23.73/6.93	YES
26.89/9.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
26.89/9.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.89/9.98	
26.89/9.98	
26.89/9.98	Termination w.r.t. Q of the given QTRS could be proven:
26.89/9.98	
26.89/9.98	(0) QTRS
26.89/9.98	(1) QTRS Reverse [EQUIVALENT, 0 ms]
26.89/9.98	(2) QTRS
26.89/9.98	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
26.89/9.98	(4) QDP
26.89/9.98	(5) QDPOrderProof [EQUIVALENT, 277 ms]
26.89/9.98	(6) QDP
26.89/9.98	(7) QDPOrderProof [EQUIVALENT, 42 ms]
26.89/9.98	(8) QDP
26.89/9.98	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
26.89/9.98	(10) TRUE
26.89/9.98	
26.89/9.98	
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(0)
26.89/9.98	Obligation:
26.89/9.98	Q restricted rewrite system:
26.89/9.98	The TRS R consists of the following rules:
26.89/9.98	
26.89/9.98	   a(x1) -> b(x1)
26.89/9.98	   b(a(b(b(x1)))) -> b(b(b(a(a(x1)))))
26.89/9.98	
26.89/9.98	Q is empty.
26.89/9.98	
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(1) QTRS Reverse (EQUIVALENT)
26.89/9.98	We applied the QTRS Reverse Processor [REVERSE].
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(2)
26.89/9.98	Obligation:
26.89/9.98	Q restricted rewrite system:
26.89/9.98	The TRS R consists of the following rules:
26.89/9.98	
26.89/9.98	   a(x1) -> b(x1)
26.89/9.98	   b(b(a(b(x1)))) -> a(a(b(b(b(x1)))))
26.89/9.98	
26.89/9.98	Q is empty.
26.89/9.98	
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(3) DependencyPairsProof (EQUIVALENT)
26.89/9.98	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(4)
26.89/9.98	Obligation:
26.89/9.98	Q DP problem:
26.89/9.98	The TRS P consists of the following rules:
26.89/9.98	
26.89/9.98	   A(x1) -> B(x1)
26.89/9.98	   B(b(a(b(x1)))) -> A(a(b(b(b(x1)))))
26.89/9.98	   B(b(a(b(x1)))) -> A(b(b(b(x1))))
26.89/9.98	   B(b(a(b(x1)))) -> B(b(b(x1)))
26.89/9.98	   B(b(a(b(x1)))) -> B(b(x1))
26.89/9.98	
26.89/9.98	The TRS R consists of the following rules:
26.89/9.98	
26.89/9.98	   a(x1) -> b(x1)
26.89/9.98	   b(b(a(b(x1)))) -> a(a(b(b(b(x1)))))
26.89/9.98	
26.89/9.98	Q is empty.
26.89/9.98	We have to consider all minimal (P,Q,R)-chains.
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(5) QDPOrderProof (EQUIVALENT)
26.89/9.98	We use the reduction pair processor [LPAR04,JAR06].
26.89/9.98	
26.89/9.98	
26.89/9.98	The following pairs can be oriented strictly and are deleted.
26.89/9.98	
26.89/9.98	   B(b(a(b(x1)))) -> B(b(b(x1)))
26.89/9.98	   B(b(a(b(x1)))) -> B(b(x1))
26.89/9.98	The remaining pairs can at least be oriented weakly.
26.89/9.98	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.89/9.98	
26.89/9.98	   <<<
26.89/9.98	 POL(A(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
26.89/9.98	>>>
26.89/9.98	
26.89/9.98	   <<<
26.89/9.98	 POL(B(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
26.89/9.98	>>>
26.89/9.98	
26.89/9.98	   <<<
26.89/9.98	 POL(b(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
26.89/9.98	>>>
26.89/9.98	
26.89/9.98	   <<<
26.89/9.98	 POL(a(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 1A, 0A]] 	* 	x_1
26.89/9.98	>>>
26.89/9.98	
26.89/9.98	
26.89/9.98	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.89/9.98	
26.89/9.98	   b(b(a(b(x1)))) -> a(a(b(b(b(x1)))))
26.89/9.98	   a(x1) -> b(x1)
26.89/9.98	
26.89/9.98	
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(6)
26.89/9.98	Obligation:
26.89/9.98	Q DP problem:
26.89/9.98	The TRS P consists of the following rules:
26.89/9.98	
26.89/9.98	   A(x1) -> B(x1)
26.89/9.98	   B(b(a(b(x1)))) -> A(a(b(b(b(x1)))))
26.89/9.98	   B(b(a(b(x1)))) -> A(b(b(b(x1))))
26.89/9.98	
26.89/9.98	The TRS R consists of the following rules:
26.89/9.98	
26.89/9.98	   a(x1) -> b(x1)
26.89/9.98	   b(b(a(b(x1)))) -> a(a(b(b(b(x1)))))
26.89/9.98	
26.89/9.98	Q is empty.
26.89/9.98	We have to consider all minimal (P,Q,R)-chains.
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(7) QDPOrderProof (EQUIVALENT)
26.89/9.98	We use the reduction pair processor [LPAR04,JAR06].
26.89/9.98	
26.89/9.98	
26.89/9.98	The following pairs can be oriented strictly and are deleted.
26.89/9.98	
26.89/9.98	   A(x1) -> B(x1)
26.89/9.98	The remaining pairs can at least be oriented weakly.
26.89/9.98	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.89/9.98	
26.89/9.98	   <<<
26.89/9.98	 POL(A(x_1)) =  	[[-I]] 	 +  	[[1A, 1A, 1A]] 	* 	x_1
26.89/9.98	>>>
26.89/9.98	
26.89/9.98	   <<<
26.89/9.98	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
26.89/9.98	>>>
26.89/9.98	
26.89/9.98	   <<<
26.89/9.98	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [1A, -I, 0A], [0A, -I, -I]] 	* 	x_1
26.89/9.98	>>>
26.89/9.98	
26.89/9.98	   <<<
26.89/9.98	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, 1A], [1A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
26.89/9.98	>>>
26.89/9.98	
26.89/9.98	
26.89/9.98	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.89/9.98	
26.89/9.98	   b(b(a(b(x1)))) -> a(a(b(b(b(x1)))))
26.89/9.98	   a(x1) -> b(x1)
26.89/9.98	
26.89/9.98	
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(8)
26.89/9.98	Obligation:
26.89/9.98	Q DP problem:
26.89/9.98	The TRS P consists of the following rules:
26.89/9.98	
26.89/9.98	   B(b(a(b(x1)))) -> A(a(b(b(b(x1)))))
26.89/9.98	   B(b(a(b(x1)))) -> A(b(b(b(x1))))
26.89/9.98	
26.89/9.98	The TRS R consists of the following rules:
26.89/9.98	
26.89/9.98	   a(x1) -> b(x1)
26.89/9.98	   b(b(a(b(x1)))) -> a(a(b(b(b(x1)))))
26.89/9.98	
26.89/9.98	Q is empty.
26.89/9.98	We have to consider all minimal (P,Q,R)-chains.
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(9) DependencyGraphProof (EQUIVALENT)
26.89/9.98	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
26.89/9.98	----------------------------------------
26.89/9.98	
26.89/9.98	(10)
26.89/9.98	TRUE
26.93/11.13	EOF