3.35/1.62	YES
3.35/1.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.35/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.35/1.63	
3.35/1.63	
3.35/1.63	Termination w.r.t. Q of the given QTRS could be proven:
3.35/1.63	
3.35/1.63	(0) QTRS
3.35/1.63	(1) QTRSRRRProof [EQUIVALENT, 37 ms]
3.35/1.63	(2) QTRS
3.35/1.63	(3) RisEmptyProof [EQUIVALENT, 0 ms]
3.35/1.63	(4) YES
3.35/1.63	
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(0)
3.35/1.63	Obligation:
3.35/1.63	Q restricted rewrite system:
3.35/1.63	The TRS R consists of the following rules:
3.35/1.63	
3.35/1.63	   first(0, X) -> nil
3.35/1.63	   first(s(X), cons(Y)) -> cons(Y)
3.35/1.63	   from(X) -> cons(X)
3.35/1.63	
3.35/1.63	The set Q consists of the following terms:
3.35/1.63	
3.35/1.63	   first(0, x0)
3.35/1.63	   first(s(x0), cons(x1))
3.35/1.63	   from(x0)
3.35/1.63	
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(1) QTRSRRRProof (EQUIVALENT)
3.35/1.63	Used ordering:
3.35/1.63	Knuth-Bendix order [KBO] with precedence:first_2 > from_1 > cons_1 > s_1 > nil > 0
3.35/1.63	
3.35/1.63	and weight map:
3.35/1.63	
3.35/1.63	   0=1
3.35/1.63	   nil=2
3.35/1.63	   s_1=1
3.35/1.63	   cons_1=1
3.35/1.63	   from_1=1
3.35/1.63	   first_2=0
3.35/1.63	
3.35/1.63	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.35/1.63	
3.35/1.63	   first(0, X) -> nil
3.35/1.63	   first(s(X), cons(Y)) -> cons(Y)
3.35/1.63	   from(X) -> cons(X)
3.35/1.63	
3.35/1.63	
3.35/1.63	
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(2)
3.35/1.63	Obligation:
3.35/1.63	Q restricted rewrite system:
3.35/1.63	R is empty.
3.35/1.63	The set Q consists of the following terms:
3.35/1.63	
3.35/1.63	   first(0, x0)
3.35/1.63	   first(s(x0), cons(x1))
3.35/1.63	   from(x0)
3.35/1.63	
3.35/1.63	
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(3) RisEmptyProof (EQUIVALENT)
3.35/1.63	The TRS R is empty. Hence, termination is trivially proven.
3.35/1.63	----------------------------------------
3.35/1.63	
3.35/1.63	(4)
3.35/1.63	YES
3.35/1.66	EOF