26.55/8.69	YES
26.55/8.71	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
26.55/8.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.55/8.71	
26.55/8.71	
26.55/8.71	Termination w.r.t. Q of the given QTRS could be proven:
26.55/8.71	
26.55/8.71	(0) QTRS
26.55/8.71	(1) QTRS Reverse [EQUIVALENT, 0 ms]
26.55/8.71	(2) QTRS
26.55/8.71	(3) QTRSRRRProof [EQUIVALENT, 51 ms]
26.55/8.71	(4) QTRS
26.55/8.71	(5) DependencyPairsProof [EQUIVALENT, 35 ms]
26.55/8.71	(6) QDP
26.55/8.71	(7) MRRProof [EQUIVALENT, 90 ms]
26.55/8.71	(8) QDP
26.55/8.71	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
26.55/8.71	(10) QDP
26.55/8.71	(11) QDPOrderProof [EQUIVALENT, 949 ms]
26.55/8.71	(12) QDP
26.55/8.71	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
26.55/8.71	(14) TRUE
26.55/8.71	
26.55/8.71	
26.55/8.71	----------------------------------------
26.55/8.71	
26.55/8.71	(0)
26.55/8.71	Obligation:
26.55/8.71	Q restricted rewrite system:
26.55/8.71	The TRS R consists of the following rules:
26.55/8.71	
26.55/8.71	   r(e(x1)) -> w(r(x1))
26.55/8.71	   i(t(x1)) -> e(r(x1))
26.55/8.71	   e(w(x1)) -> r(i(x1))
26.55/8.71	   t(e(x1)) -> r(e(x1))
26.55/8.71	   w(r(x1)) -> i(t(x1))
26.55/8.71	   e(r(x1)) -> e(w(x1))
26.55/8.71	   r(i(t(e(r(x1))))) -> e(w(r(i(t(e(x1))))))
26.55/8.71	
26.55/8.71	Q is empty.
26.55/8.71	
26.55/8.71	----------------------------------------
26.55/8.71	
26.55/8.71	(1) QTRS Reverse (EQUIVALENT)
26.55/8.71	We applied the QTRS Reverse Processor [REVERSE].
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(2)
26.55/8.72	Obligation:
26.55/8.72	Q restricted rewrite system:
26.55/8.72	The TRS R consists of the following rules:
26.55/8.72	
26.55/8.72	   e(r(x1)) -> r(w(x1))
26.55/8.72	   t(i(x1)) -> r(e(x1))
26.55/8.72	   w(e(x1)) -> i(r(x1))
26.55/8.72	   e(t(x1)) -> e(r(x1))
26.55/8.72	   r(w(x1)) -> t(i(x1))
26.55/8.72	   r(e(x1)) -> w(e(x1))
26.55/8.72	   r(e(t(i(r(x1))))) -> e(t(i(r(w(e(x1))))))
26.55/8.72	
26.55/8.72	Q is empty.
26.55/8.72	
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(3) QTRSRRRProof (EQUIVALENT)
26.55/8.72	Used ordering:
26.55/8.72	Polynomial interpretation [POLO]:
26.55/8.72	
26.55/8.72	   POL(e(x_1)) = 1 + x_1
26.55/8.72	   POL(i(x_1)) = x_1
26.55/8.72	   POL(r(x_1)) = 2 + x_1
26.55/8.72	   POL(t(x_1)) = 3 + x_1
26.55/8.72	   POL(w(x_1)) = 1 + x_1
26.55/8.72	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
26.55/8.72	
26.55/8.72	   e(t(x1)) -> e(r(x1))
26.55/8.72	   r(e(x1)) -> w(e(x1))
26.55/8.72	
26.55/8.72	
26.55/8.72	
26.55/8.72	
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(4)
26.55/8.72	Obligation:
26.55/8.72	Q restricted rewrite system:
26.55/8.72	The TRS R consists of the following rules:
26.55/8.72	
26.55/8.72	   e(r(x1)) -> r(w(x1))
26.55/8.72	   t(i(x1)) -> r(e(x1))
26.55/8.72	   w(e(x1)) -> i(r(x1))
26.55/8.72	   r(w(x1)) -> t(i(x1))
26.55/8.72	   r(e(t(i(r(x1))))) -> e(t(i(r(w(e(x1))))))
26.55/8.72	
26.55/8.72	Q is empty.
26.55/8.72	
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(5) DependencyPairsProof (EQUIVALENT)
26.55/8.72	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(6)
26.55/8.72	Obligation:
26.55/8.72	Q DP problem:
26.55/8.72	The TRS P consists of the following rules:
26.55/8.72	
26.55/8.72	   E(r(x1)) -> R(w(x1))
26.55/8.72	   E(r(x1)) -> W(x1)
26.55/8.72	   T(i(x1)) -> R(e(x1))
26.55/8.72	   T(i(x1)) -> E(x1)
26.55/8.72	   W(e(x1)) -> R(x1)
26.55/8.72	   R(w(x1)) -> T(i(x1))
26.55/8.72	   R(e(t(i(r(x1))))) -> E(t(i(r(w(e(x1))))))
26.55/8.72	   R(e(t(i(r(x1))))) -> T(i(r(w(e(x1)))))
26.55/8.72	   R(e(t(i(r(x1))))) -> R(w(e(x1)))
26.55/8.72	   R(e(t(i(r(x1))))) -> W(e(x1))
26.55/8.72	   R(e(t(i(r(x1))))) -> E(x1)
26.55/8.72	
26.55/8.72	The TRS R consists of the following rules:
26.55/8.72	
26.55/8.72	   e(r(x1)) -> r(w(x1))
26.55/8.72	   t(i(x1)) -> r(e(x1))
26.55/8.72	   w(e(x1)) -> i(r(x1))
26.55/8.72	   r(w(x1)) -> t(i(x1))
26.55/8.72	   r(e(t(i(r(x1))))) -> e(t(i(r(w(e(x1))))))
26.55/8.72	
26.55/8.72	Q is empty.
26.55/8.72	We have to consider all minimal (P,Q,R)-chains.
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(7) MRRProof (EQUIVALENT)
26.55/8.72	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.
26.55/8.72	
26.55/8.72	Strictly oriented dependency pairs:
26.55/8.72	
26.55/8.72	   E(r(x1)) -> W(x1)
26.55/8.72	   T(i(x1)) -> E(x1)
26.55/8.72	   R(e(t(i(r(x1))))) -> T(i(r(w(e(x1)))))
26.55/8.72	   R(e(t(i(r(x1))))) -> R(w(e(x1)))
26.55/8.72	   R(e(t(i(r(x1))))) -> W(e(x1))
26.55/8.72	   R(e(t(i(r(x1))))) -> E(x1)
26.55/8.72	
26.55/8.72	
26.55/8.72	Used ordering: Polynomial interpretation [POLO]:
26.55/8.72	
26.55/8.72	   POL(E(x_1)) = 1 + x_1
26.55/8.72	   POL(R(x_1)) = 2 + x_1
26.55/8.72	   POL(T(x_1)) = 3 + x_1
26.55/8.72	   POL(W(x_1)) = 1 + x_1
26.55/8.72	   POL(e(x_1)) = 1 + x_1
26.55/8.72	   POL(i(x_1)) = x_1
26.55/8.72	   POL(r(x_1)) = 2 + x_1
26.55/8.72	   POL(t(x_1)) = 3 + x_1
26.55/8.72	   POL(w(x_1)) = 1 + x_1
26.55/8.72	
26.55/8.72	
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(8)
26.55/8.72	Obligation:
26.55/8.72	Q DP problem:
26.55/8.72	The TRS P consists of the following rules:
26.55/8.72	
26.55/8.72	   E(r(x1)) -> R(w(x1))
26.55/8.72	   T(i(x1)) -> R(e(x1))
26.55/8.72	   W(e(x1)) -> R(x1)
26.55/8.72	   R(w(x1)) -> T(i(x1))
26.55/8.72	   R(e(t(i(r(x1))))) -> E(t(i(r(w(e(x1))))))
26.55/8.72	
26.55/8.72	The TRS R consists of the following rules:
26.55/8.72	
26.55/8.72	   e(r(x1)) -> r(w(x1))
26.55/8.72	   t(i(x1)) -> r(e(x1))
26.55/8.72	   w(e(x1)) -> i(r(x1))
26.55/8.72	   r(w(x1)) -> t(i(x1))
26.55/8.72	   r(e(t(i(r(x1))))) -> e(t(i(r(w(e(x1))))))
26.55/8.72	
26.55/8.72	Q is empty.
26.55/8.72	We have to consider all minimal (P,Q,R)-chains.
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(9) DependencyGraphProof (EQUIVALENT)
26.55/8.72	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(10)
26.55/8.72	Obligation:
26.55/8.72	Q DP problem:
26.55/8.72	The TRS P consists of the following rules:
26.55/8.72	
26.55/8.72	   R(w(x1)) -> T(i(x1))
26.55/8.72	   T(i(x1)) -> R(e(x1))
26.55/8.72	   R(e(t(i(r(x1))))) -> E(t(i(r(w(e(x1))))))
26.55/8.72	   E(r(x1)) -> R(w(x1))
26.55/8.72	
26.55/8.72	The TRS R consists of the following rules:
26.55/8.72	
26.55/8.72	   e(r(x1)) -> r(w(x1))
26.55/8.72	   t(i(x1)) -> r(e(x1))
26.55/8.72	   w(e(x1)) -> i(r(x1))
26.55/8.72	   r(w(x1)) -> t(i(x1))
26.55/8.72	   r(e(t(i(r(x1))))) -> e(t(i(r(w(e(x1))))))
26.55/8.72	
26.55/8.72	Q is empty.
26.55/8.72	We have to consider all minimal (P,Q,R)-chains.
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(11) QDPOrderProof (EQUIVALENT)
26.55/8.72	We use the reduction pair processor [LPAR04,JAR06].
26.55/8.72	
26.55/8.72	
26.55/8.72	The following pairs can be oriented strictly and are deleted.
26.55/8.72	
26.55/8.72	   E(r(x1)) -> R(w(x1))
26.55/8.72	The remaining pairs can at least be oriented weakly.
26.55/8.72	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
26.55/8.72	
26.55/8.72	   <<<
26.55/8.72	 POL(R(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
26.55/8.72	>>>
26.55/8.72	
26.55/8.72	   <<<
26.55/8.72	 POL(w(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 1A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
26.55/8.72	>>>
26.55/8.72	
26.55/8.72	   <<<
26.55/8.72	 POL(T(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
26.55/8.72	>>>
26.55/8.72	
26.55/8.72	   <<<
26.55/8.72	 POL(i(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, -I], [0A, -I, -I], [0A, -I, 0A]] 	* 	x_1
26.55/8.72	>>>
26.55/8.72	
26.55/8.72	   <<<
26.55/8.72	 POL(e(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 0A]] 	* 	x_1
26.55/8.72	>>>
26.55/8.72	
26.55/8.72	   <<<
26.55/8.72	 POL(t(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
26.55/8.72	>>>
26.55/8.72	
26.55/8.72	   <<<
26.55/8.72	 POL(r(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, 0A, -I], [0A, 1A, 0A], [-I, 0A, 0A]] 	* 	x_1
26.55/8.72	>>>
26.55/8.72	
26.55/8.72	   <<<
26.55/8.72	 POL(E(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
26.55/8.72	>>>
26.55/8.72	
26.55/8.72	
26.55/8.72	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.55/8.72	
26.55/8.72	   r(w(x1)) -> t(i(x1))
26.55/8.72	   t(i(x1)) -> r(e(x1))
26.55/8.72	   r(e(t(i(r(x1))))) -> e(t(i(r(w(e(x1))))))
26.55/8.72	   e(r(x1)) -> r(w(x1))
26.55/8.72	   w(e(x1)) -> i(r(x1))
26.55/8.72	
26.55/8.72	
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(12)
26.55/8.72	Obligation:
26.55/8.72	Q DP problem:
26.55/8.72	The TRS P consists of the following rules:
26.55/8.72	
26.55/8.72	   R(w(x1)) -> T(i(x1))
26.55/8.72	   T(i(x1)) -> R(e(x1))
26.55/8.72	   R(e(t(i(r(x1))))) -> E(t(i(r(w(e(x1))))))
26.55/8.72	
26.55/8.72	The TRS R consists of the following rules:
26.55/8.72	
26.55/8.72	   e(r(x1)) -> r(w(x1))
26.55/8.72	   t(i(x1)) -> r(e(x1))
26.55/8.72	   w(e(x1)) -> i(r(x1))
26.55/8.72	   r(w(x1)) -> t(i(x1))
26.55/8.72	   r(e(t(i(r(x1))))) -> e(t(i(r(w(e(x1))))))
26.55/8.72	
26.55/8.72	Q is empty.
26.55/8.72	We have to consider all minimal (P,Q,R)-chains.
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(13) DependencyGraphProof (EQUIVALENT)
26.55/8.72	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
26.55/8.72	----------------------------------------
26.55/8.72	
26.55/8.72	(14)
26.55/8.72	TRUE
26.85/9.72	EOF