17.36/5.19	YES
17.46/5.22	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.46/5.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.46/5.22	
17.46/5.22	
17.46/5.22	Termination w.r.t. Q of the given QTRS could be proven:
17.46/5.22	
17.46/5.22	(0) QTRS
17.46/5.22	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
17.46/5.22	(2) QDP
17.46/5.22	(3) MRRProof [EQUIVALENT, 50 ms]
17.46/5.22	(4) QDP
17.46/5.22	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
17.46/5.22	(6) QDP
17.46/5.22	(7) QDPOrderProof [EQUIVALENT, 70 ms]
17.46/5.22	(8) QDP
17.46/5.22	(9) PisEmptyProof [EQUIVALENT, 0 ms]
17.46/5.22	(10) YES
17.46/5.22	
17.46/5.22	
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(0)
17.46/5.22	Obligation:
17.46/5.22	Q restricted rewrite system:
17.46/5.22	The TRS R consists of the following rules:
17.46/5.22	
17.46/5.22	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
17.46/5.22	   1(0(1(1(x1)))) -> 0(0(0(1(x1))))
17.46/5.22	
17.46/5.22	Q is empty.
17.46/5.22	
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(1) DependencyPairsProof (EQUIVALENT)
17.46/5.22	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(2)
17.46/5.22	Obligation:
17.46/5.22	Q DP problem:
17.46/5.22	The TRS P consists of the following rules:
17.46/5.22	
17.46/5.22	   0^1(0(0(0(x1)))) -> 0^1(1(1(1(x1))))
17.46/5.22	   0^1(0(0(0(x1)))) -> 1^1(1(1(x1)))
17.46/5.22	   0^1(0(0(0(x1)))) -> 1^1(1(x1))
17.46/5.22	   0^1(0(0(0(x1)))) -> 1^1(x1)
17.46/5.22	   1^1(0(1(1(x1)))) -> 0^1(0(0(1(x1))))
17.46/5.22	   1^1(0(1(1(x1)))) -> 0^1(0(1(x1)))
17.46/5.22	   1^1(0(1(1(x1)))) -> 0^1(1(x1))
17.46/5.22	
17.46/5.22	The TRS R consists of the following rules:
17.46/5.22	
17.46/5.22	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
17.46/5.22	   1(0(1(1(x1)))) -> 0(0(0(1(x1))))
17.46/5.22	
17.46/5.22	Q is empty.
17.46/5.22	We have to consider all minimal (P,Q,R)-chains.
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(3) MRRProof (EQUIVALENT)
17.46/5.22	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.
17.46/5.22	
17.46/5.22	Strictly oriented dependency pairs:
17.46/5.22	
17.46/5.22	   0^1(0(0(0(x1)))) -> 1^1(1(1(x1)))
17.46/5.22	   0^1(0(0(0(x1)))) -> 1^1(1(x1))
17.46/5.22	   0^1(0(0(0(x1)))) -> 1^1(x1)
17.46/5.22	   1^1(0(1(1(x1)))) -> 0^1(0(1(x1)))
17.46/5.22	   1^1(0(1(1(x1)))) -> 0^1(1(x1))
17.46/5.22	
17.46/5.22	
17.46/5.22	Used ordering: Polynomial interpretation [POLO]:
17.46/5.22	
17.46/5.22	   POL(0(x_1)) = 1 + x_1
17.46/5.22	   POL(0^1(x_1)) = x_1
17.46/5.22	   POL(1(x_1)) = 1 + x_1
17.46/5.22	   POL(1^1(x_1)) = x_1
17.46/5.22	
17.46/5.22	
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(4)
17.46/5.22	Obligation:
17.46/5.22	Q DP problem:
17.46/5.22	The TRS P consists of the following rules:
17.46/5.22	
17.46/5.22	   0^1(0(0(0(x1)))) -> 0^1(1(1(1(x1))))
17.46/5.22	   1^1(0(1(1(x1)))) -> 0^1(0(0(1(x1))))
17.46/5.22	
17.46/5.22	The TRS R consists of the following rules:
17.46/5.22	
17.46/5.22	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
17.46/5.22	   1(0(1(1(x1)))) -> 0(0(0(1(x1))))
17.46/5.22	
17.46/5.22	Q is empty.
17.46/5.22	We have to consider all minimal (P,Q,R)-chains.
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(5) DependencyGraphProof (EQUIVALENT)
17.46/5.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(6)
17.46/5.22	Obligation:
17.46/5.22	Q DP problem:
17.46/5.22	The TRS P consists of the following rules:
17.46/5.22	
17.46/5.22	   0^1(0(0(0(x1)))) -> 0^1(1(1(1(x1))))
17.46/5.22	
17.46/5.22	The TRS R consists of the following rules:
17.46/5.22	
17.46/5.22	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
17.46/5.22	   1(0(1(1(x1)))) -> 0(0(0(1(x1))))
17.46/5.22	
17.46/5.22	Q is empty.
17.46/5.22	We have to consider all minimal (P,Q,R)-chains.
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(7) QDPOrderProof (EQUIVALENT)
17.46/5.22	We use the reduction pair processor [LPAR04,JAR06].
17.46/5.22	
17.46/5.22	
17.46/5.22	The following pairs can be oriented strictly and are deleted.
17.46/5.22	
17.46/5.22	   0^1(0(0(0(x1)))) -> 0^1(1(1(1(x1))))
17.46/5.22	The remaining pairs can at least be oriented weakly.
17.46/5.22	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
17.46/5.22	
17.46/5.22	   <<<
17.46/5.22	 POL(0^1(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
17.46/5.22	>>>
17.46/5.22	
17.46/5.22	   <<<
17.46/5.22	 POL(0(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [1A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
17.46/5.22	>>>
17.46/5.22	
17.46/5.22	   <<<
17.46/5.22	 POL(1(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [1A, 0A, 1A], [-I, -I, 0A]] 	* 	x_1
17.46/5.22	>>>
17.46/5.22	
17.46/5.22	
17.46/5.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.46/5.22	
17.46/5.22	   1(0(1(1(x1)))) -> 0(0(0(1(x1))))
17.46/5.22	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
17.46/5.22	
17.46/5.22	
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(8)
17.46/5.22	Obligation:
17.46/5.22	Q DP problem:
17.46/5.22	P is empty.
17.46/5.22	The TRS R consists of the following rules:
17.46/5.22	
17.46/5.22	   0(0(0(0(x1)))) -> 0(1(1(1(x1))))
17.46/5.22	   1(0(1(1(x1)))) -> 0(0(0(1(x1))))
17.46/5.22	
17.46/5.22	Q is empty.
17.46/5.22	We have to consider all minimal (P,Q,R)-chains.
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(9) PisEmptyProof (EQUIVALENT)
17.46/5.22	The TRS P is empty. Hence, there is no (P,Q,R) chain.
17.46/5.22	----------------------------------------
17.46/5.22	
17.46/5.22	(10)
17.46/5.22	YES
17.46/5.27	EOF