35.54/9.83	YES
35.70/9.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
35.70/9.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
35.70/9.87	
35.70/9.87	
35.70/9.87	Termination w.r.t. Q of the given QTRS could be proven:
35.70/9.87	
35.70/9.87	(0) QTRS
35.70/9.87	(1) DependencyPairsProof [EQUIVALENT, 15 ms]
35.70/9.87	(2) QDP
35.70/9.87	(3) MRRProof [EQUIVALENT, 68 ms]
35.70/9.87	(4) QDP
35.70/9.87	(5) QDPOrderProof [EQUIVALENT, 107 ms]
35.70/9.87	(6) QDP
35.70/9.87	(7) PisEmptyProof [EQUIVALENT, 0 ms]
35.70/9.87	(8) YES
35.70/9.87	
35.70/9.87	
35.70/9.87	----------------------------------------
35.70/9.87	
35.70/9.87	(0)
35.70/9.87	Obligation:
35.70/9.87	Q restricted rewrite system:
35.70/9.87	The TRS R consists of the following rules:
35.70/9.87	
35.70/9.87	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
35.70/9.87	   1(1(0(1(x1)))) -> 0(0(0(0(x1))))
35.70/9.87	
35.70/9.87	Q is empty.
35.70/9.87	
35.70/9.87	----------------------------------------
35.70/9.87	
35.70/9.87	(1) DependencyPairsProof (EQUIVALENT)
35.70/9.87	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
35.70/9.87	----------------------------------------
35.70/9.87	
35.70/9.87	(2)
35.70/9.87	Obligation:
35.70/9.87	Q DP problem:
35.70/9.87	The TRS P consists of the following rules:
35.70/9.87	
35.70/9.87	   0^1(0(0(0(x1)))) -> 0^1(1(1(1(x1))))
35.70/9.87	   0^1(0(0(0(x1)))) -> 1^1(1(1(x1)))
35.70/9.87	   0^1(0(0(0(x1)))) -> 1^1(1(x1))
35.70/9.87	   0^1(0(0(0(x1)))) -> 1^1(x1)
35.70/9.87	   1^1(1(0(1(x1)))) -> 0^1(0(0(0(x1))))
35.70/9.87	   1^1(1(0(1(x1)))) -> 0^1(0(0(x1)))
35.70/9.87	   1^1(1(0(1(x1)))) -> 0^1(0(x1))
35.70/9.87	   1^1(1(0(1(x1)))) -> 0^1(x1)
35.70/9.87	
35.70/9.87	The TRS R consists of the following rules:
35.70/9.87	
35.70/9.87	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
35.70/9.87	   1(1(0(1(x1)))) -> 0(0(0(0(x1))))
35.70/9.87	
35.70/9.87	Q is empty.
35.70/9.87	We have to consider all minimal (P,Q,R)-chains.
35.70/9.87	----------------------------------------
35.70/9.87	
35.70/9.87	(3) MRRProof (EQUIVALENT)
35.70/9.87	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.
35.70/9.87	
35.70/9.87	Strictly oriented dependency pairs:
35.70/9.87	
35.70/9.87	   0^1(0(0(0(x1)))) -> 1^1(1(1(x1)))
35.70/9.87	   0^1(0(0(0(x1)))) -> 1^1(1(x1))
35.70/9.87	   0^1(0(0(0(x1)))) -> 1^1(x1)
35.70/9.87	   1^1(1(0(1(x1)))) -> 0^1(0(0(0(x1))))
35.70/9.87	   1^1(1(0(1(x1)))) -> 0^1(0(0(x1)))
35.70/9.87	   1^1(1(0(1(x1)))) -> 0^1(0(x1))
35.70/9.87	   1^1(1(0(1(x1)))) -> 0^1(x1)
35.70/9.87	
35.70/9.87	
35.70/9.87	Used ordering: Polynomial interpretation [POLO]:
35.70/9.87	
35.70/9.87	   POL(0(x_1)) = 2 + x_1
35.70/9.87	   POL(0^1(x_1)) = 1 + x_1
35.70/9.87	   POL(1(x_1)) = 2 + x_1
35.70/9.87	   POL(1^1(x_1)) = 2 + x_1
35.70/9.87	
35.70/9.87	
35.70/9.87	----------------------------------------
35.70/9.87	
35.70/9.87	(4)
35.70/9.87	Obligation:
35.70/9.87	Q DP problem:
35.70/9.87	The TRS P consists of the following rules:
35.70/9.87	
35.70/9.87	   0^1(0(0(0(x1)))) -> 0^1(1(1(1(x1))))
35.70/9.87	
35.70/9.87	The TRS R consists of the following rules:
35.70/9.87	
35.70/9.87	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
35.70/9.87	   1(1(0(1(x1)))) -> 0(0(0(0(x1))))
35.70/9.87	
35.70/9.87	Q is empty.
35.70/9.87	We have to consider all minimal (P,Q,R)-chains.
35.70/9.87	----------------------------------------
35.70/9.87	
35.70/9.87	(5) QDPOrderProof (EQUIVALENT)
35.70/9.87	We use the reduction pair processor [LPAR04,JAR06].
35.70/9.87	
35.70/9.87	
35.70/9.87	The following pairs can be oriented strictly and are deleted.
35.70/9.87	
35.70/9.87	   0^1(0(0(0(x1)))) -> 0^1(1(1(1(x1))))
35.70/9.87	The remaining pairs can at least be oriented weakly.
35.70/9.87	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
35.70/9.87	
35.70/9.87	   <<<
35.70/9.87	 POL(0^1(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
35.70/9.87	>>>
35.70/9.87	
35.70/9.87	   <<<
35.70/9.87	 POL(0(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 1A, 0A], [0A, -I, 0A], [-I, 0A, -I]] 	* 	x_1
35.70/9.87	>>>
35.70/9.87	
35.70/9.87	   <<<
35.70/9.87	 POL(1(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, -I, 0A], [1A, 0A, 1A], [0A, -I, -I]] 	* 	x_1
35.70/9.87	>>>
35.70/9.87	
35.70/9.87	
35.70/9.87	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.70/9.87	
35.70/9.87	   1(1(0(1(x1)))) -> 0(0(0(0(x1))))
35.70/9.87	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
35.70/9.87	
35.70/9.87	
35.70/9.87	----------------------------------------
35.70/9.87	
35.70/9.87	(6)
35.70/9.87	Obligation:
35.70/9.87	Q DP problem:
35.70/9.87	P is empty.
35.70/9.87	The TRS R consists of the following rules:
35.70/9.87	
35.70/9.87	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
35.70/9.87	   1(1(0(1(x1)))) -> 0(0(0(0(x1))))
35.70/9.87	
35.70/9.87	Q is empty.
35.70/9.87	We have to consider all minimal (P,Q,R)-chains.
35.70/9.87	----------------------------------------
35.70/9.87	
35.70/9.87	(7) PisEmptyProof (EQUIVALENT)
35.70/9.87	The TRS P is empty. Hence, there is no (P,Q,R) chain.
35.70/9.87	----------------------------------------
35.70/9.87	
35.70/9.87	(8)
35.70/9.87	YES
35.92/9.96	EOF