11.93/4.01	YES
11.93/4.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
11.93/4.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.93/4.03	
11.93/4.03	
11.93/4.03	Termination w.r.t. Q of the given QTRS could be proven:
11.93/4.03	
11.93/4.03	(0) QTRS
11.93/4.03	(1) DependencyPairsProof [EQUIVALENT, 2 ms]
11.93/4.03	(2) QDP
11.93/4.03	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
11.93/4.03	(4) QDP
11.93/4.03	(5) QDPOrderProof [EQUIVALENT, 61 ms]
11.93/4.03	(6) QDP
11.93/4.03	(7) PisEmptyProof [EQUIVALENT, 0 ms]
11.93/4.03	(8) YES
11.93/4.03	
11.93/4.03	
11.93/4.03	----------------------------------------
11.93/4.03	
11.93/4.03	(0)
11.93/4.03	Obligation:
11.93/4.03	Q restricted rewrite system:
11.93/4.03	The TRS R consists of the following rules:
11.93/4.03	
11.93/4.03	   a(x1) -> b(c(x1))
11.93/4.03	   a(c(b(x1))) -> b(a(a(c(x1))))
11.93/4.03	   b(b(x1)) -> x1
11.93/4.03	
11.93/4.03	Q is empty.
11.93/4.03	
11.93/4.03	----------------------------------------
11.93/4.03	
11.93/4.03	(1) DependencyPairsProof (EQUIVALENT)
11.93/4.03	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.93/4.03	----------------------------------------
11.93/4.03	
11.93/4.03	(2)
11.93/4.03	Obligation:
11.93/4.03	Q DP problem:
11.93/4.03	The TRS P consists of the following rules:
11.93/4.03	
11.93/4.03	   A(x1) -> B(c(x1))
11.93/4.03	   A(c(b(x1))) -> B(a(a(c(x1))))
11.93/4.03	   A(c(b(x1))) -> A(a(c(x1)))
11.93/4.03	   A(c(b(x1))) -> A(c(x1))
11.93/4.03	
11.93/4.03	The TRS R consists of the following rules:
11.93/4.03	
11.93/4.03	   a(x1) -> b(c(x1))
11.93/4.03	   a(c(b(x1))) -> b(a(a(c(x1))))
11.93/4.03	   b(b(x1)) -> x1
11.93/4.03	
11.93/4.03	Q is empty.
11.93/4.03	We have to consider all minimal (P,Q,R)-chains.
11.93/4.03	----------------------------------------
11.93/4.03	
11.93/4.03	(3) DependencyGraphProof (EQUIVALENT)
11.93/4.03	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
11.93/4.03	----------------------------------------
11.93/4.03	
11.93/4.03	(4)
11.93/4.03	Obligation:
11.93/4.03	Q DP problem:
11.93/4.03	The TRS P consists of the following rules:
11.93/4.03	
11.93/4.03	   A(c(b(x1))) -> A(c(x1))
11.93/4.03	   A(c(b(x1))) -> A(a(c(x1)))
11.93/4.03	
11.93/4.03	The TRS R consists of the following rules:
11.93/4.03	
11.93/4.03	   a(x1) -> b(c(x1))
11.93/4.03	   a(c(b(x1))) -> b(a(a(c(x1))))
11.93/4.03	   b(b(x1)) -> x1
11.93/4.03	
11.93/4.03	Q is empty.
11.93/4.03	We have to consider all minimal (P,Q,R)-chains.
11.93/4.03	----------------------------------------
11.93/4.03	
11.93/4.03	(5) QDPOrderProof (EQUIVALENT)
11.93/4.03	We use the reduction pair processor [LPAR04,JAR06].
11.93/4.03	
11.93/4.03	
11.93/4.03	The following pairs can be oriented strictly and are deleted.
11.93/4.03	
11.93/4.03	   A(c(b(x1))) -> A(c(x1))
11.93/4.03	   A(c(b(x1))) -> A(a(c(x1)))
11.93/4.03	The remaining pairs can at least be oriented weakly.
11.93/4.03	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
11.93/4.03	
11.93/4.03	   <<<
11.93/4.03	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
11.93/4.03	>>>
11.93/4.03	
11.93/4.03	   <<<
11.93/4.03	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, -I, 0A], [0A, -I, 1A], [-I, -I, -I]] 	* 	x_1
11.93/4.03	>>>
11.93/4.03	
11.93/4.03	   <<<
11.93/4.03	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [0A, -I, 0A], [0A, -I, 1A]] 	* 	x_1
11.93/4.03	>>>
11.93/4.03	
11.93/4.03	   <<<
11.93/4.03	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
11.93/4.03	>>>
11.93/4.03	
11.93/4.03	
11.93/4.03	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.93/4.03	
11.93/4.03	   a(x1) -> b(c(x1))
11.93/4.03	   a(c(b(x1))) -> b(a(a(c(x1))))
11.93/4.03	   b(b(x1)) -> x1
11.93/4.03	
11.93/4.03	
11.93/4.03	----------------------------------------
11.93/4.03	
11.93/4.03	(6)
11.93/4.03	Obligation:
11.93/4.03	Q DP problem:
11.93/4.03	P is empty.
11.93/4.03	The TRS R consists of the following rules:
11.93/4.03	
11.93/4.03	   a(x1) -> b(c(x1))
11.93/4.03	   a(c(b(x1))) -> b(a(a(c(x1))))
11.93/4.03	   b(b(x1)) -> x1
11.93/4.03	
11.93/4.03	Q is empty.
11.93/4.03	We have to consider all minimal (P,Q,R)-chains.
11.93/4.03	----------------------------------------
11.93/4.03	
11.93/4.03	(7) PisEmptyProof (EQUIVALENT)
11.93/4.03	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.93/4.03	----------------------------------------
11.93/4.03	
11.93/4.03	(8)
11.93/4.03	YES
12.18/4.09	EOF