3.08/1.60	YES
3.08/1.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.08/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.08/1.61	
3.08/1.61	
3.08/1.61	Termination w.r.t. Q of the given QTRS could be proven:
3.08/1.61	
3.08/1.61	(0) QTRS
3.08/1.61	(1) QTRSRRRProof [EQUIVALENT, 0 ms]
3.08/1.61	(2) QTRS
3.08/1.61	(3) RisEmptyProof [EQUIVALENT, 1 ms]
3.08/1.61	(4) YES
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(0)
3.08/1.61	Obligation:
3.08/1.61	Q restricted rewrite system:
3.08/1.61	The TRS R consists of the following rules:
3.08/1.61	
3.08/1.61	   fst(0, Z) -> nil
3.08/1.61	   fst(s, cons(Y)) -> cons(Y)
3.08/1.61	   from(X) -> cons(X)
3.08/1.61	   add(0, X) -> X
3.08/1.61	   add(s, Y) -> s
3.08/1.61	   len(nil) -> 0
3.08/1.61	   len(cons(X)) -> s
3.08/1.61	
3.08/1.61	The set Q consists of the following terms:
3.08/1.61	
3.08/1.61	   fst(0, x0)
3.08/1.61	   fst(s, cons(x0))
3.08/1.61	   from(x0)
3.08/1.61	   add(0, x0)
3.08/1.61	   add(s, x0)
3.08/1.61	   len(nil)
3.08/1.61	   len(cons(x0))
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(1) QTRSRRRProof (EQUIVALENT)
3.08/1.61	Used ordering:
3.08/1.61	Knuth-Bendix order [KBO] with precedence:nil > len_1 > s > add_2 > from_1 > cons_1 > 0 > fst_2
3.08/1.61	
3.08/1.61	and weight map:
3.08/1.61	
3.08/1.61	   0=2
3.08/1.61	   nil=1
3.08/1.61	   s=3
3.08/1.61	   cons_1=1
3.08/1.61	   from_1=2
3.08/1.61	   len_1=1
3.08/1.61	   fst_2=0
3.08/1.61	   add_2=0
3.08/1.61	
3.08/1.61	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.08/1.61	
3.08/1.61	   fst(0, Z) -> nil
3.08/1.61	   fst(s, cons(Y)) -> cons(Y)
3.08/1.61	   from(X) -> cons(X)
3.08/1.61	   add(0, X) -> X
3.08/1.61	   add(s, Y) -> s
3.08/1.61	   len(nil) -> 0
3.08/1.61	   len(cons(X)) -> s
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(2)
3.08/1.61	Obligation:
3.08/1.61	Q restricted rewrite system:
3.08/1.61	R is empty.
3.08/1.61	The set Q consists of the following terms:
3.08/1.61	
3.08/1.61	   fst(0, x0)
3.08/1.61	   fst(s, cons(x0))
3.08/1.61	   from(x0)
3.08/1.61	   add(0, x0)
3.08/1.61	   add(s, x0)
3.08/1.61	   len(nil)
3.08/1.61	   len(cons(x0))
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(3) RisEmptyProof (EQUIVALENT)
3.08/1.61	The TRS R is empty. Hence, termination is trivially proven.
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(4)
3.08/1.61	YES
3.37/1.63	EOF