31.69/9.03	YES
33.77/9.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
33.77/9.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.77/9.58	
33.77/9.58	
33.77/9.58	Termination w.r.t. Q of the given QTRS could be proven:
33.77/9.58	
33.77/9.58	(0) QTRS
33.77/9.58	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
33.77/9.58	(2) QDP
33.77/9.58	(3) QDPOrderProof [EQUIVALENT, 140 ms]
33.77/9.58	(4) QDP
33.77/9.58	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
33.77/9.58	(6) QDP
33.77/9.58	(7) QDPOrderProof [EQUIVALENT, 58 ms]
33.77/9.58	(8) QDP
33.77/9.58	(9) PisEmptyProof [EQUIVALENT, 0 ms]
33.77/9.58	(10) YES
33.77/9.58	
33.77/9.58	
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(0)
33.77/9.58	Obligation:
33.77/9.58	Q restricted rewrite system:
33.77/9.58	The TRS R consists of the following rules:
33.77/9.58	
33.77/9.58	   a(x1) -> x1
33.77/9.58	   a(b(b(x1))) -> b(b(b(a(c(x1)))))
33.77/9.58	   c(b(x1)) -> a(x1)
33.77/9.58	
33.77/9.58	Q is empty.
33.77/9.58	
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(1) DependencyPairsProof (EQUIVALENT)
33.77/9.58	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(2)
33.77/9.58	Obligation:
33.77/9.58	Q DP problem:
33.77/9.58	The TRS P consists of the following rules:
33.77/9.58	
33.77/9.58	   A(b(b(x1))) -> A(c(x1))
33.77/9.58	   A(b(b(x1))) -> C(x1)
33.77/9.58	   C(b(x1)) -> A(x1)
33.77/9.58	
33.77/9.58	The TRS R consists of the following rules:
33.77/9.58	
33.77/9.58	   a(x1) -> x1
33.77/9.58	   a(b(b(x1))) -> b(b(b(a(c(x1)))))
33.77/9.58	   c(b(x1)) -> a(x1)
33.77/9.58	
33.77/9.58	Q is empty.
33.77/9.58	We have to consider all minimal (P,Q,R)-chains.
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(3) QDPOrderProof (EQUIVALENT)
33.77/9.58	We use the reduction pair processor [LPAR04,JAR06].
33.77/9.58	
33.77/9.58	
33.77/9.58	The following pairs can be oriented strictly and are deleted.
33.77/9.58	
33.77/9.58	   A(b(b(x1))) -> C(x1)
33.77/9.58	The remaining pairs can at least be oriented weakly.
33.77/9.58	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.77/9.58	
33.77/9.58	   <<<
33.77/9.58	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
33.77/9.58	>>>
33.77/9.58	
33.77/9.58	   <<<
33.77/9.58	 POL(b(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [0A, -I, 0A], [1A, 0A, 0A]] 	* 	x_1
33.77/9.58	>>>
33.77/9.58	
33.77/9.58	   <<<
33.77/9.58	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
33.77/9.58	>>>
33.77/9.58	
33.77/9.58	   <<<
33.77/9.58	 POL(C(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
33.77/9.58	>>>
33.77/9.58	
33.77/9.58	   <<<
33.77/9.58	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
33.77/9.58	>>>
33.77/9.58	
33.77/9.58	
33.77/9.58	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.77/9.58	
33.77/9.58	   c(b(x1)) -> a(x1)
33.77/9.58	   a(b(b(x1))) -> b(b(b(a(c(x1)))))
33.77/9.58	   a(x1) -> x1
33.77/9.58	
33.77/9.58	
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(4)
33.77/9.58	Obligation:
33.77/9.58	Q DP problem:
33.77/9.58	The TRS P consists of the following rules:
33.77/9.58	
33.77/9.58	   A(b(b(x1))) -> A(c(x1))
33.77/9.58	   C(b(x1)) -> A(x1)
33.77/9.58	
33.77/9.58	The TRS R consists of the following rules:
33.77/9.58	
33.77/9.58	   a(x1) -> x1
33.77/9.58	   a(b(b(x1))) -> b(b(b(a(c(x1)))))
33.77/9.58	   c(b(x1)) -> a(x1)
33.77/9.58	
33.77/9.58	Q is empty.
33.77/9.58	We have to consider all minimal (P,Q,R)-chains.
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(5) DependencyGraphProof (EQUIVALENT)
33.77/9.58	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(6)
33.77/9.58	Obligation:
33.77/9.58	Q DP problem:
33.77/9.58	The TRS P consists of the following rules:
33.77/9.58	
33.77/9.58	   A(b(b(x1))) -> A(c(x1))
33.77/9.58	
33.77/9.58	The TRS R consists of the following rules:
33.77/9.58	
33.77/9.58	   a(x1) -> x1
33.77/9.58	   a(b(b(x1))) -> b(b(b(a(c(x1)))))
33.77/9.58	   c(b(x1)) -> a(x1)
33.77/9.58	
33.77/9.58	Q is empty.
33.77/9.58	We have to consider all minimal (P,Q,R)-chains.
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(7) QDPOrderProof (EQUIVALENT)
33.77/9.58	We use the reduction pair processor [LPAR04,JAR06].
33.77/9.58	
33.77/9.58	
33.77/9.58	The following pairs can be oriented strictly and are deleted.
33.77/9.58	
33.77/9.58	   A(b(b(x1))) -> A(c(x1))
33.77/9.58	The remaining pairs can at least be oriented weakly.
33.77/9.58	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.77/9.58	
33.77/9.58	   <<<
33.77/9.58	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
33.77/9.58	>>>
33.77/9.58	
33.77/9.58	   <<<
33.77/9.58	 POL(b(x_1)) =  	[[-I], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 1A], [0A, -I, -I]] 	* 	x_1
33.77/9.58	>>>
33.77/9.58	
33.77/9.58	   <<<
33.77/9.58	 POL(c(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] 	* 	x_1
33.77/9.58	>>>
33.77/9.58	
33.77/9.58	   <<<
33.77/9.58	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [-I, 0A, 0A]] 	* 	x_1
33.77/9.58	>>>
33.77/9.58	
33.77/9.58	
33.77/9.58	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.77/9.58	
33.77/9.58	   c(b(x1)) -> a(x1)
33.77/9.58	   a(b(b(x1))) -> b(b(b(a(c(x1)))))
33.77/9.58	   a(x1) -> x1
33.77/9.58	
33.77/9.58	
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(8)
33.77/9.58	Obligation:
33.77/9.58	Q DP problem:
33.77/9.58	P is empty.
33.77/9.58	The TRS R consists of the following rules:
33.77/9.58	
33.77/9.58	   a(x1) -> x1
33.77/9.58	   a(b(b(x1))) -> b(b(b(a(c(x1)))))
33.77/9.58	   c(b(x1)) -> a(x1)
33.77/9.58	
33.77/9.58	Q is empty.
33.77/9.58	We have to consider all minimal (P,Q,R)-chains.
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(9) PisEmptyProof (EQUIVALENT)
33.77/9.58	The TRS P is empty. Hence, there is no (P,Q,R) chain.
33.77/9.58	----------------------------------------
33.77/9.58	
33.77/9.58	(10)
33.77/9.58	YES
34.17/10.86	EOF