3.43/1.59	YES
3.43/1.60	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.43/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.43/1.60	
3.43/1.60	
3.43/1.60	Termination w.r.t. Q of the given QTRS could be proven:
3.43/1.60	
3.43/1.60	(0) QTRS
3.43/1.60	(1) QTRSRRRProof [EQUIVALENT, 24 ms]
3.43/1.60	(2) QTRS
3.43/1.60	(3) RisEmptyProof [EQUIVALENT, 0 ms]
3.43/1.60	(4) YES
3.43/1.60	
3.43/1.60	
3.43/1.60	----------------------------------------
3.43/1.60	
3.43/1.60	(0)
3.43/1.60	Obligation:
3.43/1.60	Q restricted rewrite system:
3.43/1.60	The TRS R consists of the following rules:
3.43/1.60	
3.43/1.60	   a__zeros -> cons(0, zeros)
3.43/1.60	   a__tail(cons(X, XS)) -> mark(XS)
3.43/1.60	   mark(zeros) -> a__zeros
3.43/1.60	   mark(tail(X)) -> a__tail(mark(X))
3.43/1.60	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.43/1.60	   mark(0) -> 0
3.43/1.60	   a__zeros -> zeros
3.43/1.60	   a__tail(X) -> tail(X)
3.43/1.60	
3.43/1.60	The set Q consists of the following terms:
3.43/1.60	
3.43/1.60	   a__zeros
3.43/1.60	   mark(zeros)
3.43/1.60	   mark(tail(x0))
3.43/1.60	   mark(cons(x0, x1))
3.43/1.60	   mark(0)
3.43/1.60	   a__tail(x0)
3.43/1.60	
3.43/1.60	
3.43/1.60	----------------------------------------
3.43/1.60	
3.43/1.60	(1) QTRSRRRProof (EQUIVALENT)
3.43/1.60	Used ordering:
3.43/1.60	Knuth-Bendix order [KBO] with precedence:mark_1 > a__tail_1 > tail_1 > a__zeros > zeros > cons_2 > 0
3.43/1.60	
3.43/1.60	and weight map:
3.43/1.60	
3.43/1.60	   a__zeros=4
3.43/1.60	   0=1
3.43/1.60	   zeros=1
3.43/1.60	   a__tail_1=2
3.43/1.60	   mark_1=3
3.43/1.60	   tail_1=2
3.43/1.60	   cons_2=1
3.43/1.60	
3.43/1.60	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.43/1.60	
3.43/1.60	   a__zeros -> cons(0, zeros)
3.43/1.60	   a__tail(cons(X, XS)) -> mark(XS)
3.43/1.60	   mark(zeros) -> a__zeros
3.43/1.60	   mark(tail(X)) -> a__tail(mark(X))
3.43/1.60	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
3.43/1.60	   mark(0) -> 0
3.43/1.60	   a__zeros -> zeros
3.43/1.60	   a__tail(X) -> tail(X)
3.43/1.60	
3.43/1.60	
3.43/1.60	
3.43/1.60	
3.43/1.60	----------------------------------------
3.43/1.60	
3.43/1.60	(2)
3.43/1.60	Obligation:
3.43/1.60	Q restricted rewrite system:
3.43/1.60	R is empty.
3.43/1.60	The set Q consists of the following terms:
3.43/1.60	
3.43/1.60	   a__zeros
3.43/1.60	   mark(zeros)
3.43/1.60	   mark(tail(x0))
3.43/1.60	   mark(cons(x0, x1))
3.43/1.60	   mark(0)
3.43/1.60	   a__tail(x0)
3.43/1.60	
3.43/1.60	
3.43/1.60	----------------------------------------
3.43/1.60	
3.43/1.60	(3) RisEmptyProof (EQUIVALENT)
3.43/1.60	The TRS R is empty. Hence, termination is trivially proven.
3.43/1.60	----------------------------------------
3.43/1.60	
3.43/1.60	(4)
3.43/1.60	YES
3.43/1.63	EOF