrite system:
33.57/9.46	The TRS R consists of the following rules:
33.57/9.46	
33.57/9.46	   a(x1) -> x1
33.57/9.46	   b(a(x1)) -> x1
33.57/9.46	   c(c(a(x1))) -> a(c(a(b(c(c(x1))))))
33.57/9.46	
33.57/9.46	Q is empty.
33.57/9.46	
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(3) DependencyPairsProof (EQUIVALENT)
33.57/9.46	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(4)
33.57/9.46	Obligation:
33.57/9.46	Q DP problem:
33.57/9.46	The TRS P consists of the following rules:
33.57/9.46	
33.57/9.46	   C(c(a(x1))) -> A(c(a(b(c(c(x1))))))
33.57/9.46	   C(c(a(x1))) -> C(a(b(c(c(x1)))))
33.57/9.46	   C(c(a(x1))) -> A(b(c(c(x1))))
33.57/9.46	   C(c(a(x1))) -> B(c(c(x1)))
33.57/9.46	   C(c(a(x1))) -> C(c(x1))
33.57/9.46	   C(c(a(x1))) -> C(x1)
33.57/9.46	
33.57/9.46	The TRS R consists of the following rules:
33.57/9.46	
33.57/9.46	   a(x1) -> x1
33.57/9.46	   b(a(x1)) -> x1
33.57/9.46	   c(c(a(x1))) -> a(c(a(b(c(c(x1))))))
33.57/9.46	
33.57/9.46	Q is empty.
33.57/9.46	We have to consider all minimal (P,Q,R)-chains.
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(5) DependencyGraphProof (EQUIVALENT)
33.57/9.46	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(6)
33.57/9.46	Obligation:
33.57/9.46	Q DP problem:
33.57/9.46	The TRS P consists of the following rules:
33.57/9.46	
33.57/9.46	   C(c(a(x1))) -> C(c(x1))
33.57/9.46	   C(c(a(x1))) -> C(a(b(c(c(x1)))))
33.57/9.46	   C(c(a(x1))) -> C(x1)
33.57/9.46	
33.57/9.46	The TRS R consists of the following rules:
33.57/9.46	
33.57/9.46	   a(x1) -> x1
33.57/9.46	   b(a(x1)) -> x1
33.57/9.46	   c(c(a(x1))) -> a(c(a(b(c(c(x1))))))
33.57/9.46	
33.57/9.46	Q is empty.
33.57/9.46	We have to consider all minimal (P,Q,R)-chains.
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(7) QDPOrderProof (EQUIVALENT)
33.57/9.46	We use the reduction pair processor [LPAR04,JAR06].
33.57/9.46	
33.57/9.46	
33.57/9.46	The following pairs can be oriented strictly and are deleted.
33.57/9.46	
33.57/9.46	   C(c(a(x1))) -> C(x1)
33.57/9.46	The remaining pairs can at least be oriented weakly.
33.57/9.46	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(C(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, 1A], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(a(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[1A, 1A, 1A], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	
33.57/9.46	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.57/9.46	
33.57/9.46	   c(c(a(x1))) -> a(c(a(b(c(c(x1))))))
33.57/9.46	   b(a(x1)) -> x1
33.57/9.46	   a(x1) -> x1
33.57/9.46	
33.57/9.46	
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(8)
33.57/9.46	Obligation:
33.57/9.46	Q DP problem:
33.57/9.46	The TRS P consists of the following rules:
33.57/9.46	
33.57/9.46	   C(c(a(x1))) -> C(c(x1))
33.57/9.46	   C(c(a(x1))) -> C(a(b(c(c(x1)))))
33.57/9.46	
33.57/9.46	The TRS R consists of the following rules:
33.57/9.46	
33.57/9.46	   a(x1) -> x1
33.57/9.46	   b(a(x1)) -> x1
33.57/9.46	   c(c(a(x1))) -> a(c(a(b(c(c(x1))))))
33.57/9.46	
33.57/9.46	Q is empty.
33.57/9.46	We have to consider all minimal (P,Q,R)-chains.
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(9) QDPOrderProof (EQUIVALENT)
33.57/9.46	We use the reduction pair processor [LPAR04,JAR06].
33.57/9.46	
33.57/9.46	
33.57/9.46	The following pairs can be oriented strictly and are deleted.
33.57/9.46	
33.57/9.46	   C(c(a(x1))) -> C(a(b(c(c(x1)))))
33.57/9.46	The remaining pairs can at least be oriented weakly.
33.57/9.46	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [0A, -I, 0A], [-I, 0A, -I]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(a(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 1A], [-I, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	
33.57/9.46	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.57/9.46	
33.57/9.46	   c(c(a(x1))) -> a(c(a(b(c(c(x1))))))
33.57/9.46	   b(a(x1)) -> x1
33.57/9.46	   a(x1) -> x1
33.57/9.46	
33.57/9.46	
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(10)
33.57/9.46	Obligation:
33.57/9.46	Q DP problem:
33.57/9.46	The TRS P consists of the following rules:
33.57/9.46	
33.57/9.46	   C(c(a(x1))) -> C(c(x1))
33.57/9.46	
33.57/9.46	The TRS R consists of the following rules:
33.57/9.46	
33.57/9.46	   a(x1) -> x1
33.57/9.46	   b(a(x1)) -> x1
33.57/9.46	   c(c(a(x1))) -> a(c(a(b(c(c(x1))))))
33.57/9.46	
33.57/9.46	Q is empty.
33.57/9.46	We have to consider all minimal (P,Q,R)-chains.
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(11) QDPOrderProof (EQUIVALENT)
33.57/9.46	We use the reduction pair processor [LPAR04,JAR06].
33.57/9.46	
33.57/9.46	
33.57/9.46	The following pairs can be oriented strictly and are deleted.
33.57/9.46	
33.57/9.46	   C(c(a(x1))) -> C(c(x1))
33.57/9.46	The remaining pairs can at least be oriented weakly.
33.57/9.46	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [1A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(a(x_1)) =  	[[-I], [1A], [0A]] 	 +  	[[0A, 0A, -I], [1A, 1A, 1A], [0A, 0A, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	   <<<
33.57/9.46	 POL(b(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
33.57/9.46	>>>
33.57/9.46	
33.57/9.46	
33.57/9.46	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.57/9.46	
33.57/9.46	   c(c(a(x1))) -> a(c(a(b(c(c(x1))))))
33.57/9.46	   b(a(x1)) -> x1
33.57/9.46	   a(x1) -> x1
33.57/9.46	
33.57/9.46	
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(12)
33.57/9.46	Obligation:
33.57/9.46	Q DP problem:
33.57/9.46	P is empty.
33.57/9.46	The TRS R consists of the following rules:
33.57/9.46	
33.57/9.46	   a(x1) -> x1
33.57/9.46	   b(a(x1)) -> x1
33.57/9.46	   c(c(a(x1))) -> a(c(a(b(c(c(x1))))))
33.57/9.46	
33.57/9.46	Q is empty.
33.57/9.46	We have to consider all minimal (P,Q,R)-chains.
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(13) PisEmptyProof (EQUIVALENT)
33.57/9.46	The TRS P is empty. Hence, there is no (P,Q,R) chain.
33.57/9.46	----------------------------------------
33.57/9.46	
33.57/9.46	(14)
33.57/9.46	YES
34.05/9.60	EOF