1023.69/261.41	YES
1024.48/261.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1024.48/261.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1024.48/261.57	
1024.48/261.57	
1024.48/261.57	Termination w.r.t. Q of the given QTRS could be proven:
1024.48/261.57	
1024.48/261.57	(0) QTRS
1024.48/261.57	(1) QTRS Reverse [EQUIVALENT, 0 ms]
1024.48/261.57	(2) QTRS
1024.48/261.57	(3) DependencyPairsProof [EQUIVALENT, 6 ms]
1024.48/261.57	(4) QDP
1024.48/261.57	(5) QDPOrderProof [EQUIVALENT, 27 ms]
1024.48/261.57	(6) QDP
1024.48/261.57	(7) QDPOrderProof [EQUIVALENT, 33.8 s]
1024.48/261.57	(8) QDP
1024.48/261.57	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
1024.48/261.57	(10) TRUE
1024.48/261.57	
1024.48/261.57	
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(0)
1024.48/261.57	Obligation:
1024.48/261.57	Q restricted rewrite system:
1024.48/261.57	The TRS R consists of the following rules:
1024.48/261.57	
1024.48/261.57	   0(0(0(0(x1)))) -> 0(0(0(1(x1))))
1024.48/261.57	   1(0(0(1(x1)))) -> 0(1(0(0(x1))))
1024.48/261.57	
1024.48/261.57	Q is empty.
1024.48/261.57	
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(1) QTRS Reverse (EQUIVALENT)
1024.48/261.57	We applied the QTRS Reverse Processor [REVERSE].
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(2)
1024.48/261.57	Obligation:
1024.48/261.57	Q restricted rewrite system:
1024.48/261.57	The TRS R consists of the following rules:
1024.48/261.57	
1024.48/261.57	   0(0(0(0(x1)))) -> 1(0(0(0(x1))))
1024.48/261.57	   1(0(0(1(x1)))) -> 0(0(1(0(x1))))
1024.48/261.57	
1024.48/261.57	Q is empty.
1024.48/261.57	
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(3) DependencyPairsProof (EQUIVALENT)
1024.48/261.57	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(4)
1024.48/261.57	Obligation:
1024.48/261.57	Q DP problem:
1024.48/261.57	The TRS P consists of the following rules:
1024.48/261.57	
1024.48/261.57	   0^1(0(0(0(x1)))) -> 1^1(0(0(0(x1))))
1024.48/261.57	   1^1(0(0(1(x1)))) -> 0^1(0(1(0(x1))))
1024.48/261.57	   1^1(0(0(1(x1)))) -> 0^1(1(0(x1)))
1024.48/261.57	   1^1(0(0(1(x1)))) -> 1^1(0(x1))
1024.48/261.57	   1^1(0(0(1(x1)))) -> 0^1(x1)
1024.48/261.57	
1024.48/261.57	The TRS R consists of the following rules:
1024.48/261.57	
1024.48/261.57	   0(0(0(0(x1)))) -> 1(0(0(0(x1))))
1024.48/261.57	   1(0(0(1(x1)))) -> 0(0(1(0(x1))))
1024.48/261.57	
1024.48/261.57	Q is empty.
1024.48/261.57	We have to consider all minimal (P,Q,R)-chains.
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(5) QDPOrderProof (EQUIVALENT)
1024.48/261.57	We use the reduction pair processor [LPAR04,JAR06].
1024.48/261.57	
1024.48/261.57	
1024.48/261.57	The following pairs can be oriented strictly and are deleted.
1024.48/261.57	
1024.48/261.57	   1^1(0(0(1(x1)))) -> 0^1(1(0(x1)))
1024.48/261.57	   1^1(0(0(1(x1)))) -> 1^1(0(x1))
1024.48/261.57	   1^1(0(0(1(x1)))) -> 0^1(x1)
1024.48/261.57	The remaining pairs can at least be oriented weakly.
1024.48/261.57	Used ordering:  Polynomial interpretation [POLO]:
1024.48/261.57	
1024.48/261.57	   POL(0(x_1)) = 1 + x_1
1024.48/261.57	   POL(0^1(x_1)) = 1 + x_1
1024.48/261.57	   POL(1(x_1)) = 1 + x_1
1024.48/261.57	   POL(1^1(x_1)) = 1 + x_1
1024.48/261.57	
1024.48/261.57	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
1024.48/261.57	
1024.48/261.57	   1(0(0(1(x1)))) -> 0(0(1(0(x1))))
1024.48/261.57	   0(0(0(0(x1)))) -> 1(0(0(0(x1))))
1024.48/261.57	
1024.48/261.57	
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(6)
1024.48/261.57	Obligation:
1024.48/261.57	Q DP problem:
1024.48/261.57	The TRS P consists of the following rules:
1024.48/261.57	
1024.48/261.57	   0^1(0(0(0(x1)))) -> 1^1(0(0(0(x1))))
1024.48/261.57	   1^1(0(0(1(x1)))) -> 0^1(0(1(0(x1))))
1024.48/261.57	
1024.48/261.57	The TRS R consists of the following rules:
1024.48/261.57	
1024.48/261.57	   0(0(0(0(x1)))) -> 1(0(0(0(x1))))
1024.48/261.57	   1(0(0(1(x1)))) -> 0(0(1(0(x1))))
1024.48/261.57	
1024.48/261.57	Q is empty.
1024.48/261.57	We have to consider all minimal (P,Q,R)-chains.
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(7) QDPOrderProof (EQUIVALENT)
1024.48/261.57	We use the reduction pair processor [LPAR04,JAR06].
1024.48/261.57	
1024.48/261.57	
1024.48/261.57	The following pairs can be oriented strictly and are deleted.
1024.48/261.57	
1024.48/261.57	   0^1(0(0(0(x1)))) -> 1^1(0(0(0(x1))))
1024.48/261.57	The remaining pairs can at least be oriented weakly.
1024.48/261.57	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
1024.48/261.57	
1024.48/261.57	   <<<
1024.48/261.57	 POL(0^1(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I, 0A, -I]] 	* 	x_1
1024.48/261.57	>>>
1024.48/261.57	
1024.48/261.57	   <<<
1024.48/261.57	 POL(0(x_1)) =  	[[0A], [-I], [0A], [-I], [0A]] 	 +  	[[0A, 0A, 0A, 0A, -I], [0A, -I, -I, -I, 0A], [1A, 0A, 1A, 0A, 0A], [0A, -I, 0A, -I, -I], [-I, 0A, 0A, 1A, -I]] 	* 	x_1
1024.48/261.57	>>>
1024.48/261.57	
1024.48/261.57	   <<<
1024.48/261.57	 POL(1^1(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I, -I, -I]] 	* 	x_1
1024.48/261.57	>>>
1024.48/261.57	
1024.48/261.57	   <<<
1024.48/261.57	 POL(1(x_1)) =  	[[0A], [0A], [0A], [0A], [0A]] 	 +  	[[-I, 0A, -I, 0A, -I], [0A, 1A, -I, 0A, 0A], [0A, 1A, -I, 0A, -I], [1A, 0A, 0A, 1A, 1A], [-I, -I, 0A, 0A, -I]] 	* 	x_1
1024.48/261.57	>>>
1024.48/261.57	
1024.48/261.57	
1024.48/261.57	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
1024.48/261.57	
1024.48/261.57	   1(0(0(1(x1)))) -> 0(0(1(0(x1))))
1024.48/261.57	   0(0(0(0(x1)))) -> 1(0(0(0(x1))))
1024.48/261.57	
1024.48/261.57	
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(8)
1024.48/261.57	Obligation:
1024.48/261.57	Q DP problem:
1024.48/261.57	The TRS P consists of the following rules:
1024.48/261.57	
1024.48/261.57	   1^1(0(0(1(x1)))) -> 0^1(0(1(0(x1))))
1024.48/261.57	
1024.48/261.57	The TRS R consists of the following rules:
1024.48/261.57	
1024.48/261.57	   0(0(0(0(x1)))) -> 1(0(0(0(x1))))
1024.48/261.57	   1(0(0(1(x1)))) -> 0(0(1(0(x1))))
1024.48/261.57	
1024.48/261.57	Q is empty.
1024.48/261.57	We have to consider all minimal (P,Q,R)-chains.
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(9) DependencyGraphProof (EQUIVALENT)
1024.48/261.57	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
1024.48/261.57	----------------------------------------
1024.48/261.57	
1024.48/261.57	(10)
1024.48/261.57	TRUE
1024.66/261.64	EOF