30.57/8.73	YES
30.57/8.76	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
30.57/8.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.57/8.76	
30.57/8.76	
30.57/8.76	Termination w.r.t. Q of the given QTRS could be proven:
30.57/8.76	
30.57/8.76	(0) QTRS
30.57/8.76	(1) DependencyPairsProof [EQUIVALENT, 6 ms]
30.57/8.76	(2) QDP
30.57/8.76	(3) DependencyGraphProof [EQUIVALENT, 7 ms]
30.57/8.76	(4) QDP
30.57/8.76	(5) QDPOrderProof [EQUIVALENT, 135 ms]
30.57/8.76	(6) QDP
30.57/8.76	(7) QDPOrderProof [EQUIVALENT, 34 ms]
30.57/8.76	(8) QDP
30.57/8.76	(9) PisEmptyProof [EQUIVALENT, 0 ms]
30.57/8.76	(10) YES
30.57/8.76	
30.57/8.76	
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(0)
30.57/8.76	Obligation:
30.57/8.76	Q restricted rewrite system:
30.57/8.76	The TRS R consists of the following rules:
30.57/8.76	
30.57/8.76	   a(x1) -> b(c(x1))
30.57/8.76	   a(b(b(x1))) -> b(b(a(a(x1))))
30.57/8.76	   b(a(x1)) -> x1
30.57/8.76	
30.57/8.76	Q is empty.
30.57/8.76	
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(1) DependencyPairsProof (EQUIVALENT)
30.57/8.76	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(2)
30.57/8.76	Obligation:
30.57/8.76	Q DP problem:
30.57/8.76	The TRS P consists of the following rules:
30.57/8.76	
30.57/8.76	   A(x1) -> B(c(x1))
30.57/8.76	   A(b(b(x1))) -> B(b(a(a(x1))))
30.57/8.76	   A(b(b(x1))) -> B(a(a(x1)))
30.57/8.76	   A(b(b(x1))) -> A(a(x1))
30.57/8.76	   A(b(b(x1))) -> A(x1)
30.57/8.76	
30.57/8.76	The TRS R consists of the following rules:
30.57/8.76	
30.57/8.76	   a(x1) -> b(c(x1))
30.57/8.76	   a(b(b(x1))) -> b(b(a(a(x1))))
30.57/8.76	   b(a(x1)) -> x1
30.57/8.76	
30.57/8.76	Q is empty.
30.57/8.76	We have to consider all minimal (P,Q,R)-chains.
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(3) DependencyGraphProof (EQUIVALENT)
30.57/8.76	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(4)
30.57/8.76	Obligation:
30.57/8.76	Q DP problem:
30.57/8.76	The TRS P consists of the following rules:
30.57/8.76	
30.57/8.76	   A(b(b(x1))) -> A(x1)
30.57/8.76	   A(b(b(x1))) -> A(a(x1))
30.57/8.76	
30.57/8.76	The TRS R consists of the following rules:
30.57/8.76	
30.57/8.76	   a(x1) -> b(c(x1))
30.57/8.76	   a(b(b(x1))) -> b(b(a(a(x1))))
30.57/8.76	   b(a(x1)) -> x1
30.57/8.76	
30.57/8.76	Q is empty.
30.57/8.76	We have to consider all minimal (P,Q,R)-chains.
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(5) QDPOrderProof (EQUIVALENT)
30.57/8.76	We use the reduction pair processor [LPAR04,JAR06].
30.57/8.76	
30.57/8.76	
30.57/8.76	The following pairs can be oriented strictly and are deleted.
30.57/8.76	
30.57/8.76	   A(b(b(x1))) -> A(x1)
30.57/8.76	The remaining pairs can at least be oriented weakly.
30.57/8.76	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.57/8.76	
30.57/8.76	   <<<
30.57/8.76	 POL(A(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
30.57/8.76	>>>
30.57/8.76	
30.57/8.76	   <<<
30.57/8.76	 POL(b(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, -I], [-I, 0A, 1A], [-I, 0A, 0A]] 	* 	x_1
30.57/8.76	>>>
30.57/8.76	
30.57/8.76	   <<<
30.57/8.76	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, -I], [-I, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
30.57/8.76	>>>
30.57/8.76	
30.57/8.76	   <<<
30.57/8.76	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
30.57/8.76	>>>
30.57/8.76	
30.57/8.76	
30.57/8.76	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.57/8.76	
30.57/8.76	   a(x1) -> b(c(x1))
30.57/8.76	   a(b(b(x1))) -> b(b(a(a(x1))))
30.57/8.76	   b(a(x1)) -> x1
30.57/8.76	
30.57/8.76	
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(6)
30.57/8.76	Obligation:
30.57/8.76	Q DP problem:
30.57/8.76	The TRS P consists of the following rules:
30.57/8.76	
30.57/8.76	   A(b(b(x1))) -> A(a(x1))
30.57/8.76	
30.57/8.76	The TRS R consists of the following rules:
30.57/8.76	
30.57/8.76	   a(x1) -> b(c(x1))
30.57/8.76	   a(b(b(x1))) -> b(b(a(a(x1))))
30.57/8.76	   b(a(x1)) -> x1
30.57/8.76	
30.57/8.76	Q is empty.
30.57/8.76	We have to consider all minimal (P,Q,R)-chains.
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(7) QDPOrderProof (EQUIVALENT)
30.57/8.76	We use the reduction pair processor [LPAR04,JAR06].
30.57/8.76	
30.57/8.76	
30.57/8.76	The following pairs can be oriented strictly and are deleted.
30.57/8.76	
30.57/8.76	   A(b(b(x1))) -> A(a(x1))
30.57/8.76	The remaining pairs can at least be oriented weakly.
30.57/8.76	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.57/8.76	
30.57/8.76	   <<<
30.57/8.76	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
30.57/8.76	>>>
30.57/8.76	
30.57/8.76	   <<<
30.57/8.76	 POL(b(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[-I, 0A, 0A], [1A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
30.57/8.76	>>>
30.57/8.76	
30.57/8.76	   <<<
30.57/8.76	 POL(a(x_1)) =  	[[1A], [1A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
30.57/8.76	>>>
30.57/8.76	
30.57/8.76	   <<<
30.57/8.76	 POL(c(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [0A, 0A, -I], [0A, -I, 0A]] 	* 	x_1
30.57/8.76	>>>
30.57/8.76	
30.57/8.76	
30.57/8.76	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.57/8.76	
30.57/8.76	   a(x1) -> b(c(x1))
30.57/8.76	   a(b(b(x1))) -> b(b(a(a(x1))))
30.57/8.76	   b(a(x1)) -> x1
30.57/8.76	
30.57/8.76	
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(8)
30.57/8.76	Obligation:
30.57/8.76	Q DP problem:
30.57/8.76	P is empty.
30.57/8.76	The TRS R consists of the following rules:
30.57/8.76	
30.57/8.76	   a(x1) -> b(c(x1))
30.57/8.76	   a(b(b(x1))) -> b(b(a(a(x1))))
30.57/8.76	   b(a(x1)) -> x1
30.57/8.76	
30.57/8.76	Q is empty.
30.57/8.76	We have to consider all minimal (P,Q,R)-chains.
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(9) PisEmptyProof (EQUIVALENT)
30.57/8.76	The TRS P is empty. Hence, there is no (P,Q,R) chain.
30.57/8.76	----------------------------------------
30.57/8.76	
30.57/8.76	(10)
30.57/8.76	YES
31.06/8.90	EOF