31.73/9.02	YES
32.19/9.11	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
32.19/9.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.19/9.11	
32.19/9.11	
32.19/9.11	Termination w.r.t. Q of the given QTRS could be proven:
32.19/9.11	
32.19/9.11	(0) QTRS
32.19/9.11	(1) DependencyPairsProof [EQUIVALENT, 5 ms]
32.19/9.11	(2) QDP
32.19/9.11	(3) QDPOrderProof [EQUIVALENT, 100 ms]
32.19/9.11	(4) QDP
32.19/9.11	(5) QDPOrderProof [EQUIVALENT, 38 ms]
32.19/9.11	(6) QDP
32.19/9.11	(7) PisEmptyProof [EQUIVALENT, 0 ms]
32.19/9.11	(8) YES
32.19/9.11	
32.19/9.11	
32.19/9.11	----------------------------------------
32.19/9.11	
32.19/9.11	(0)
32.19/9.11	Obligation:
32.19/9.11	Q restricted rewrite system:
32.19/9.11	The TRS R consists of the following rules:
32.19/9.11	
32.19/9.11	   a(x1) -> x1
32.19/9.11	   a(x1) -> b(c(b(x1)))
32.19/9.11	   a(b(b(x1))) -> b(b(a(a(x1))))
32.19/9.11	
32.19/9.11	Q is empty.
32.19/9.11	
32.19/9.11	----------------------------------------
32.19/9.11	
32.19/9.11	(1) DependencyPairsProof (EQUIVALENT)
32.19/9.11	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.19/9.11	----------------------------------------
32.19/9.11	
32.19/9.11	(2)
32.19/9.11	Obligation:
32.19/9.11	Q DP problem:
32.19/9.11	The TRS P consists of the following rules:
32.19/9.11	
32.19/9.11	   A(b(b(x1))) -> A(a(x1))
32.19/9.11	   A(b(b(x1))) -> A(x1)
32.19/9.11	
32.19/9.11	The TRS R consists of the following rules:
32.19/9.11	
32.19/9.11	   a(x1) -> x1
32.19/9.11	   a(x1) -> b(c(b(x1)))
32.19/9.11	   a(b(b(x1))) -> b(b(a(a(x1))))
32.19/9.11	
32.19/9.11	Q is empty.
32.19/9.11	We have to consider all minimal (P,Q,R)-chains.
32.19/9.11	----------------------------------------
32.19/9.11	
32.19/9.11	(3) QDPOrderProof (EQUIVALENT)
32.19/9.11	We use the reduction pair processor [LPAR04,JAR06].
32.19/9.11	
32.19/9.11	
32.19/9.11	The following pairs can be oriented strictly and are deleted.
32.19/9.11	
32.19/9.11	   A(b(b(x1))) -> A(x1)
32.19/9.11	The remaining pairs can at least be oriented weakly.
32.19/9.11	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.19/9.11	
32.19/9.11	   <<<
32.19/9.11	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
32.19/9.11	>>>
32.19/9.11	
32.19/9.11	   <<<
32.19/9.11	 POL(b(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-I, 1A, -I], [0A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
32.19/9.11	>>>
32.19/9.11	
32.19/9.11	   <<<
32.19/9.11	 POL(a(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [0A, 1A, 0A]] 	* 	x_1
32.19/9.11	>>>
32.19/9.11	
32.19/9.11	   <<<
32.19/9.11	 POL(c(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
32.19/9.11	>>>
32.19/9.11	
32.19/9.11	
32.19/9.11	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.19/9.11	
32.19/9.11	   a(x1) -> x1
32.19/9.11	   a(x1) -> b(c(b(x1)))
32.19/9.11	   a(b(b(x1))) -> b(b(a(a(x1))))
32.19/9.11	
32.19/9.11	
32.19/9.11	----------------------------------------
32.19/9.11	
32.19/9.11	(4)
32.19/9.11	Obligation:
32.19/9.11	Q DP problem:
32.19/9.11	The TRS P consists of the following rules:
32.19/9.11	
32.19/9.11	   A(b(b(x1))) -> A(a(x1))
32.19/9.11	
32.19/9.11	The TRS R consists of the following rules:
32.19/9.11	
32.19/9.11	   a(x1) -> x1
32.19/9.11	   a(x1) -> b(c(b(x1)))
32.19/9.11	   a(b(b(x1))) -> b(b(a(a(x1))))
32.19/9.11	
32.19/9.11	Q is empty.
32.19/9.11	We have to consider all minimal (P,Q,R)-chains.
32.19/9.11	----------------------------------------
32.19/9.11	
32.19/9.11	(5) QDPOrderProof (EQUIVALENT)
32.19/9.11	We use the reduction pair processor [LPAR04,JAR06].
32.19/9.11	
32.19/9.11	
32.19/9.11	The following pairs can be oriented strictly and are deleted.
32.19/9.11	
32.19/9.11	   A(b(b(x1))) -> A(a(x1))
32.19/9.11	The remaining pairs can at least be oriented weakly.
32.19/9.11	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
32.19/9.11	
32.19/9.11	   <<<
32.19/9.11	 POL(A(x_1)) =  	[[-I]] 	 +  	[[1A, 0A, 0A]] 	* 	x_1
32.19/9.11	>>>
32.19/9.11	
32.19/9.11	   <<<
32.19/9.11	 POL(b(x_1)) =  	[[-I], [1A], [-I]] 	 +  	[[0A, 0A, -I], [1A, 0A, 0A], [0A, -I, -I]] 	* 	x_1
32.19/9.11	>>>
32.19/9.11	
32.19/9.11	   <<<
32.19/9.11	 POL(a(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
32.19/9.11	>>>
32.19/9.11	
32.19/9.11	   <<<
32.19/9.11	 POL(c(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
32.19/9.11	>>>
32.19/9.11	
32.19/9.11	
32.19/9.11	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
32.19/9.11	
32.19/9.11	   a(x1) -> x1
32.19/9.11	   a(x1) -> b(c(b(x1)))
32.19/9.11	   a(b(b(x1))) -> b(b(a(a(x1))))
32.19/9.11	
32.19/9.11	
32.19/9.11	----------------------------------------
32.19/9.11	
32.19/9.11	(6)
32.19/9.11	Obligation:
32.19/9.11	Q DP problem:
32.19/9.11	P is empty.
32.19/9.11	The TRS R consists of the following rules:
32.19/9.11	
32.19/9.11	   a(x1) -> x1
32.19/9.11	   a(x1) -> b(c(b(x1)))
32.19/9.11	   a(b(b(x1))) -> b(b(a(a(x1))))
32.19/9.11	
32.19/9.11	Q is empty.
32.19/9.11	We have to consider all minimal (P,Q,R)-chains.
32.19/9.11	----------------------------------------
32.19/9.11	
32.19/9.11	(7) PisEmptyProof (EQUIVALENT)
32.19/9.11	The TRS P is empty. Hence, there is no (P,Q,R) chain.
32.19/9.11	----------------------------------------
32.19/9.11	
32.19/9.11	(8)
32.19/9.11	YES
32.56/9.29	EOF