24.52/7.08	YES
25.65/7.38	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
25.65/7.38	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
25.65/7.38	
25.65/7.38	
25.65/7.38	Termination w.r.t. Q of the given QTRS could be proven:
25.65/7.38	
25.65/7.38	(0) QTRS
25.65/7.38	(1) QTRS Reverse [EQUIVALENT, 0 ms]
25.65/7.38	(2) QTRS
25.65/7.38	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
25.65/7.38	(4) QDP
25.65/7.38	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
25.65/7.38	(6) QDP
25.65/7.38	(7) QDPOrderProof [EQUIVALENT, 143 ms]
25.65/7.38	(8) QDP
25.65/7.38	(9) QDPOrderProof [EQUIVALENT, 0 ms]
25.65/7.38	(10) QDP
25.65/7.38	(11) PisEmptyProof [EQUIVALENT, 0 ms]
25.65/7.38	(12) YES
25.65/7.38	
25.65/7.38	
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(0)
25.65/7.38	Obligation:
25.65/7.38	Q restricted rewrite system:
25.65/7.38	The TRS R consists of the following rules:
25.65/7.38	
25.65/7.38	   a(x1) -> x1
25.65/7.38	   a(b(x1)) -> x1
25.65/7.38	   a(c(c(x1))) -> c(b(c(a(c(a(x1))))))
25.65/7.38	
25.65/7.38	Q is empty.
25.65/7.38	
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(1) QTRS Reverse (EQUIVALENT)
25.65/7.38	We applied the QTRS Reverse Processor [REVERSE].
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(2)
25.65/7.38	Obligation:
25.65/7.38	Q restricted rewrite system:
25.65/7.38	The TRS R consists of the following rules:
25.65/7.38	
25.65/7.38	   a(x1) -> x1
25.65/7.38	   b(a(x1)) -> x1
25.65/7.38	   c(c(a(x1))) -> a(c(a(c(b(c(x1))))))
25.65/7.38	
25.65/7.38	Q is empty.
25.65/7.38	
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(3) DependencyPairsProof (EQUIVALENT)
25.65/7.38	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(4)
25.65/7.38	Obligation:
25.65/7.38	Q DP problem:
25.65/7.38	The TRS P consists of the following rules:
25.65/7.38	
25.65/7.38	   C(c(a(x1))) -> A(c(a(c(b(c(x1))))))
25.65/7.38	   C(c(a(x1))) -> C(a(c(b(c(x1)))))
25.65/7.38	   C(c(a(x1))) -> A(c(b(c(x1))))
25.65/7.38	   C(c(a(x1))) -> C(b(c(x1)))
25.65/7.38	   C(c(a(x1))) -> B(c(x1))
25.65/7.38	   C(c(a(x1))) -> C(x1)
25.65/7.38	
25.65/7.38	The TRS R consists of the following rules:
25.65/7.38	
25.65/7.38	   a(x1) -> x1
25.65/7.38	   b(a(x1)) -> x1
25.65/7.38	   c(c(a(x1))) -> a(c(a(c(b(c(x1))))))
25.65/7.38	
25.65/7.38	Q is empty.
25.65/7.38	We have to consider all minimal (P,Q,R)-chains.
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(5) DependencyGraphProof (EQUIVALENT)
25.65/7.38	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(6)
25.65/7.38	Obligation:
25.65/7.38	Q DP problem:
25.65/7.38	The TRS P consists of the following rules:
25.65/7.38	
25.65/7.38	   C(c(a(x1))) -> C(b(c(x1)))
25.65/7.38	   C(c(a(x1))) -> C(a(c(b(c(x1)))))
25.65/7.38	   C(c(a(x1))) -> C(x1)
25.65/7.38	
25.65/7.38	The TRS R consists of the following rules:
25.65/7.38	
25.65/7.38	   a(x1) -> x1
25.65/7.38	   b(a(x1)) -> x1
25.65/7.38	   c(c(a(x1))) -> a(c(a(c(b(c(x1))))))
25.65/7.38	
25.65/7.38	Q is empty.
25.65/7.38	We have to consider all minimal (P,Q,R)-chains.
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(7) QDPOrderProof (EQUIVALENT)
25.65/7.38	We use the reduction pair processor [LPAR04,JAR06].
25.65/7.38	
25.65/7.38	
25.65/7.38	The following pairs can be oriented strictly and are deleted.
25.65/7.38	
25.65/7.38	   C(c(a(x1))) -> C(b(c(x1)))
25.65/7.38	The remaining pairs can at least be oriented weakly.
25.65/7.38	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
25.65/7.38	
25.65/7.38	   <<<
25.65/7.38	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
25.65/7.38	>>>
25.65/7.38	
25.65/7.38	   <<<
25.65/7.38	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
25.65/7.38	>>>
25.65/7.38	
25.65/7.38	   <<<
25.65/7.38	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
25.65/7.38	>>>
25.65/7.38	
25.65/7.38	   <<<
25.65/7.38	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
25.65/7.38	>>>
25.65/7.38	
25.65/7.38	
25.65/7.38	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.65/7.38	
25.65/7.38	   c(c(a(x1))) -> a(c(a(c(b(c(x1))))))
25.65/7.38	   b(a(x1)) -> x1
25.65/7.38	   a(x1) -> x1
25.65/7.38	
25.65/7.38	
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(8)
25.65/7.38	Obligation:
25.65/7.38	Q DP problem:
25.65/7.38	The TRS P consists of the following rules:
25.65/7.38	
25.65/7.38	   C(c(a(x1))) -> C(a(c(b(c(x1)))))
25.65/7.38	   C(c(a(x1))) -> C(x1)
25.65/7.38	
25.65/7.38	The TRS R consists of the following rules:
25.65/7.38	
25.65/7.38	   a(x1) -> x1
25.65/7.38	   b(a(x1)) -> x1
25.65/7.38	   c(c(a(x1))) -> a(c(a(c(b(c(x1))))))
25.65/7.38	
25.65/7.38	Q is empty.
25.65/7.38	We have to consider all minimal (P,Q,R)-chains.
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(9) QDPOrderProof (EQUIVALENT)
25.65/7.38	We use the reduction pair processor [LPAR04,JAR06].
25.65/7.38	
25.65/7.38	
25.65/7.38	The following pairs can be oriented strictly and are deleted.
25.65/7.38	
25.65/7.38	   C(c(a(x1))) -> C(a(c(b(c(x1)))))
25.65/7.38	   C(c(a(x1))) -> C(x1)
25.65/7.38	The remaining pairs can at least be oriented weakly.
25.65/7.38	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
25.65/7.38	
25.65/7.38	   <<<
25.65/7.38	 POL(C(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
25.65/7.38	>>>
25.65/7.38	
25.65/7.38	   <<<
25.65/7.38	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [-I, 0A, 0A], [0A, 1A, -I]] 	* 	x_1
25.65/7.38	>>>
25.65/7.38	
25.65/7.38	   <<<
25.65/7.38	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
25.65/7.38	>>>
25.65/7.38	
25.65/7.38	   <<<
25.65/7.38	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
25.65/7.38	>>>
25.65/7.38	
25.65/7.38	
25.65/7.38	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.65/7.38	
25.65/7.38	   c(c(a(x1))) -> a(c(a(c(b(c(x1))))))
25.65/7.38	   b(a(x1)) -> x1
25.65/7.38	   a(x1) -> x1
25.65/7.38	
25.65/7.38	
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(10)
25.65/7.38	Obligation:
25.65/7.38	Q DP problem:
25.65/7.38	P is empty.
25.65/7.38	The TRS R consists of the following rules:
25.65/7.38	
25.65/7.38	   a(x1) -> x1
25.65/7.38	   b(a(x1)) -> x1
25.65/7.38	   c(c(a(x1))) -> a(c(a(c(b(c(x1))))))
25.65/7.38	
25.65/7.38	Q is empty.
25.65/7.38	We have to consider all minimal (P,Q,R)-chains.
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(11) PisEmptyProof (EQUIVALENT)
25.65/7.38	The TRS P is empty. Hence, there is no (P,Q,R) chain.
25.65/7.38	----------------------------------------
25.65/7.38	
25.65/7.38	(12)
25.65/7.38	YES
25.88/7.45	EOF