30.15/8.73	YES
30.15/8.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
30.15/8.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
30.15/8.73	
30.15/8.73	
30.15/8.73	Termination w.r.t. Q of the given QTRS could be proven:
30.15/8.73	
30.15/8.73	(0) QTRS
30.15/8.73	(1) QTRS Reverse [EQUIVALENT, 0 ms]
30.15/8.73	(2) QTRS
30.15/8.73	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
30.15/8.73	(4) QDP
30.15/8.73	(5) QDPOrderProof [EQUIVALENT, 24 ms]
30.15/8.73	(6) QDP
30.15/8.73	(7) QDPOrderProof [EQUIVALENT, 50 ms]
30.15/8.73	(8) QDP
30.15/8.73	(9) PisEmptyProof [EQUIVALENT, 0 ms]
30.15/8.73	(10) YES
30.15/8.73	
30.15/8.73	
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(0)
30.15/8.73	Obligation:
30.15/8.73	Q restricted rewrite system:
30.15/8.73	The TRS R consists of the following rules:
30.15/8.73	
30.15/8.73	   0(0(0(0(x1)))) -> 0(1(0(1(x1))))
30.15/8.73	   1(0(1(0(x1)))) -> 0(1(0(0(x1))))
30.15/8.73	
30.15/8.73	Q is empty.
30.15/8.73	
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(1) QTRS Reverse (EQUIVALENT)
30.15/8.73	We applied the QTRS Reverse Processor [REVERSE].
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(2)
30.15/8.73	Obligation:
30.15/8.73	Q restricted rewrite system:
30.15/8.73	The TRS R consists of the following rules:
30.15/8.73	
30.15/8.73	   0(0(0(0(x1)))) -> 1(0(1(0(x1))))
30.15/8.73	   0(1(0(1(x1)))) -> 0(0(1(0(x1))))
30.15/8.73	
30.15/8.73	Q is empty.
30.15/8.73	
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(3) DependencyPairsProof (EQUIVALENT)
30.15/8.73	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(4)
30.15/8.73	Obligation:
30.15/8.73	Q DP problem:
30.15/8.73	The TRS P consists of the following rules:
30.15/8.73	
30.15/8.73	   0^1(0(0(0(x1)))) -> 0^1(1(0(x1)))
30.15/8.73	   0^1(1(0(1(x1)))) -> 0^1(0(1(0(x1))))
30.15/8.73	   0^1(1(0(1(x1)))) -> 0^1(1(0(x1)))
30.15/8.73	   0^1(1(0(1(x1)))) -> 0^1(x1)
30.15/8.73	
30.15/8.73	The TRS R consists of the following rules:
30.15/8.73	
30.15/8.73	   0(0(0(0(x1)))) -> 1(0(1(0(x1))))
30.15/8.73	   0(1(0(1(x1)))) -> 0(0(1(0(x1))))
30.15/8.73	
30.15/8.73	Q is empty.
30.15/8.73	We have to consider all minimal (P,Q,R)-chains.
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(5) QDPOrderProof (EQUIVALENT)
30.15/8.73	We use the reduction pair processor [LPAR04,JAR06].
30.15/8.73	
30.15/8.73	
30.15/8.73	The following pairs can be oriented strictly and are deleted.
30.15/8.73	
30.15/8.73	   0^1(0(0(0(x1)))) -> 0^1(1(0(x1)))
30.15/8.73	   0^1(1(0(1(x1)))) -> 0^1(1(0(x1)))
30.15/8.73	   0^1(1(0(1(x1)))) -> 0^1(x1)
30.15/8.73	The remaining pairs can at least be oriented weakly.
30.15/8.73	Used ordering:  Polynomial interpretation [POLO]:
30.15/8.73	
30.15/8.73	   POL(0(x_1)) = 1 + x_1
30.15/8.73	   POL(0^1(x_1)) = x_1
30.15/8.73	   POL(1(x_1)) = 1 + x_1
30.15/8.73	
30.15/8.73	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.15/8.73	
30.15/8.73	   0(0(0(0(x1)))) -> 1(0(1(0(x1))))
30.15/8.73	   0(1(0(1(x1)))) -> 0(0(1(0(x1))))
30.15/8.73	
30.15/8.73	
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(6)
30.15/8.73	Obligation:
30.15/8.73	Q DP problem:
30.15/8.73	The TRS P consists of the following rules:
30.15/8.73	
30.15/8.73	   0^1(1(0(1(x1)))) -> 0^1(0(1(0(x1))))
30.15/8.73	
30.15/8.73	The TRS R consists of the following rules:
30.15/8.73	
30.15/8.73	   0(0(0(0(x1)))) -> 1(0(1(0(x1))))
30.15/8.73	   0(1(0(1(x1)))) -> 0(0(1(0(x1))))
30.15/8.73	
30.15/8.73	Q is empty.
30.15/8.73	We have to consider all minimal (P,Q,R)-chains.
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(7) QDPOrderProof (EQUIVALENT)
30.15/8.73	We use the reduction pair processor [LPAR04,JAR06].
30.15/8.73	
30.15/8.73	
30.15/8.73	The following pairs can be oriented strictly and are deleted.
30.15/8.73	
30.15/8.73	   0^1(1(0(1(x1)))) -> 0^1(0(1(0(x1))))
30.15/8.73	The remaining pairs can at least be oriented weakly.
30.15/8.73	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
30.15/8.73	
30.15/8.73	   <<<
30.15/8.73	 POL(0^1(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
30.15/8.73	>>>
30.15/8.73	
30.15/8.73	   <<<
30.15/8.73	 POL(1(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 0A], [-I, -I, 0A], [-I, -I, -I]] 	* 	x_1
30.15/8.73	>>>
30.15/8.73	
30.15/8.73	   <<<
30.15/8.73	 POL(0(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [1A, 0A, 0A]] 	* 	x_1
30.15/8.73	>>>
30.15/8.73	
30.15/8.73	
30.15/8.73	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
30.15/8.73	
30.15/8.73	   0(0(0(0(x1)))) -> 1(0(1(0(x1))))
30.15/8.73	   0(1(0(1(x1)))) -> 0(0(1(0(x1))))
30.15/8.73	
30.15/8.73	
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(8)
30.15/8.73	Obligation:
30.15/8.73	Q DP problem:
30.15/8.73	P is empty.
30.15/8.73	The TRS R consists of the following rules:
30.15/8.73	
30.15/8.73	   0(0(0(0(x1)))) -> 1(0(1(0(x1))))
30.15/8.73	   0(1(0(1(x1)))) -> 0(0(1(0(x1))))
30.15/8.73	
30.15/8.73	Q is empty.
30.15/8.73	We have to consider all minimal (P,Q,R)-chains.
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(9) PisEmptyProof (EQUIVALENT)
30.15/8.73	The TRS P is empty. Hence, there is no (P,Q,R) chain.
30.15/8.73	----------------------------------------
30.15/8.73	
30.15/8.73	(10)
30.15/8.73	YES
30.49/8.80	EOF