12.69/4.08	YES
12.92/4.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.92/4.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.92/4.16	
12.92/4.16	
12.92/4.16	Termination w.r.t. Q of the given QTRS could be proven:
12.92/4.16	
12.92/4.16	(0) QTRS
12.92/4.16	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
12.92/4.16	(2) QDP
12.92/4.16	(3) DependencyGraphProof [EQUIVALENT, 1 ms]
12.92/4.16	(4) QDP
12.92/4.16	(5) QDPOrderProof [EQUIVALENT, 151 ms]
12.92/4.16	(6) QDP
12.92/4.16	(7) PisEmptyProof [EQUIVALENT, 0 ms]
12.92/4.16	(8) YES
12.92/4.16	
12.92/4.16	
12.92/4.16	----------------------------------------
12.92/4.16	
12.92/4.16	(0)
12.92/4.16	Obligation:
12.92/4.16	Q restricted rewrite system:
12.92/4.16	The TRS R consists of the following rules:
12.92/4.16	
12.92/4.16	   a(x1) -> x1
12.92/4.16	   a(b(b(x1))) -> b(b(b(a(a(c(x1))))))
12.92/4.16	   c(b(x1)) -> x1
12.92/4.16	
12.92/4.16	Q is empty.
12.92/4.16	
12.92/4.16	----------------------------------------
12.92/4.16	
12.92/4.16	(1) DependencyPairsProof (EQUIVALENT)
12.92/4.16	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
12.92/4.16	----------------------------------------
12.92/4.16	
12.92/4.16	(2)
12.92/4.16	Obligation:
12.92/4.16	Q DP problem:
12.92/4.16	The TRS P consists of the following rules:
12.92/4.16	
12.92/4.16	   A(b(b(x1))) -> A(a(c(x1)))
12.92/4.16	   A(b(b(x1))) -> A(c(x1))
12.92/4.16	   A(b(b(x1))) -> C(x1)
12.92/4.16	
12.92/4.16	The TRS R consists of the following rules:
12.92/4.16	
12.92/4.16	   a(x1) -> x1
12.92/4.16	   a(b(b(x1))) -> b(b(b(a(a(c(x1))))))
12.92/4.16	   c(b(x1)) -> x1
12.92/4.16	
12.92/4.16	Q is empty.
12.92/4.16	We have to consider all minimal (P,Q,R)-chains.
12.92/4.16	----------------------------------------
12.92/4.16	
12.92/4.16	(3) DependencyGraphProof (EQUIVALENT)
12.92/4.16	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
12.92/4.16	----------------------------------------
12.92/4.16	
12.92/4.16	(4)
12.92/4.16	Obligation:
12.92/4.16	Q DP problem:
12.92/4.16	The TRS P consists of the following rules:
12.92/4.16	
12.92/4.16	   A(b(b(x1))) -> A(c(x1))
12.92/4.16	   A(b(b(x1))) -> A(a(c(x1)))
12.92/4.16	
12.92/4.16	The TRS R consists of the following rules:
12.92/4.16	
12.92/4.16	   a(x1) -> x1
12.92/4.16	   a(b(b(x1))) -> b(b(b(a(a(c(x1))))))
12.92/4.16	   c(b(x1)) -> x1
12.92/4.16	
12.92/4.16	Q is empty.
12.92/4.16	We have to consider all minimal (P,Q,R)-chains.
12.92/4.16	----------------------------------------
12.92/4.16	
12.92/4.16	(5) QDPOrderProof (EQUIVALENT)
12.92/4.16	We use the reduction pair processor [LPAR04,JAR06].
12.92/4.16	
12.92/4.16	
12.92/4.16	The following pairs can be oriented strictly and are deleted.
12.92/4.16	
12.92/4.16	   A(b(b(x1))) -> A(c(x1))
12.92/4.16	   A(b(b(x1))) -> A(a(c(x1)))
12.92/4.16	The remaining pairs can at least be oriented weakly.
12.92/4.16	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
12.92/4.16	
12.92/4.16	   <<<
12.92/4.16	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
12.92/4.16	>>>
12.92/4.16	
12.92/4.16	   <<<
12.92/4.16	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 1A], [0A, -I, 0A]] 	* 	x_1
12.92/4.16	>>>
12.92/4.16	
12.92/4.16	   <<<
12.92/4.16	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
12.92/4.16	>>>
12.92/4.16	
12.92/4.16	   <<<
12.92/4.16	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
12.92/4.16	>>>
12.92/4.16	
12.92/4.16	
12.92/4.16	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
12.92/4.16	
12.92/4.16	   c(b(x1)) -> x1
12.92/4.16	   a(x1) -> x1
12.92/4.16	   a(b(b(x1))) -> b(b(b(a(a(c(x1))))))
12.92/4.16	
12.92/4.16	
12.92/4.16	----------------------------------------
12.92/4.16	
12.92/4.16	(6)
12.92/4.16	Obligation:
12.92/4.16	Q DP problem:
12.92/4.16	P is empty.
12.92/4.16	The TRS R consists of the following rules:
12.92/4.16	
12.92/4.16	   a(x1) -> x1
12.92/4.16	   a(b(b(x1))) -> b(b(b(a(a(c(x1))))))
12.92/4.16	   c(b(x1)) -> x1
12.92/4.16	
12.92/4.16	Q is empty.
12.92/4.16	We have to consider all minimal (P,Q,R)-chains.
12.92/4.16	----------------------------------------
12.92/4.16	
12.92/4.16	(7) PisEmptyProof (EQUIVALENT)
12.92/4.16	The TRS P is empty. Hence, there is no (P,Q,R) chain.
12.92/4.16	----------------------------------------
12.92/4.16	
12.92/4.16	(8)
12.92/4.16	YES
13.17/4.30	EOF