12.74/4.38	YES
12.74/4.46	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.74/4.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.74/4.46	
12.74/4.46	
12.74/4.46	Termination w.r.t. Q of the given QTRS could be proven:
12.74/4.46	
12.74/4.46	(0) QTRS
12.74/4.46	(1) QTRS Reverse [EQUIVALENT, 0 ms]
12.74/4.46	(2) QTRS
12.74/4.46	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
12.74/4.46	(4) QDP
12.74/4.46	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
12.74/4.46	(6) QDP
12.74/4.46	(7) QDPOrderProof [EQUIVALENT, 175 ms]
12.74/4.46	(8) QDP
12.74/4.46	(9) PisEmptyProof [EQUIVALENT, 0 ms]
12.74/4.46	(10) YES
12.74/4.46	
12.74/4.46	
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(0)
12.74/4.46	Obligation:
12.74/4.46	Q restricted rewrite system:
12.74/4.46	The TRS R consists of the following rules:
12.74/4.46	
12.74/4.46	   a(x1) -> x1
12.74/4.46	   a(b(x1)) -> c(b(c(b(c(a(x1))))))
12.74/4.46	   c(c(x1)) -> a(x1)
12.74/4.46	
12.74/4.46	Q is empty.
12.74/4.46	
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(1) QTRS Reverse (EQUIVALENT)
12.74/4.46	We applied the QTRS Reverse Processor [REVERSE].
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(2)
12.74/4.46	Obligation:
12.74/4.46	Q restricted rewrite system:
12.74/4.46	The TRS R consists of the following rules:
12.74/4.46	
12.74/4.46	   a(x1) -> x1
12.74/4.46	   b(a(x1)) -> a(c(b(c(b(c(x1))))))
12.74/4.46	   c(c(x1)) -> a(x1)
12.74/4.46	
12.74/4.46	Q is empty.
12.74/4.46	
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(3) DependencyPairsProof (EQUIVALENT)
12.74/4.46	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(4)
12.74/4.46	Obligation:
12.74/4.46	Q DP problem:
12.74/4.46	The TRS P consists of the following rules:
12.74/4.46	
12.74/4.46	   B(a(x1)) -> A(c(b(c(b(c(x1))))))
12.74/4.46	   B(a(x1)) -> C(b(c(b(c(x1)))))
12.74/4.46	   B(a(x1)) -> B(c(b(c(x1))))
12.74/4.46	   B(a(x1)) -> C(b(c(x1)))
12.74/4.46	   B(a(x1)) -> B(c(x1))
12.74/4.46	   B(a(x1)) -> C(x1)
12.74/4.46	   C(c(x1)) -> A(x1)
12.74/4.46	
12.74/4.46	The TRS R consists of the following rules:
12.74/4.46	
12.74/4.46	   a(x1) -> x1
12.74/4.46	   b(a(x1)) -> a(c(b(c(b(c(x1))))))
12.74/4.46	   c(c(x1)) -> a(x1)
12.74/4.46	
12.74/4.46	Q is empty.
12.74/4.46	We have to consider all minimal (P,Q,R)-chains.
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(5) DependencyGraphProof (EQUIVALENT)
12.74/4.46	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(6)
12.74/4.46	Obligation:
12.74/4.46	Q DP problem:
12.74/4.46	The TRS P consists of the following rules:
12.74/4.46	
12.74/4.46	   B(a(x1)) -> B(c(x1))
12.74/4.46	   B(a(x1)) -> B(c(b(c(x1))))
12.74/4.46	
12.74/4.46	The TRS R consists of the following rules:
12.74/4.46	
12.74/4.46	   a(x1) -> x1
12.74/4.46	   b(a(x1)) -> a(c(b(c(b(c(x1))))))
12.74/4.46	   c(c(x1)) -> a(x1)
12.74/4.46	
12.74/4.46	Q is empty.
12.74/4.46	We have to consider all minimal (P,Q,R)-chains.
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(7) QDPOrderProof (EQUIVALENT)
12.74/4.46	We use the reduction pair processor [LPAR04,JAR06].
12.74/4.46	
12.74/4.46	
12.74/4.46	The following pairs can be oriented strictly and are deleted.
12.74/4.46	
12.74/4.46	   B(a(x1)) -> B(c(x1))
12.74/4.46	   B(a(x1)) -> B(c(b(c(x1))))
12.74/4.46	The remaining pairs can at least be oriented weakly.
12.74/4.46	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
12.74/4.46	
12.74/4.46	   <<<
12.74/4.46	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
12.74/4.46	>>>
12.74/4.46	
12.74/4.46	   <<<
12.74/4.46	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, 0A, -I], [1A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
12.74/4.46	>>>
12.74/4.46	
12.74/4.46	   <<<
12.74/4.46	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
12.74/4.46	>>>
12.74/4.46	
12.74/4.46	   <<<
12.74/4.46	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, -I], [0A, 0A, 0A]] 	* 	x_1
12.74/4.46	>>>
12.74/4.46	
12.74/4.46	
12.74/4.46	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.74/4.46	
12.74/4.46	   c(c(x1)) -> a(x1)
12.74/4.46	   b(a(x1)) -> a(c(b(c(b(c(x1))))))
12.74/4.46	   a(x1) -> x1
12.74/4.46	
12.74/4.46	
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(8)
12.74/4.46	Obligation:
12.74/4.46	Q DP problem:
12.74/4.46	P is empty.
12.74/4.46	The TRS R consists of the following rules:
12.74/4.46	
12.74/4.46	   a(x1) -> x1
12.74/4.46	   b(a(x1)) -> a(c(b(c(b(c(x1))))))
12.74/4.46	   c(c(x1)) -> a(x1)
12.74/4.46	
12.74/4.46	Q is empty.
12.74/4.46	We have to consider all minimal (P,Q,R)-chains.
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(9) PisEmptyProof (EQUIVALENT)
12.74/4.46	The TRS P is empty. Hence, there is no (P,Q,R) chain.
12.74/4.46	----------------------------------------
12.74/4.46	
12.74/4.46	(10)
12.74/4.46	YES
13.22/4.53	EOF