49.45/13.47	YES
49.45/13.50	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
49.45/13.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
49.45/13.50	
49.45/13.50	
49.45/13.50	Termination w.r.t. Q of the given QTRS could be proven:
49.45/13.50	
49.45/13.50	(0) QTRS
49.45/13.50	(1) DependencyPairsProof [EQUIVALENT, 17 ms]
49.45/13.50	(2) QDP
49.45/13.50	(3) MRRProof [EQUIVALENT, 93 ms]
49.45/13.50	(4) QDP
49.45/13.50	(5) QDPOrderProof [EQUIVALENT, 1916 ms]
49.45/13.50	(6) QDP
49.45/13.50	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
49.45/13.50	(8) TRUE
49.45/13.50	
49.45/13.50	
49.45/13.50	----------------------------------------
49.45/13.50	
49.45/13.50	(0)
49.45/13.50	Obligation:
49.45/13.50	Q restricted rewrite system:
49.45/13.50	The TRS R consists of the following rules:
49.45/13.50	
49.45/13.50	   b(b(b(b(x1)))) -> a(a(b(b(x1))))
49.45/13.50	   a(b(a(a(x1)))) -> b(a(a(b(x1))))
49.45/13.50	
49.45/13.50	Q is empty.
49.45/13.50	
49.45/13.50	----------------------------------------
49.45/13.50	
49.45/13.50	(1) DependencyPairsProof (EQUIVALENT)
49.45/13.50	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
49.45/13.50	----------------------------------------
49.45/13.50	
49.45/13.50	(2)
49.45/13.50	Obligation:
49.45/13.50	Q DP problem:
49.45/13.50	The TRS P consists of the following rules:
49.45/13.50	
49.45/13.50	   B(b(b(b(x1)))) -> A(a(b(b(x1))))
49.45/13.50	   B(b(b(b(x1)))) -> A(b(b(x1)))
49.45/13.50	   A(b(a(a(x1)))) -> B(a(a(b(x1))))
49.45/13.50	   A(b(a(a(x1)))) -> A(a(b(x1)))
49.45/13.50	   A(b(a(a(x1)))) -> A(b(x1))
49.45/13.50	   A(b(a(a(x1)))) -> B(x1)
49.45/13.50	
49.45/13.50	The TRS R consists of the following rules:
49.45/13.50	
49.45/13.50	   b(b(b(b(x1)))) -> a(a(b(b(x1))))
49.45/13.50	   a(b(a(a(x1)))) -> b(a(a(b(x1))))
49.45/13.50	
49.45/13.50	Q is empty.
49.45/13.50	We have to consider all minimal (P,Q,R)-chains.
49.45/13.50	----------------------------------------
49.45/13.50	
49.45/13.50	(3) MRRProof (EQUIVALENT)
49.45/13.50	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
49.45/13.50	
49.45/13.50	Strictly oriented dependency pairs:
49.45/13.50	
49.45/13.50	   B(b(b(b(x1)))) -> A(b(b(x1)))
49.45/13.50	   A(b(a(a(x1)))) -> A(a(b(x1)))
49.45/13.50	   A(b(a(a(x1)))) -> A(b(x1))
49.45/13.50	   A(b(a(a(x1)))) -> B(x1)
49.45/13.50	
49.45/13.50	
49.45/13.50	Used ordering: Polynomial interpretation [POLO]:
49.45/13.50	
49.45/13.50	   POL(A(x_1)) = 1 + x_1
49.45/13.50	   POL(B(x_1)) = 1 + x_1
49.45/13.50	   POL(a(x_1)) = 1 + x_1
49.45/13.50	   POL(b(x_1)) = 1 + x_1
49.45/13.50	
49.45/13.50	
49.45/13.50	----------------------------------------
49.45/13.50	
49.45/13.50	(4)
49.45/13.50	Obligation:
49.45/13.50	Q DP problem:
49.45/13.50	The TRS P consists of the following rules:
49.45/13.50	
49.45/13.50	   B(b(b(b(x1)))) -> A(a(b(b(x1))))
49.45/13.50	   A(b(a(a(x1)))) -> B(a(a(b(x1))))
49.45/13.50	
49.45/13.50	The TRS R consists of the following rules:
49.45/13.50	
49.45/13.50	   b(b(b(b(x1)))) -> a(a(b(b(x1))))
49.45/13.50	   a(b(a(a(x1)))) -> b(a(a(b(x1))))
49.45/13.50	
49.45/13.50	Q is empty.
49.45/13.50	We have to consider all minimal (P,Q,R)-chains.
49.45/13.50	----------------------------------------
49.45/13.50	
49.45/13.50	(5) QDPOrderProof (EQUIVALENT)
49.45/13.50	We use the reduction pair processor [LPAR04,JAR06].
49.45/13.50	
49.45/13.50	
49.45/13.50	The following pairs can be oriented strictly and are deleted.
49.45/13.50	
49.45/13.50	   B(b(b(b(x1)))) -> A(a(b(b(x1))))
49.45/13.50	The remaining pairs can at least be oriented weakly.
49.45/13.50	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
49.45/13.50	
49.45/13.50	   <<<
49.45/13.50	 POL(B(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
49.45/13.50	>>>
49.45/13.50	
49.45/13.50	   <<<
49.45/13.50	 POL(b(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[1A, 0A, 0A], [0A, -I, -I], [0A, 1A, -I]] 	* 	x_1
49.45/13.50	>>>
49.45/13.50	
49.45/13.50	   <<<
49.45/13.50	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
49.45/13.50	>>>
49.45/13.50	
49.45/13.50	   <<<
49.45/13.50	 POL(a(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [0A, 1A, 1A]] 	* 	x_1
49.45/13.50	>>>
49.45/13.50	
49.45/13.50	
49.45/13.50	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
49.45/13.50	
49.45/13.50	   a(b(a(a(x1)))) -> b(a(a(b(x1))))
49.45/13.50	   b(b(b(b(x1)))) -> a(a(b(b(x1))))
49.45/13.50	
49.45/13.50	
49.45/13.50	----------------------------------------
49.45/13.50	
49.45/13.50	(6)
49.45/13.50	Obligation:
49.45/13.50	Q DP problem:
49.45/13.50	The TRS P consists of the following rules:
49.45/13.50	
49.45/13.50	   A(b(a(a(x1)))) -> B(a(a(b(x1))))
49.45/13.50	
49.45/13.50	The TRS R consists of the following rules:
49.45/13.50	
49.45/13.50	   b(b(b(b(x1)))) -> a(a(b(b(x1))))
49.45/13.50	   a(b(a(a(x1)))) -> b(a(a(b(x1))))
49.45/13.50	
49.45/13.50	Q is empty.
49.45/13.50	We have to consider all minimal (P,Q,R)-chains.
49.45/13.50	----------------------------------------
49.45/13.50	
49.45/13.50	(7) DependencyGraphProof (EQUIVALENT)
49.45/13.50	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
49.45/13.50	----------------------------------------
49.45/13.50	
49.45/13.50	(8)
49.45/13.50	TRUE
49.71/13.59	EOF