263.68/67.87	YES
264.10/67.94	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
264.10/67.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
264.10/67.94	
264.10/67.94	
264.10/67.94	Termination w.r.t. Q of the given QTRS could be proven:
264.10/67.94	
264.10/67.94	(0) QTRS
264.10/67.94	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
264.10/67.94	(2) QDP
264.10/67.94	(3) MRRProof [EQUIVALENT, 95 ms]
264.10/67.94	(4) QDP
264.10/67.94	(5) QDPOrderProof [EQUIVALENT, 609 ms]
264.10/67.94	(6) QDP
264.10/67.94	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
264.10/67.94	(8) TRUE
264.10/67.94	
264.10/67.94	
264.10/67.94	----------------------------------------
264.10/67.94	
264.10/67.94	(0)
264.10/67.94	Obligation:
264.10/67.94	Q restricted rewrite system:
264.10/67.94	The TRS R consists of the following rules:
264.10/67.94	
264.10/67.94	   0(0(0(0(x1)))) -> 1(0(1(1(x1))))
264.10/67.94	   1(0(0(1(x1)))) -> 0(0(0(0(x1))))
264.10/67.94	
264.10/67.94	Q is empty.
264.10/67.94	
264.10/67.94	----------------------------------------
264.10/67.94	
264.10/67.94	(1) DependencyPairsProof (EQUIVALENT)
264.10/67.94	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
264.10/67.94	----------------------------------------
264.10/67.94	
264.10/67.94	(2)
264.10/67.94	Obligation:
264.10/67.94	Q DP problem:
264.10/67.94	The TRS P consists of the following rules:
264.10/67.94	
264.10/67.94	   0^1(0(0(0(x1)))) -> 1^1(0(1(1(x1))))
264.10/67.94	   0^1(0(0(0(x1)))) -> 0^1(1(1(x1)))
264.10/67.94	   0^1(0(0(0(x1)))) -> 1^1(1(x1))
264.10/67.94	   0^1(0(0(0(x1)))) -> 1^1(x1)
264.10/67.94	   1^1(0(0(1(x1)))) -> 0^1(0(0(0(x1))))
264.10/67.94	   1^1(0(0(1(x1)))) -> 0^1(0(0(x1)))
264.10/67.94	   1^1(0(0(1(x1)))) -> 0^1(0(x1))
264.10/67.94	   1^1(0(0(1(x1)))) -> 0^1(x1)
264.10/67.94	
264.10/67.94	The TRS R consists of the following rules:
264.10/67.94	
264.10/67.94	   0(0(0(0(x1)))) -> 1(0(1(1(x1))))
264.10/67.94	   1(0(0(1(x1)))) -> 0(0(0(0(x1))))
264.10/67.94	
264.10/67.94	Q is empty.
264.10/67.94	We have to consider all minimal (P,Q,R)-chains.
264.10/67.94	----------------------------------------
264.10/67.94	
264.10/67.94	(3) MRRProof (EQUIVALENT)
264.10/67.94	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.
264.10/67.94	
264.10/67.94	Strictly oriented dependency pairs:
264.10/67.94	
264.10/67.94	   0^1(0(0(0(x1)))) -> 0^1(1(1(x1)))
264.10/67.94	   0^1(0(0(0(x1)))) -> 1^1(1(x1))
264.10/67.94	   0^1(0(0(0(x1)))) -> 1^1(x1)
264.10/67.94	   1^1(0(0(1(x1)))) -> 0^1(0(0(x1)))
264.10/67.94	   1^1(0(0(1(x1)))) -> 0^1(0(x1))
264.10/67.94	   1^1(0(0(1(x1)))) -> 0^1(x1)
264.10/67.94	
264.10/67.94	
264.10/67.94	Used ordering: Polynomial interpretation [POLO]:
264.10/67.94	
264.10/67.94	   POL(0(x_1)) = 1 + x_1
264.10/67.94	   POL(0^1(x_1)) = x_1
264.10/67.94	   POL(1(x_1)) = 1 + x_1
264.10/67.94	   POL(1^1(x_1)) = x_1
264.10/67.94	
264.10/67.94	
264.10/67.94	----------------------------------------
264.10/67.94	
264.10/67.94	(4)
264.10/67.94	Obligation:
264.10/67.94	Q DP problem:
264.10/67.94	The TRS P consists of the following rules:
264.10/67.94	
264.10/67.94	   0^1(0(0(0(x1)))) -> 1^1(0(1(1(x1))))
264.10/67.94	   1^1(0(0(1(x1)))) -> 0^1(0(0(0(x1))))
264.10/67.94	
264.10/67.94	The TRS R consists of the following rules:
264.10/67.94	
264.10/67.94	   0(0(0(0(x1)))) -> 1(0(1(1(x1))))
264.10/67.94	   1(0(0(1(x1)))) -> 0(0(0(0(x1))))
264.10/67.94	
264.10/67.94	Q is empty.
264.10/67.94	We have to consider all minimal (P,Q,R)-chains.
264.10/67.94	----------------------------------------
264.10/67.94	
264.10/67.94	(5) QDPOrderProof (EQUIVALENT)
264.10/67.94	We use the reduction pair processor [LPAR04,JAR06].
264.10/67.94	
264.10/67.94	
264.10/67.94	The following pairs can be oriented strictly and are deleted.
264.10/67.94	
264.10/67.94	   0^1(0(0(0(x1)))) -> 1^1(0(1(1(x1))))
264.10/67.94	The remaining pairs can at least be oriented weakly.
264.10/67.94	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
264.10/67.94	
264.10/67.94	   <<<
264.10/67.94	 POL(0^1(x_1)) =  	[[-I]] 	 +  	[[1A, 0A, 0A, 0A, 0A]] 	* 	x_1
264.10/67.94	>>>
264.10/67.94	
264.10/67.94	   <<<
264.10/67.94	 POL(0(x_1)) =  	[[-I], [-I], [-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A, -I, -I], [0A, 0A, -I, -I, -I], [0A, 0A, 0A, 0A, 0A], [0A, 0A, -I, 0A, 1A], [-I, 0A, -I, -I, 0A]] 	* 	x_1
264.10/67.94	>>>
264.10/67.94	
264.10/67.94	   <<<
264.10/67.94	 POL(1^1(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A, 0A, 1A]] 	* 	x_1
264.10/67.94	>>>
264.10/67.94	
264.10/67.94	   <<<
264.10/67.94	 POL(1(x_1)) =  	[[-I], [-I], [-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A, 0A, 1A], [-I, -I, -I, -I, 0A], [0A, 0A, 0A, 0A, 1A], [0A, 0A, 0A, 0A, 1A], [-I, -I, -I, -I, 0A]] 	* 	x_1
264.10/67.94	>>>
264.10/67.94	
264.10/67.94	
264.10/67.94	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
264.10/67.94	
264.10/67.94	   1(0(0(1(x1)))) -> 0(0(0(0(x1))))
264.10/67.94	   0(0(0(0(x1)))) -> 1(0(1(1(x1))))
264.10/67.94	
264.10/67.94	
264.10/67.94	----------------------------------------
264.10/67.94	
264.10/67.94	(6)
264.10/67.94	Obligation:
264.10/67.94	Q DP problem:
264.10/67.94	The TRS P consists of the following rules:
264.10/67.94	
264.10/67.94	   1^1(0(0(1(x1)))) -> 0^1(0(0(0(x1))))
264.10/67.94	
264.10/67.94	The TRS R consists of the following rules:
264.10/67.94	
264.10/67.94	   0(0(0(0(x1)))) -> 1(0(1(1(x1))))
264.10/67.94	   1(0(0(1(x1)))) -> 0(0(0(0(x1))))
264.10/67.94	
264.10/67.94	Q is empty.
264.10/67.94	We have to consider all minimal (P,Q,R)-chains.
264.10/67.94	----------------------------------------
264.10/67.94	
264.10/67.94	(7) DependencyGraphProof (EQUIVALENT)
264.10/67.94	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
264.10/67.94	----------------------------------------
264.10/67.94	
264.10/67.94	(8)
264.10/67.94	TRUE
264.23/67.99	EOF