43.21/11.94	YES
43.50/12.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
43.50/12.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
43.50/12.04	
43.50/12.04	
43.50/12.04	Termination w.r.t. Q of the given QTRS could be proven:
43.50/12.04	
43.50/12.04	(0) QTRS
43.50/12.04	(1) DependencyPairsProof [EQUIVALENT, 22 ms]
43.50/12.04	(2) QDP
43.50/12.04	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
43.50/12.04	(4) QDP
43.50/12.04	(5) QDPOrderProof [EQUIVALENT, 64 ms]
43.50/12.04	(6) QDP
43.50/12.04	(7) PisEmptyProof [EQUIVALENT, 0 ms]
43.50/12.04	(8) YES
43.50/12.04	
43.50/12.04	
43.50/12.04	----------------------------------------
43.50/12.04	
43.50/12.04	(0)
43.50/12.04	Obligation:
43.50/12.04	Q restricted rewrite system:
43.50/12.04	The TRS R consists of the following rules:
43.50/12.04	
43.50/12.04	   a(x1) -> b(x1)
43.50/12.04	   a(b(x1)) -> b(a(c(a(x1))))
43.50/12.04	   b(x1) -> c(x1)
43.50/12.04	   c(c(x1)) -> x1
43.50/12.04	
43.50/12.04	Q is empty.
43.50/12.04	
43.50/12.04	----------------------------------------
43.50/12.04	
43.50/12.04	(1) DependencyPairsProof (EQUIVALENT)
43.50/12.04	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
43.50/12.04	----------------------------------------
43.50/12.04	
43.50/12.04	(2)
43.50/12.04	Obligation:
43.50/12.04	Q DP problem:
43.50/12.04	The TRS P consists of the following rules:
43.50/12.04	
43.50/12.04	   A(x1) -> B(x1)
43.50/12.04	   A(b(x1)) -> B(a(c(a(x1))))
43.50/12.04	   A(b(x1)) -> A(c(a(x1)))
43.50/12.04	   A(b(x1)) -> C(a(x1))
43.50/12.04	   A(b(x1)) -> A(x1)
43.50/12.04	   B(x1) -> C(x1)
43.50/12.04	
43.50/12.04	The TRS R consists of the following rules:
43.50/12.04	
43.50/12.04	   a(x1) -> b(x1)
43.50/12.04	   a(b(x1)) -> b(a(c(a(x1))))
43.50/12.04	   b(x1) -> c(x1)
43.50/12.04	   c(c(x1)) -> x1
43.50/12.04	
43.50/12.04	Q is empty.
43.50/12.04	We have to consider all minimal (P,Q,R)-chains.
43.50/12.04	----------------------------------------
43.50/12.04	
43.50/12.04	(3) DependencyGraphProof (EQUIVALENT)
43.50/12.04	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
43.50/12.04	----------------------------------------
43.50/12.04	
43.50/12.04	(4)
43.50/12.04	Obligation:
43.50/12.04	Q DP problem:
43.50/12.04	The TRS P consists of the following rules:
43.50/12.04	
43.50/12.04	   A(b(x1)) -> A(x1)
43.50/12.04	   A(b(x1)) -> A(c(a(x1)))
43.50/12.04	
43.50/12.04	The TRS R consists of the following rules:
43.50/12.04	
43.50/12.04	   a(x1) -> b(x1)
43.50/12.04	   a(b(x1)) -> b(a(c(a(x1))))
43.50/12.04	   b(x1) -> c(x1)
43.50/12.04	   c(c(x1)) -> x1
43.50/12.04	
43.50/12.04	Q is empty.
43.50/12.04	We have to consider all minimal (P,Q,R)-chains.
43.50/12.04	----------------------------------------
43.50/12.04	
43.50/12.04	(5) QDPOrderProof (EQUIVALENT)
43.50/12.04	We use the reduction pair processor [LPAR04,JAR06].
43.50/12.04	
43.50/12.04	
43.50/12.04	The following pairs can be oriented strictly and are deleted.
43.50/12.04	
43.50/12.04	   A(b(x1)) -> A(x1)
43.50/12.04	   A(b(x1)) -> A(c(a(x1)))
43.50/12.04	The remaining pairs can at least be oriented weakly.
43.50/12.04	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
43.50/12.04	
43.50/12.04	   <<<
43.50/12.04	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
43.50/12.04	>>>
43.50/12.04	
43.50/12.04	   <<<
43.50/12.04	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[-I, 0A, -I], [0A, 1A, -I], [0A, 0A, 0A]] 	* 	x_1
43.50/12.04	>>>
43.50/12.04	
43.50/12.04	   <<<
43.50/12.04	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
43.50/12.04	>>>
43.50/12.04	
43.50/12.04	   <<<
43.50/12.04	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, -I], [0A, 0A, 0A]] 	* 	x_1
43.50/12.04	>>>
43.50/12.04	
43.50/12.04	
43.50/12.04	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.50/12.04	
43.50/12.04	   a(x1) -> b(x1)
43.50/12.04	   a(b(x1)) -> b(a(c(a(x1))))
43.50/12.04	   c(c(x1)) -> x1
43.50/12.04	   b(x1) -> c(x1)
43.50/12.04	
43.50/12.04	
43.50/12.04	----------------------------------------
43.50/12.04	
43.50/12.04	(6)
43.50/12.04	Obligation:
43.50/12.04	Q DP problem:
43.50/12.04	P is empty.
43.50/12.04	The TRS R consists of the following rules:
43.50/12.04	
43.50/12.04	   a(x1) -> b(x1)
43.50/12.04	   a(b(x1)) -> b(a(c(a(x1))))
43.50/12.04	   b(x1) -> c(x1)
43.50/12.04	   c(c(x1)) -> x1
43.50/12.04	
43.50/12.04	Q is empty.
43.50/12.04	We have to consider all minimal (P,Q,R)-chains.
43.50/12.04	----------------------------------------
43.50/12.04	
43.50/12.04	(7) PisEmptyProof (EQUIVALENT)
43.50/12.04	The TRS P is empty. Hence, there is no (P,Q,R) chain.
43.50/12.04	----------------------------------------
43.50/12.04	
43.50/12.04	(8)
43.50/12.04	YES
43.91/12.11	EOF