321.92/82.95	YES
321.92/83.00	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
321.92/83.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
321.92/83.00	
321.92/83.00	
321.92/83.00	Termination w.r.t. Q of the given QTRS could be proven:
321.92/83.00	
321.92/83.00	(0) QTRS
321.92/83.00	(1) QTRS Reverse [EQUIVALENT, 0 ms]
321.92/83.00	(2) QTRS
321.92/83.00	(3) DependencyPairsProof [EQUIVALENT, 19 ms]
321.92/83.00	(4) QDP
321.92/83.00	(5) MRRProof [EQUIVALENT, 0 ms]
321.92/83.00	(6) QDP
321.92/83.00	(7) QDPOrderProof [EQUIVALENT, 2389 ms]
321.92/83.00	(8) QDP
321.92/83.00	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
321.92/83.00	(10) TRUE
321.92/83.00	
321.92/83.00	
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(0)
321.92/83.00	Obligation:
321.92/83.00	Q restricted rewrite system:
321.92/83.00	The TRS R consists of the following rules:
321.92/83.00	
321.92/83.00	   a(a(a(x1))) -> b(b(b(x1)))
321.92/83.00	   b(b(b(b(x1)))) -> a(b(b(a(x1))))
321.92/83.00	
321.92/83.00	Q is empty.
321.92/83.00	
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(1) QTRS Reverse (EQUIVALENT)
321.92/83.00	We applied the QTRS Reverse Processor [REVERSE].
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(2)
321.92/83.00	Obligation:
321.92/83.00	Q restricted rewrite system:
321.92/83.00	The TRS R consists of the following rules:
321.92/83.00	
321.92/83.00	   a(a(a(x1))) -> b(b(b(x1)))
321.92/83.00	   b(b(b(b(x1)))) -> a(b(b(a(x1))))
321.92/83.00	
321.92/83.00	Q is empty.
321.92/83.00	
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(3) DependencyPairsProof (EQUIVALENT)
321.92/83.00	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(4)
321.92/83.00	Obligation:
321.92/83.00	Q DP problem:
321.92/83.00	The TRS P consists of the following rules:
321.92/83.00	
321.92/83.00	   A(a(a(x1))) -> B(b(b(x1)))
321.92/83.00	   A(a(a(x1))) -> B(b(x1))
321.92/83.00	   A(a(a(x1))) -> B(x1)
321.92/83.00	   B(b(b(b(x1)))) -> A(b(b(a(x1))))
321.92/83.00	   B(b(b(b(x1)))) -> B(b(a(x1)))
321.92/83.00	   B(b(b(b(x1)))) -> B(a(x1))
321.92/83.00	   B(b(b(b(x1)))) -> A(x1)
321.92/83.00	
321.92/83.00	The TRS R consists of the following rules:
321.92/83.00	
321.92/83.00	   a(a(a(x1))) -> b(b(b(x1)))
321.92/83.00	   b(b(b(b(x1)))) -> a(b(b(a(x1))))
321.92/83.00	
321.92/83.00	Q is empty.
321.92/83.00	We have to consider all minimal (P,Q,R)-chains.
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(5) MRRProof (EQUIVALENT)
321.92/83.00	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.
321.92/83.00	
321.92/83.00	Strictly oriented dependency pairs:
321.92/83.00	
321.92/83.00	   A(a(a(x1))) -> B(b(x1))
321.92/83.00	   A(a(a(x1))) -> B(x1)
321.92/83.00	   B(b(b(b(x1)))) -> B(b(a(x1)))
321.92/83.00	   B(b(b(b(x1)))) -> B(a(x1))
321.92/83.00	   B(b(b(b(x1)))) -> A(x1)
321.92/83.00	
321.92/83.00	
321.92/83.00	Used ordering: Polynomial interpretation [POLO]:
321.92/83.00	
321.92/83.00	   POL(A(x_1)) = 2*x_1
321.92/83.00	   POL(B(x_1)) = 2*x_1
321.92/83.00	   POL(a(x_1)) = 2 + 2*x_1
321.92/83.00	   POL(b(x_1)) = 2 + 2*x_1
321.92/83.00	
321.92/83.00	
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(6)
321.92/83.00	Obligation:
321.92/83.00	Q DP problem:
321.92/83.00	The TRS P consists of the following rules:
321.92/83.00	
321.92/83.00	   A(a(a(x1))) -> B(b(b(x1)))
321.92/83.00	   B(b(b(b(x1)))) -> A(b(b(a(x1))))
321.92/83.00	
321.92/83.00	The TRS R consists of the following rules:
321.92/83.00	
321.92/83.00	   a(a(a(x1))) -> b(b(b(x1)))
321.92/83.00	   b(b(b(b(x1)))) -> a(b(b(a(x1))))
321.92/83.00	
321.92/83.00	Q is empty.
321.92/83.00	We have to consider all minimal (P,Q,R)-chains.
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(7) QDPOrderProof (EQUIVALENT)
321.92/83.00	We use the reduction pair processor [LPAR04,JAR06].
321.92/83.00	
321.92/83.00	
321.92/83.00	The following pairs can be oriented strictly and are deleted.
321.92/83.00	
321.92/83.00	   A(a(a(x1))) -> B(b(b(x1)))
321.92/83.00	The remaining pairs can at least be oriented weakly.
321.92/83.00	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
321.92/83.00	
321.92/83.00	   <<<
321.92/83.00	 POL(A(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A, -I, 0A]] 	* 	x_1
321.92/83.00	>>>
321.92/83.00	
321.92/83.00	   <<<
321.92/83.00	 POL(a(x_1)) =  	[[0A], [0A], [1A], [0A], [-I]] 	 +  	[[-I, 0A, -I, -I, 0A], [1A, -I, 1A, -I, 0A], [-I, 1A, 0A, 0A, 1A], [0A, 0A, -I, -I, 0A], [0A, 0A, 0A, -I, 0A]] 	* 	x_1
321.92/83.00	>>>
321.92/83.00	
321.92/83.00	   <<<
321.92/83.00	 POL(B(x_1)) =  	[[1A]] 	 +  	[[0A, -I, -I, 0A, 0A]] 	* 	x_1
321.92/83.00	>>>
321.92/83.00	
321.92/83.00	   <<<
321.92/83.00	 POL(b(x_1)) =  	[[0A], [0A], [-I], [-I], [-I]] 	 +  	[[-I, 0A, 0A, -I, 0A], [0A, -I, -I, -I, 1A], [0A, -I, 0A, 0A, 1A], [1A, 0A, 0A, -I, -I], [0A, -I, -I, 0A, 0A]] 	* 	x_1
321.92/83.00	>>>
321.92/83.00	
321.92/83.00	
321.92/83.00	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
321.92/83.00	
321.92/83.00	   b(b(b(b(x1)))) -> a(b(b(a(x1))))
321.92/83.00	   a(a(a(x1))) -> b(b(b(x1)))
321.92/83.00	
321.92/83.00	
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(8)
321.92/83.00	Obligation:
321.92/83.00	Q DP problem:
321.92/83.00	The TRS P consists of the following rules:
321.92/83.00	
321.92/83.00	   B(b(b(b(x1)))) -> A(b(b(a(x1))))
321.92/83.00	
321.92/83.00	The TRS R consists of the following rules:
321.92/83.00	
321.92/83.00	   a(a(a(x1))) -> b(b(b(x1)))
321.92/83.00	   b(b(b(b(x1)))) -> a(b(b(a(x1))))
321.92/83.00	
321.92/83.00	Q is empty.
321.92/83.00	We have to consider all minimal (P,Q,R)-chains.
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(9) DependencyGraphProof (EQUIVALENT)
321.92/83.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
321.92/83.00	----------------------------------------
321.92/83.00	
321.92/83.00	(10)
321.92/83.00	TRUE
322.35/83.07	EOF