33.24/9.36	YES
33.55/9.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
33.55/9.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
33.55/9.43	
33.55/9.43	
33.55/9.43	Termination w.r.t. Q of the given QTRS could be proven:
33.55/9.43	
33.55/9.43	(0) QTRS
33.55/9.43	(1) QTRS Reverse [EQUIVALENT, 0 ms]
33.55/9.43	(2) QTRS
33.55/9.43	(3) DependencyPairsProof [EQUIVALENT, 16 ms]
33.55/9.43	(4) QDP
33.55/9.43	(5) QDPOrderProof [EQUIVALENT, 1759 ms]
33.55/9.43	(6) QDP
33.55/9.43	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
33.55/9.43	(8) TRUE
33.55/9.43	
33.55/9.43	
33.55/9.43	----------------------------------------
33.55/9.43	
33.55/9.43	(0)
33.55/9.43	Obligation:
33.55/9.43	Q restricted rewrite system:
33.55/9.43	The TRS R consists of the following rules:
33.55/9.43	
33.55/9.43	   a(a(a(x1))) -> b(x1)
33.55/9.43	   a(b(b(x1))) -> b(a(a(b(a(x1)))))
33.55/9.43	
33.55/9.43	Q is empty.
33.55/9.43	
33.55/9.43	----------------------------------------
33.55/9.43	
33.55/9.43	(1) QTRS Reverse (EQUIVALENT)
33.55/9.43	We applied the QTRS Reverse Processor [REVERSE].
33.55/9.43	----------------------------------------
33.55/9.43	
33.55/9.43	(2)
33.55/9.43	Obligation:
33.55/9.43	Q restricted rewrite system:
33.55/9.43	The TRS R consists of the following rules:
33.55/9.43	
33.55/9.43	   a(a(a(x1))) -> b(x1)
33.55/9.43	   b(b(a(x1))) -> a(b(a(a(b(x1)))))
33.55/9.43	
33.55/9.43	Q is empty.
33.55/9.43	
33.55/9.43	----------------------------------------
33.55/9.43	
33.55/9.43	(3) DependencyPairsProof (EQUIVALENT)
33.55/9.43	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
33.55/9.43	----------------------------------------
33.55/9.43	
33.55/9.43	(4)
33.55/9.43	Obligation:
33.55/9.43	Q DP problem:
33.55/9.43	The TRS P consists of the following rules:
33.55/9.43	
33.55/9.43	   A(a(a(x1))) -> B(x1)
33.55/9.43	   B(b(a(x1))) -> A(b(a(a(b(x1)))))
33.55/9.43	   B(b(a(x1))) -> B(a(a(b(x1))))
33.55/9.43	   B(b(a(x1))) -> A(a(b(x1)))
33.55/9.43	   B(b(a(x1))) -> A(b(x1))
33.55/9.43	   B(b(a(x1))) -> B(x1)
33.55/9.43	
33.55/9.43	The TRS R consists of the following rules:
33.55/9.43	
33.55/9.43	   a(a(a(x1))) -> b(x1)
33.55/9.43	   b(b(a(x1))) -> a(b(a(a(b(x1)))))
33.55/9.43	
33.55/9.43	Q is empty.
33.55/9.43	We have to consider all minimal (P,Q,R)-chains.
33.55/9.43	----------------------------------------
33.55/9.43	
33.55/9.43	(5) QDPOrderProof (EQUIVALENT)
33.55/9.43	We use the reduction pair processor [LPAR04,JAR06].
33.55/9.43	
33.55/9.43	
33.55/9.43	The following pairs can be oriented strictly and are deleted.
33.55/9.43	
33.55/9.43	   B(b(a(x1))) -> A(b(a(a(b(x1)))))
33.55/9.43	   B(b(a(x1))) -> B(a(a(b(x1))))
33.55/9.43	   B(b(a(x1))) -> A(a(b(x1)))
33.55/9.43	   B(b(a(x1))) -> A(b(x1))
33.55/9.43	   B(b(a(x1))) -> B(x1)
33.55/9.43	The remaining pairs can at least be oriented weakly.
33.55/9.43	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
33.55/9.43	
33.55/9.43	   <<<
33.55/9.43	 POL(A(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
33.55/9.43	>>>
33.55/9.43	
33.55/9.43	   <<<
33.55/9.43	 POL(a(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[1A, -I, 0A], [0A, -I, -I], [0A, 1A, -I]] 	* 	x_1
33.55/9.43	>>>
33.55/9.43	
33.55/9.43	   <<<
33.55/9.43	 POL(B(x_1)) =  	[[1A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
33.55/9.43	>>>
33.55/9.43	
33.55/9.43	   <<<
33.55/9.43	 POL(b(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, 0A, -I], [-I, 0A, 1A], [-I, 1A, -I]] 	* 	x_1
33.55/9.43	>>>
33.55/9.43	
33.55/9.43	
33.55/9.43	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
33.55/9.43	
33.55/9.43	   b(b(a(x1))) -> a(b(a(a(b(x1)))))
33.55/9.43	   a(a(a(x1))) -> b(x1)
33.55/9.43	
33.55/9.43	
33.55/9.43	----------------------------------------
33.55/9.43	
33.55/9.43	(6)
33.55/9.43	Obligation:
33.55/9.43	Q DP problem:
33.55/9.43	The TRS P consists of the following rules:
33.55/9.43	
33.55/9.43	   A(a(a(x1))) -> B(x1)
33.55/9.43	
33.55/9.43	The TRS R consists of the following rules:
33.55/9.43	
33.55/9.43	   a(a(a(x1))) -> b(x1)
33.55/9.43	   b(b(a(x1))) -> a(b(a(a(b(x1)))))
33.55/9.43	
33.55/9.43	Q is empty.
33.55/9.43	We have to consider all minimal (P,Q,R)-chains.
33.55/9.43	----------------------------------------
33.55/9.43	
33.55/9.43	(7) DependencyGraphProof (EQUIVALENT)
33.55/9.43	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
33.55/9.43	----------------------------------------
33.55/9.44	
33.55/9.44	(8)
33.55/9.44	TRUE
33.97/9.56	EOF