16.77/5.28	YES
17.34/5.45	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.34/5.45	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.34/5.45	
17.34/5.45	
17.34/5.45	Termination w.r.t. Q of the given QTRS could be proven:
17.34/5.45	
17.34/5.45	(0) QTRS
17.34/5.45	(1) QTRS Reverse [EQUIVALENT, 0 ms]
17.34/5.45	(2) QTRS
17.34/5.45	(3) DependencyPairsProof [EQUIVALENT, 3 ms]
17.34/5.45	(4) QDP
17.34/5.45	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
17.34/5.45	(6) QDP
17.34/5.45	(7) QDPOrderProof [EQUIVALENT, 142 ms]
17.34/5.45	(8) QDP
17.34/5.45	(9) PisEmptyProof [EQUIVALENT, 0 ms]
17.34/5.45	(10) YES
17.34/5.45	
17.34/5.45	
17.34/5.45	----------------------------------------
17.34/5.45	
17.34/5.45	(0)
17.34/5.45	Obligation:
17.34/5.45	Q restricted rewrite system:
17.34/5.45	The TRS R consists of the following rules:
17.34/5.45	
17.34/5.45	   a(x1) -> x1
17.34/5.45	   a(b(x1)) -> a(c(b(b(c(x1)))))
17.34/5.45	   b(x1) -> x1
17.34/5.45	   c(c(x1)) -> a(x1)
17.34/5.45	
17.34/5.45	Q is empty.
17.34/5.45	
17.34/5.45	----------------------------------------
17.34/5.45	
17.34/5.45	(1) QTRS Reverse (EQUIVALENT)
17.34/5.45	We applied the QTRS Reverse Processor [REVERSE].
17.34/5.45	----------------------------------------
17.34/5.45	
17.34/5.45	(2)
17.34/5.45	Obligation:
17.34/5.45	Q restricted rewrite system:
17.34/5.45	The TRS R consists of the following rules:
17.34/5.45	
17.34/5.45	   a(x1) -> x1
17.34/5.45	   b(a(x1)) -> c(b(b(c(a(x1)))))
17.34/5.45	   b(x1) -> x1
17.34/5.45	   c(c(x1)) -> a(x1)
17.34/5.46	
17.34/5.46	Q is empty.
17.34/5.46	
17.34/5.46	----------------------------------------
17.34/5.46	
17.34/5.46	(3) DependencyPairsProof (EQUIVALENT)
17.34/5.46	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.34/5.46	----------------------------------------
17.34/5.46	
17.34/5.46	(4)
17.34/5.46	Obligation:
17.34/5.46	Q DP problem:
17.34/5.46	The TRS P consists of the following rules:
17.34/5.46	
17.34/5.46	   B(a(x1)) -> C(b(b(c(a(x1)))))
17.34/5.46	   B(a(x1)) -> B(b(c(a(x1))))
17.34/5.46	   B(a(x1)) -> B(c(a(x1)))
17.34/5.46	   B(a(x1)) -> C(a(x1))
17.34/5.46	   C(c(x1)) -> A(x1)
17.34/5.46	
17.34/5.46	The TRS R consists of the following rules:
17.34/5.46	
17.34/5.46	   a(x1) -> x1
17.34/5.46	   b(a(x1)) -> c(b(b(c(a(x1)))))
17.34/5.46	   b(x1) -> x1
17.34/5.46	   c(c(x1)) -> a(x1)
17.34/5.46	
17.34/5.46	Q is empty.
17.34/5.46	We have to consider all minimal (P,Q,R)-chains.
17.34/5.46	----------------------------------------
17.34/5.46	
17.34/5.46	(5) DependencyGraphProof (EQUIVALENT)
17.34/5.46	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
17.34/5.46	----------------------------------------
17.34/5.46	
17.34/5.46	(6)
17.34/5.46	Obligation:
17.34/5.46	Q DP problem:
17.34/5.46	The TRS P consists of the following rules:
17.34/5.46	
17.34/5.46	   B(a(x1)) -> B(c(a(x1)))
17.34/5.46	   B(a(x1)) -> B(b(c(a(x1))))
17.34/5.46	
17.34/5.46	The TRS R consists of the following rules:
17.34/5.46	
17.34/5.46	   a(x1) -> x1
17.34/5.46	   b(a(x1)) -> c(b(b(c(a(x1)))))
17.34/5.46	   b(x1) -> x1
17.34/5.46	   c(c(x1)) -> a(x1)
17.34/5.46	
17.34/5.46	Q is empty.
17.34/5.46	We have to consider all minimal (P,Q,R)-chains.
17.34/5.46	----------------------------------------
17.34/5.46	
17.34/5.46	(7) QDPOrderProof (EQUIVALENT)
17.34/5.46	We use the reduction pair processor [LPAR04,JAR06].
17.34/5.46	
17.34/5.46	
17.34/5.46	The following pairs can be oriented strictly and are deleted.
17.34/5.46	
17.34/5.46	   B(a(x1)) -> B(c(a(x1)))
17.34/5.46	   B(a(x1)) -> B(b(c(a(x1))))
17.34/5.46	The remaining pairs can at least be oriented weakly.
17.34/5.46	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
17.34/5.46	
17.34/5.46	   <<<
17.34/5.46	 POL(B(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
17.34/5.46	>>>
17.34/5.46	
17.34/5.46	   <<<
17.34/5.46	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
17.34/5.46	>>>
17.34/5.46	
17.34/5.46	   <<<
17.34/5.46	 POL(c(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
17.34/5.46	>>>
17.34/5.46	
17.34/5.46	   <<<
17.34/5.46	 POL(b(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
17.34/5.46	>>>
17.34/5.46	
17.34/5.46	
17.34/5.46	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.34/5.46	
17.34/5.46	   a(x1) -> x1
17.34/5.46	   c(c(x1)) -> a(x1)
17.34/5.46	   b(a(x1)) -> c(b(b(c(a(x1)))))
17.34/5.46	   b(x1) -> x1
17.34/5.46	
17.34/5.46	
17.34/5.46	----------------------------------------
17.34/5.46	
17.34/5.46	(8)
17.34/5.46	Obligation:
17.34/5.46	Q DP problem:
17.34/5.46	P is empty.
17.34/5.46	The TRS R consists of the following rules:
17.34/5.46	
17.34/5.46	   a(x1) -> x1
17.34/5.46	   b(a(x1)) -> c(b(b(c(a(x1)))))
17.34/5.46	   b(x1) -> x1
17.34/5.46	   c(c(x1)) -> a(x1)
17.34/5.46	
17.34/5.46	Q is empty.
17.34/5.46	We have to consider all minimal (P,Q,R)-chains.
17.34/5.46	----------------------------------------
17.34/5.46	
17.34/5.46	(9) PisEmptyProof (EQUIVALENT)
17.34/5.46	The TRS P is empty. Hence, there is no (P,Q,R) chain.
17.34/5.46	----------------------------------------
17.34/5.46	
17.34/5.46	(10)
17.34/5.46	YES
17.56/5.53	EOF