20.54/6.19	YES
20.85/6.23	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.85/6.23	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.85/6.23	
20.85/6.23	
20.85/6.23	Termination w.r.t. Q of the given QTRS could be proven:
20.85/6.23	
20.85/6.23	(0) QTRS
20.85/6.23	(1) QTRS Reverse [EQUIVALENT, 0 ms]
20.85/6.23	(2) QTRS
20.85/6.23	(3) DependencyPairsProof [EQUIVALENT, 21 ms]
20.85/6.23	(4) QDP
20.85/6.23	(5) QDPOrderProof [EQUIVALENT, 68 ms]
20.85/6.23	(6) QDP
20.85/6.23	(7) QDPOrderProof [EQUIVALENT, 77 ms]
20.85/6.23	(8) QDP
20.85/6.23	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
20.85/6.23	(10) TRUE
20.85/6.23	
20.85/6.23	
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(0)
20.85/6.23	Obligation:
20.85/6.23	Q restricted rewrite system:
20.85/6.23	The TRS R consists of the following rules:
20.85/6.23	
20.85/6.23	   0(0(0(0(x1)))) -> 1(0(1(1(x1))))
20.85/6.23	   0(1(0(1(x1)))) -> 0(0(1(0(x1))))
20.85/6.23	
20.85/6.23	Q is empty.
20.85/6.23	
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(1) QTRS Reverse (EQUIVALENT)
20.85/6.23	We applied the QTRS Reverse Processor [REVERSE].
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(2)
20.85/6.23	Obligation:
20.85/6.23	Q restricted rewrite system:
20.85/6.23	The TRS R consists of the following rules:
20.85/6.23	
20.85/6.23	   0(0(0(0(x1)))) -> 1(1(0(1(x1))))
20.85/6.23	   1(0(1(0(x1)))) -> 0(1(0(0(x1))))
20.85/6.23	
20.85/6.23	Q is empty.
20.85/6.23	
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(3) DependencyPairsProof (EQUIVALENT)
20.85/6.23	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(4)
20.85/6.23	Obligation:
20.85/6.23	Q DP problem:
20.85/6.23	The TRS P consists of the following rules:
20.85/6.23	
20.85/6.23	   0^1(0(0(0(x1)))) -> 1^1(1(0(1(x1))))
20.85/6.23	   0^1(0(0(0(x1)))) -> 1^1(0(1(x1)))
20.85/6.23	   0^1(0(0(0(x1)))) -> 0^1(1(x1))
20.85/6.23	   0^1(0(0(0(x1)))) -> 1^1(x1)
20.85/6.23	   1^1(0(1(0(x1)))) -> 0^1(1(0(0(x1))))
20.85/6.23	   1^1(0(1(0(x1)))) -> 1^1(0(0(x1)))
20.85/6.23	   1^1(0(1(0(x1)))) -> 0^1(0(x1))
20.85/6.23	
20.85/6.23	The TRS R consists of the following rules:
20.85/6.23	
20.85/6.23	   0(0(0(0(x1)))) -> 1(1(0(1(x1))))
20.85/6.23	   1(0(1(0(x1)))) -> 0(1(0(0(x1))))
20.85/6.23	
20.85/6.23	Q is empty.
20.85/6.23	We have to consider all minimal (P,Q,R)-chains.
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(5) QDPOrderProof (EQUIVALENT)
20.85/6.23	We use the reduction pair processor [LPAR04,JAR06].
20.85/6.23	
20.85/6.23	
20.85/6.23	The following pairs can be oriented strictly and are deleted.
20.85/6.23	
20.85/6.23	   0^1(0(0(0(x1)))) -> 1^1(0(1(x1)))
20.85/6.23	   0^1(0(0(0(x1)))) -> 0^1(1(x1))
20.85/6.23	   0^1(0(0(0(x1)))) -> 1^1(x1)
20.85/6.23	   1^1(0(1(0(x1)))) -> 1^1(0(0(x1)))
20.85/6.23	   1^1(0(1(0(x1)))) -> 0^1(0(x1))
20.85/6.23	The remaining pairs can at least be oriented weakly.
20.85/6.23	Used ordering:  Polynomial interpretation [POLO]:
20.85/6.23	
20.85/6.23	   POL(0(x_1)) = 1 + x_1
20.85/6.23	   POL(0^1(x_1)) = 1 + x_1
20.85/6.23	   POL(1(x_1)) = 1 + x_1
20.85/6.23	   POL(1^1(x_1)) = 1 + x_1
20.85/6.23	
20.85/6.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.85/6.23	
20.85/6.23	   1(0(1(0(x1)))) -> 0(1(0(0(x1))))
20.85/6.23	   0(0(0(0(x1)))) -> 1(1(0(1(x1))))
20.85/6.23	
20.85/6.23	
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(6)
20.85/6.23	Obligation:
20.85/6.23	Q DP problem:
20.85/6.23	The TRS P consists of the following rules:
20.85/6.23	
20.85/6.23	   0^1(0(0(0(x1)))) -> 1^1(1(0(1(x1))))
20.85/6.23	   1^1(0(1(0(x1)))) -> 0^1(1(0(0(x1))))
20.85/6.23	
20.85/6.23	The TRS R consists of the following rules:
20.85/6.23	
20.85/6.23	   0(0(0(0(x1)))) -> 1(1(0(1(x1))))
20.85/6.23	   1(0(1(0(x1)))) -> 0(1(0(0(x1))))
20.85/6.23	
20.85/6.23	Q is empty.
20.85/6.23	We have to consider all minimal (P,Q,R)-chains.
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(7) QDPOrderProof (EQUIVALENT)
20.85/6.23	We use the reduction pair processor [LPAR04,JAR06].
20.85/6.23	
20.85/6.23	
20.85/6.23	The following pairs can be oriented strictly and are deleted.
20.85/6.23	
20.85/6.23	   1^1(0(1(0(x1)))) -> 0^1(1(0(0(x1))))
20.85/6.23	The remaining pairs can at least be oriented weakly.
20.85/6.23	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
20.85/6.23	
20.85/6.23	   <<<
20.85/6.23	 POL(0^1(x_1)) =  	[[0A]] 	 +  	[[-I, -I, 0A]] 	* 	x_1
20.85/6.23	>>>
20.85/6.23	
20.85/6.23	   <<<
20.85/6.23	 POL(0(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 1A, 1A], [0A, -I, 0A], [0A, 0A, 1A]] 	* 	x_1
20.85/6.23	>>>
20.85/6.23	
20.85/6.23	   <<<
20.85/6.23	 POL(1^1(x_1)) =  	[[1A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
20.85/6.23	>>>
20.85/6.23	
20.85/6.23	   <<<
20.85/6.23	 POL(1(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[1A, 0A, 0A], [0A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
20.85/6.23	>>>
20.85/6.23	
20.85/6.23	
20.85/6.23	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.85/6.23	
20.85/6.23	   1(0(1(0(x1)))) -> 0(1(0(0(x1))))
20.85/6.23	   0(0(0(0(x1)))) -> 1(1(0(1(x1))))
20.85/6.23	
20.85/6.23	
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(8)
20.85/6.23	Obligation:
20.85/6.23	Q DP problem:
20.85/6.23	The TRS P consists of the following rules:
20.85/6.23	
20.85/6.23	   0^1(0(0(0(x1)))) -> 1^1(1(0(1(x1))))
20.85/6.23	
20.85/6.23	The TRS R consists of the following rules:
20.85/6.23	
20.85/6.23	   0(0(0(0(x1)))) -> 1(1(0(1(x1))))
20.85/6.23	   1(0(1(0(x1)))) -> 0(1(0(0(x1))))
20.85/6.23	
20.85/6.23	Q is empty.
20.85/6.23	We have to consider all minimal (P,Q,R)-chains.
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(9) DependencyGraphProof (EQUIVALENT)
20.85/6.23	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
20.85/6.23	----------------------------------------
20.85/6.23	
20.85/6.23	(10)
20.85/6.23	TRUE
20.99/6.28	EOF