45.07/12.41	YES
45.53/12.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
45.53/12.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
45.53/12.45	
45.53/12.45	
45.53/12.45	Termination w.r.t. Q of the given QTRS could be proven:
45.53/12.45	
45.53/12.45	(0) QTRS
45.53/12.45	(1) QTRS Reverse [EQUIVALENT, 0 ms]
45.53/12.45	(2) QTRS
45.53/12.45	(3) DependencyPairsProof [EQUIVALENT, 10 ms]
45.53/12.45	(4) QDP
45.53/12.45	(5) DependencyGraphProof [EQUIVALENT, 1 ms]
45.53/12.45	(6) QDP
45.53/12.45	(7) QDPOrderProof [EQUIVALENT, 174 ms]
45.53/12.45	(8) QDP
45.53/12.45	(9) QDPOrderProof [EQUIVALENT, 35 ms]
45.53/12.45	(10) QDP
45.53/12.45	(11) PisEmptyProof [EQUIVALENT, 0 ms]
45.53/12.45	(12) YES
45.53/12.45	
45.53/12.45	
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(0)
45.53/12.45	Obligation:
45.53/12.45	Q restricted rewrite system:
45.53/12.45	The TRS R consists of the following rules:
45.53/12.45	
45.53/12.45	   a(x1) -> b(x1)
45.53/12.45	   b(a(c(x1))) -> c(c(b(a(a(x1)))))
45.53/12.45	   c(b(x1)) -> x1
45.53/12.45	
45.53/12.45	Q is empty.
45.53/12.45	
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(1) QTRS Reverse (EQUIVALENT)
45.53/12.45	We applied the QTRS Reverse Processor [REVERSE].
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(2)
45.53/12.45	Obligation:
45.53/12.45	Q restricted rewrite system:
45.53/12.45	The TRS R consists of the following rules:
45.53/12.45	
45.53/12.45	   a(x1) -> b(x1)
45.53/12.45	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
45.53/12.45	   b(c(x1)) -> x1
45.53/12.45	
45.53/12.45	Q is empty.
45.53/12.45	
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(3) DependencyPairsProof (EQUIVALENT)
45.53/12.45	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(4)
45.53/12.45	Obligation:
45.53/12.45	Q DP problem:
45.53/12.45	The TRS P consists of the following rules:
45.53/12.45	
45.53/12.45	   A(x1) -> B(x1)
45.53/12.45	   C(a(b(x1))) -> A(a(b(c(c(x1)))))
45.53/12.45	   C(a(b(x1))) -> A(b(c(c(x1))))
45.53/12.45	   C(a(b(x1))) -> B(c(c(x1)))
45.53/12.45	   C(a(b(x1))) -> C(c(x1))
45.53/12.45	   C(a(b(x1))) -> C(x1)
45.53/12.45	
45.53/12.45	The TRS R consists of the following rules:
45.53/12.45	
45.53/12.45	   a(x1) -> b(x1)
45.53/12.45	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
45.53/12.45	   b(c(x1)) -> x1
45.53/12.45	
45.53/12.45	Q is empty.
45.53/12.45	We have to consider all minimal (P,Q,R)-chains.
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(5) DependencyGraphProof (EQUIVALENT)
45.53/12.45	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(6)
45.53/12.45	Obligation:
45.53/12.45	Q DP problem:
45.53/12.45	The TRS P consists of the following rules:
45.53/12.45	
45.53/12.45	   C(a(b(x1))) -> C(x1)
45.53/12.45	   C(a(b(x1))) -> C(c(x1))
45.53/12.45	
45.53/12.45	The TRS R consists of the following rules:
45.53/12.45	
45.53/12.45	   a(x1) -> b(x1)
45.53/12.45	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
45.53/12.45	   b(c(x1)) -> x1
45.53/12.45	
45.53/12.45	Q is empty.
45.53/12.45	We have to consider all minimal (P,Q,R)-chains.
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(7) QDPOrderProof (EQUIVALENT)
45.53/12.45	We use the reduction pair processor [LPAR04,JAR06].
45.53/12.45	
45.53/12.45	
45.53/12.45	The following pairs can be oriented strictly and are deleted.
45.53/12.45	
45.53/12.45	   C(a(b(x1))) -> C(x1)
45.53/12.45	The remaining pairs can at least be oriented weakly.
45.53/12.45	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
45.53/12.45	
45.53/12.45	   <<<
45.53/12.45	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
45.53/12.45	>>>
45.53/12.45	
45.53/12.45	   <<<
45.53/12.45	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
45.53/12.45	>>>
45.53/12.45	
45.53/12.45	   <<<
45.53/12.45	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
45.53/12.45	>>>
45.53/12.45	
45.53/12.45	   <<<
45.53/12.45	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
45.53/12.45	>>>
45.53/12.45	
45.53/12.45	
45.53/12.45	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
45.53/12.45	
45.53/12.45	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
45.53/12.45	   b(c(x1)) -> x1
45.53/12.45	   a(x1) -> b(x1)
45.53/12.45	
45.53/12.45	
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(8)
45.53/12.45	Obligation:
45.53/12.45	Q DP problem:
45.53/12.45	The TRS P consists of the following rules:
45.53/12.45	
45.53/12.45	   C(a(b(x1))) -> C(c(x1))
45.53/12.45	
45.53/12.45	The TRS R consists of the following rules:
45.53/12.45	
45.53/12.45	   a(x1) -> b(x1)
45.53/12.45	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
45.53/12.45	   b(c(x1)) -> x1
45.53/12.45	
45.53/12.45	Q is empty.
45.53/12.45	We have to consider all minimal (P,Q,R)-chains.
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(9) QDPOrderProof (EQUIVALENT)
45.53/12.45	We use the reduction pair processor [LPAR04,JAR06].
45.53/12.45	
45.53/12.45	
45.53/12.45	The following pairs can be oriented strictly and are deleted.
45.53/12.45	
45.53/12.45	   C(a(b(x1))) -> C(c(x1))
45.53/12.45	The remaining pairs can at least be oriented weakly.
45.53/12.45	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
45.53/12.45	
45.53/12.45	   <<<
45.53/12.45	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
45.53/12.45	>>>
45.53/12.45	
45.53/12.45	   <<<
45.53/12.45	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [1A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
45.53/12.45	>>>
45.53/12.45	
45.53/12.45	   <<<
45.53/12.45	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
45.53/12.45	>>>
45.53/12.45	
45.53/12.45	   <<<
45.53/12.45	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
45.53/12.45	>>>
45.53/12.45	
45.53/12.45	
45.53/12.45	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
45.53/12.45	
45.53/12.45	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
45.53/12.45	   b(c(x1)) -> x1
45.53/12.45	   a(x1) -> b(x1)
45.53/12.45	
45.53/12.45	
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(10)
45.53/12.45	Obligation:
45.53/12.45	Q DP problem:
45.53/12.45	P is empty.
45.53/12.45	The TRS R consists of the following rules:
45.53/12.45	
45.53/12.45	   a(x1) -> b(x1)
45.53/12.45	   c(a(b(x1))) -> a(a(b(c(c(x1)))))
45.53/12.45	   b(c(x1)) -> x1
45.53/12.45	
45.53/12.45	Q is empty.
45.53/12.45	We have to consider all minimal (P,Q,R)-chains.
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(11) PisEmptyProof (EQUIVALENT)
45.53/12.45	The TRS P is empty. Hence, there is no (P,Q,R) chain.
45.53/12.45	----------------------------------------
45.53/12.45	
45.53/12.45	(12)
45.53/12.45	YES
45.79/12.58	EOF