3.15/1.63	YES
3.15/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.15/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.15/1.64	
3.15/1.64	
3.15/1.64	Termination w.r.t. Q of the given QTRS could be proven:
3.15/1.64	
3.15/1.64	(0) QTRS
3.15/1.64	(1) QTRSRRRProof [EQUIVALENT, 26 ms]
3.15/1.64	(2) QTRS
3.15/1.64	(3) RisEmptyProof [EQUIVALENT, 0 ms]
3.15/1.64	(4) YES
3.15/1.64	
3.15/1.64	
3.15/1.64	----------------------------------------
3.15/1.64	
3.15/1.64	(0)
3.15/1.64	Obligation:
3.15/1.64	Q restricted rewrite system:
3.15/1.64	The TRS R consists of the following rules:
3.15/1.64	
3.15/1.64	   terms(N) -> cons(recip(sqr(N)))
3.15/1.64	   sqr(0) -> 0
3.15/1.64	   sqr(s) -> s
3.15/1.64	   dbl(0) -> 0
3.15/1.64	   dbl(s) -> s
3.15/1.64	   add(0, X) -> X
3.15/1.64	   add(s, Y) -> s
3.15/1.64	   first(0, X) -> nil
3.15/1.64	   first(s, cons(Y)) -> cons(Y)
3.15/1.64	
3.15/1.64	The set Q consists of the following terms:
3.15/1.64	
3.15/1.64	   terms(x0)
3.15/1.64	   sqr(0)
3.15/1.64	   sqr(s)
3.15/1.64	   dbl(0)
3.15/1.64	   dbl(s)
3.15/1.64	   add(0, x0)
3.15/1.64	   add(s, x0)
3.15/1.64	   first(0, x0)
3.15/1.64	   first(s, cons(x0))
3.15/1.64	
3.15/1.64	
3.15/1.64	----------------------------------------
3.15/1.64	
3.15/1.64	(1) QTRSRRRProof (EQUIVALENT)
3.15/1.64	Used ordering:
3.15/1.64	Knuth-Bendix order [KBO] with precedence:dbl_1 > first_2 > nil > add_2 > s > 0 > recip_1 > sqr_1 > terms_1 > cons_1
3.15/1.64	
3.15/1.64	and weight map:
3.15/1.64	
3.15/1.64	   0=1
3.15/1.64	   s=1
3.15/1.64	   nil=2
3.15/1.64	   terms_1=3
3.15/1.64	   cons_1=1
3.15/1.64	   recip_1=1
3.15/1.64	   sqr_1=1
3.15/1.64	   dbl_1=0
3.15/1.64	   add_2=0
3.15/1.64	   first_2=0
3.15/1.64	
3.15/1.64	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.15/1.64	
3.15/1.64	   terms(N) -> cons(recip(sqr(N)))
3.15/1.64	   sqr(0) -> 0
3.15/1.64	   sqr(s) -> s
3.15/1.64	   dbl(0) -> 0
3.15/1.64	   dbl(s) -> s
3.15/1.64	   add(0, X) -> X
3.15/1.64	   add(s, Y) -> s
3.15/1.64	   first(0, X) -> nil
3.15/1.64	   first(s, cons(Y)) -> cons(Y)
3.15/1.64	
3.15/1.64	
3.15/1.64	
3.15/1.64	
3.15/1.64	----------------------------------------
3.15/1.64	
3.15/1.64	(2)
3.15/1.64	Obligation:
3.15/1.64	Q restricted rewrite system:
3.15/1.64	R is empty.
3.15/1.64	The set Q consists of the following terms:
3.15/1.64	
3.15/1.64	   terms(x0)
3.15/1.64	   sqr(0)
3.15/1.64	   sqr(s)
3.15/1.64	   dbl(0)
3.15/1.64	   dbl(s)
3.15/1.64	   add(0, x0)
3.15/1.64	   add(s, x0)
3.15/1.64	   first(0, x0)
3.15/1.64	   first(s, cons(x0))
3.15/1.64	
3.15/1.64	
3.15/1.64	----------------------------------------
3.15/1.64	
3.15/1.64	(3) RisEmptyProof (EQUIVALENT)
3.15/1.64	The TRS R is empty. Hence, termination is trivially proven.
3.15/1.64	----------------------------------------
3.15/1.64	
3.15/1.64	(4)
3.15/1.64	YES
3.15/1.65	EOF