18.19/5.57	YES
18.45/5.61	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
18.45/5.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
18.45/5.61	
18.45/5.61	
18.45/5.61	Termination w.r.t. Q of the given QTRS could be proven:
18.45/5.61	
18.45/5.61	(0) QTRS
18.45/5.61	(1) QTRS Reverse [EQUIVALENT, 0 ms]
18.45/5.61	(2) QTRS
18.45/5.61	(3) DependencyPairsProof [EQUIVALENT, 5 ms]
18.45/5.61	(4) QDP
18.45/5.61	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
18.45/5.61	(6) QDP
18.45/5.61	(7) QDPOrderProof [EQUIVALENT, 131 ms]
18.45/5.61	(8) QDP
18.45/5.61	(9) PisEmptyProof [EQUIVALENT, 0 ms]
18.45/5.61	(10) YES
18.45/5.61	
18.45/5.61	
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(0)
18.45/5.61	Obligation:
18.45/5.61	Q restricted rewrite system:
18.45/5.61	The TRS R consists of the following rules:
18.45/5.61	
18.45/5.61	   a(x1) -> x1
18.45/5.61	   a(b(x1)) -> b(b(b(a(c(x1)))))
18.45/5.61	   b(x1) -> x1
18.45/5.61	   c(c(x1)) -> a(x1)
18.45/5.61	
18.45/5.61	Q is empty.
18.45/5.61	
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(1) QTRS Reverse (EQUIVALENT)
18.45/5.61	We applied the QTRS Reverse Processor [REVERSE].
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(2)
18.45/5.61	Obligation:
18.45/5.61	Q restricted rewrite system:
18.45/5.61	The TRS R consists of the following rules:
18.45/5.61	
18.45/5.61	   a(x1) -> x1
18.45/5.61	   b(a(x1)) -> c(a(b(b(b(x1)))))
18.45/5.61	   b(x1) -> x1
18.45/5.61	   c(c(x1)) -> a(x1)
18.45/5.61	
18.45/5.61	Q is empty.
18.45/5.61	
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(3) DependencyPairsProof (EQUIVALENT)
18.45/5.61	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(4)
18.45/5.61	Obligation:
18.45/5.61	Q DP problem:
18.45/5.61	The TRS P consists of the following rules:
18.45/5.61	
18.45/5.61	   B(a(x1)) -> C(a(b(b(b(x1)))))
18.45/5.61	   B(a(x1)) -> A(b(b(b(x1))))
18.45/5.61	   B(a(x1)) -> B(b(b(x1)))
18.45/5.61	   B(a(x1)) -> B(b(x1))
18.45/5.61	   B(a(x1)) -> B(x1)
18.45/5.61	   C(c(x1)) -> A(x1)
18.45/5.61	
18.45/5.61	The TRS R consists of the following rules:
18.45/5.61	
18.45/5.61	   a(x1) -> x1
18.45/5.61	   b(a(x1)) -> c(a(b(b(b(x1)))))
18.45/5.61	   b(x1) -> x1
18.45/5.61	   c(c(x1)) -> a(x1)
18.45/5.61	
18.45/5.61	Q is empty.
18.45/5.61	We have to consider all minimal (P,Q,R)-chains.
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(5) DependencyGraphProof (EQUIVALENT)
18.45/5.61	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(6)
18.45/5.61	Obligation:
18.45/5.61	Q DP problem:
18.45/5.61	The TRS P consists of the following rules:
18.45/5.61	
18.45/5.61	   B(a(x1)) -> B(b(x1))
18.45/5.61	   B(a(x1)) -> B(b(b(x1)))
18.45/5.61	   B(a(x1)) -> B(x1)
18.45/5.61	
18.45/5.61	The TRS R consists of the following rules:
18.45/5.61	
18.45/5.61	   a(x1) -> x1
18.45/5.61	   b(a(x1)) -> c(a(b(b(b(x1)))))
18.45/5.61	   b(x1) -> x1
18.45/5.61	   c(c(x1)) -> a(x1)
18.45/5.61	
18.45/5.61	Q is empty.
18.45/5.61	We have to consider all minimal (P,Q,R)-chains.
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(7) QDPOrderProof (EQUIVALENT)
18.45/5.61	We use the reduction pair processor [LPAR04,JAR06].
18.45/5.61	
18.45/5.61	
18.45/5.61	The following pairs can be oriented strictly and are deleted.
18.45/5.61	
18.45/5.61	   B(a(x1)) -> B(b(x1))
18.45/5.61	   B(a(x1)) -> B(b(b(x1)))
18.45/5.61	   B(a(x1)) -> B(x1)
18.45/5.61	The remaining pairs can at least be oriented weakly.
18.45/5.61	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
18.45/5.61	
18.45/5.61	   <<<
18.45/5.61	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
18.45/5.61	>>>
18.45/5.61	
18.45/5.61	   <<<
18.45/5.61	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [1A, 1A, 1A]] 	* 	x_1
18.45/5.61	>>>
18.45/5.61	
18.45/5.61	   <<<
18.45/5.61	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
18.45/5.61	>>>
18.45/5.61	
18.45/5.61	   <<<
18.45/5.61	 POL(c(x_1)) =  	[[-I], [0A], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [1A, 0A, 0A]] 	* 	x_1
18.45/5.61	>>>
18.45/5.61	
18.45/5.61	
18.45/5.61	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
18.45/5.61	
18.45/5.61	   b(a(x1)) -> c(a(b(b(b(x1)))))
18.45/5.61	   b(x1) -> x1
18.45/5.61	   a(x1) -> x1
18.45/5.61	   c(c(x1)) -> a(x1)
18.45/5.61	
18.45/5.61	
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(8)
18.45/5.61	Obligation:
18.45/5.61	Q DP problem:
18.45/5.61	P is empty.
18.45/5.61	The TRS R consists of the following rules:
18.45/5.61	
18.45/5.61	   a(x1) -> x1
18.45/5.61	   b(a(x1)) -> c(a(b(b(b(x1)))))
18.45/5.61	   b(x1) -> x1
18.45/5.61	   c(c(x1)) -> a(x1)
18.45/5.61	
18.45/5.61	Q is empty.
18.45/5.61	We have to consider all minimal (P,Q,R)-chains.
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(9) PisEmptyProof (EQUIVALENT)
18.45/5.61	The TRS P is empty. Hence, there is no (P,Q,R) chain.
18.45/5.61	----------------------------------------
18.45/5.61	
18.45/5.61	(10)
18.45/5.61	YES
18.70/5.68	EOF