40.39/11.32	YES
40.47/11.34	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
40.47/11.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.47/11.34	
40.47/11.34	
40.47/11.34	Termination w.r.t. Q of the given QTRS could be proven:
40.47/11.34	
40.47/11.34	(0) QTRS
40.47/11.34	(1) DependencyPairsProof [EQUIVALENT, 20 ms]
40.47/11.34	(2) QDP
40.47/11.34	(3) DependencyGraphProof [EQUIVALENT, 4 ms]
40.47/11.34	(4) QDP
40.47/11.34	(5) QDPOrderProof [EQUIVALENT, 140 ms]
40.47/11.34	(6) QDP
40.47/11.34	(7) QDPOrderProof [EQUIVALENT, 44 ms]
40.47/11.34	(8) QDP
40.47/11.34	(9) PisEmptyProof [EQUIVALENT, 0 ms]
40.47/11.34	(10) YES
40.47/11.34	
40.47/11.34	
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(0)
40.47/11.34	Obligation:
40.47/11.34	Q restricted rewrite system:
40.47/11.34	The TRS R consists of the following rules:
40.47/11.34	
40.47/11.34	   a(x1) -> x1
40.47/11.34	   a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
40.47/11.34	   c(b(x1)) -> x1
40.47/11.34	
40.47/11.34	Q is empty.
40.47/11.34	
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(1) DependencyPairsProof (EQUIVALENT)
40.47/11.34	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(2)
40.47/11.34	Obligation:
40.47/11.34	Q DP problem:
40.47/11.34	The TRS P consists of the following rules:
40.47/11.34	
40.47/11.34	   A(b(b(x1))) -> C(a(a(x1)))
40.47/11.34	   A(b(b(x1))) -> A(a(x1))
40.47/11.34	   A(b(b(x1))) -> A(x1)
40.47/11.34	
40.47/11.34	The TRS R consists of the following rules:
40.47/11.34	
40.47/11.34	   a(x1) -> x1
40.47/11.34	   a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
40.47/11.34	   c(b(x1)) -> x1
40.47/11.34	
40.47/11.34	Q is empty.
40.47/11.34	We have to consider all minimal (P,Q,R)-chains.
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(3) DependencyGraphProof (EQUIVALENT)
40.47/11.34	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(4)
40.47/11.34	Obligation:
40.47/11.34	Q DP problem:
40.47/11.34	The TRS P consists of the following rules:
40.47/11.34	
40.47/11.34	   A(b(b(x1))) -> A(x1)
40.47/11.34	   A(b(b(x1))) -> A(a(x1))
40.47/11.34	
40.47/11.34	The TRS R consists of the following rules:
40.47/11.34	
40.47/11.34	   a(x1) -> x1
40.47/11.34	   a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
40.47/11.34	   c(b(x1)) -> x1
40.47/11.34	
40.47/11.34	Q is empty.
40.47/11.34	We have to consider all minimal (P,Q,R)-chains.
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(5) QDPOrderProof (EQUIVALENT)
40.47/11.34	We use the reduction pair processor [LPAR04,JAR06].
40.47/11.34	
40.47/11.34	
40.47/11.34	The following pairs can be oriented strictly and are deleted.
40.47/11.34	
40.47/11.34	   A(b(b(x1))) -> A(x1)
40.47/11.34	The remaining pairs can at least be oriented weakly.
40.47/11.34	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.47/11.34	
40.47/11.34	   <<<
40.47/11.34	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
40.47/11.34	>>>
40.47/11.34	
40.47/11.34	   <<<
40.47/11.34	 POL(b(x_1)) =  	[[0A], [1A], [-I]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
40.47/11.34	>>>
40.47/11.34	
40.47/11.34	   <<<
40.47/11.34	 POL(a(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
40.47/11.34	>>>
40.47/11.34	
40.47/11.34	   <<<
40.47/11.34	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
40.47/11.34	>>>
40.47/11.34	
40.47/11.34	
40.47/11.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.47/11.34	
40.47/11.34	   a(x1) -> x1
40.47/11.34	   a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
40.47/11.34	   c(b(x1)) -> x1
40.47/11.34	
40.47/11.34	
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(6)
40.47/11.34	Obligation:
40.47/11.34	Q DP problem:
40.47/11.34	The TRS P consists of the following rules:
40.47/11.34	
40.47/11.34	   A(b(b(x1))) -> A(a(x1))
40.47/11.34	
40.47/11.34	The TRS R consists of the following rules:
40.47/11.34	
40.47/11.34	   a(x1) -> x1
40.47/11.34	   a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
40.47/11.34	   c(b(x1)) -> x1
40.47/11.34	
40.47/11.34	Q is empty.
40.47/11.34	We have to consider all minimal (P,Q,R)-chains.
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(7) QDPOrderProof (EQUIVALENT)
40.47/11.34	We use the reduction pair processor [LPAR04,JAR06].
40.47/11.34	
40.47/11.34	
40.47/11.34	The following pairs can be oriented strictly and are deleted.
40.47/11.34	
40.47/11.34	   A(b(b(x1))) -> A(a(x1))
40.47/11.34	The remaining pairs can at least be oriented weakly.
40.47/11.34	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.47/11.34	
40.47/11.34	   <<<
40.47/11.34	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
40.47/11.34	>>>
40.47/11.34	
40.47/11.34	   <<<
40.47/11.34	 POL(b(x_1)) =  	[[1A], [0A], [-I]] 	 +  	[[-I, 0A, 1A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
40.47/11.34	>>>
40.47/11.34	
40.47/11.34	   <<<
40.47/11.34	 POL(a(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, 1A], [0A, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
40.47/11.34	>>>
40.47/11.34	
40.47/11.34	   <<<
40.47/11.34	 POL(c(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, -I, 0A], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
40.47/11.34	>>>
40.47/11.34	
40.47/11.34	
40.47/11.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.47/11.34	
40.47/11.34	   a(x1) -> x1
40.47/11.34	   a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
40.47/11.34	   c(b(x1)) -> x1
40.47/11.34	
40.47/11.34	
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(8)
40.47/11.34	Obligation:
40.47/11.34	Q DP problem:
40.47/11.34	P is empty.
40.47/11.34	The TRS R consists of the following rules:
40.47/11.34	
40.47/11.34	   a(x1) -> x1
40.47/11.34	   a(b(b(x1))) -> b(b(b(c(a(a(x1))))))
40.47/11.34	   c(b(x1)) -> x1
40.47/11.34	
40.47/11.34	Q is empty.
40.47/11.34	We have to consider all minimal (P,Q,R)-chains.
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(9) PisEmptyProof (EQUIVALENT)
40.47/11.34	The TRS P is empty. Hence, there is no (P,Q,R) chain.
40.47/11.34	----------------------------------------
40.47/11.34	
40.47/11.34	(10)
40.47/11.34	YES
40.88/11.52	EOF