3.36/1.66	YES
3.36/1.66	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.36/1.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.36/1.66	
3.36/1.66	
3.36/1.66	Termination w.r.t. Q of the given QTRS could be proven:
3.36/1.66	
3.36/1.66	(0) QTRS
3.36/1.66	(1) QTRSRRRProof [EQUIVALENT, 57 ms]
3.36/1.66	(2) QTRS
3.36/1.66	(3) RisEmptyProof [EQUIVALENT, 0 ms]
3.36/1.66	(4) YES
3.36/1.66	
3.36/1.66	
3.36/1.66	----------------------------------------
3.36/1.66	
3.36/1.66	(0)
3.36/1.66	Obligation:
3.36/1.66	Q restricted rewrite system:
3.36/1.66	The TRS R consists of the following rules:
3.36/1.66	
3.36/1.66	   terms(N) -> cons(recip(sqr(N)))
3.36/1.66	   sqr(0) -> 0
3.36/1.66	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
3.36/1.66	   dbl(0) -> 0
3.36/1.66	   dbl(s(X)) -> s(s(dbl(X)))
3.36/1.66	   add(0, X) -> X
3.36/1.66	   add(s(X), Y) -> s(add(X, Y))
3.36/1.66	   first(0, X) -> nil
3.36/1.66	   first(s(X), cons(Y)) -> cons(Y)
3.36/1.66	
3.36/1.66	The set Q consists of the following terms:
3.36/1.66	
3.36/1.66	   terms(x0)
3.36/1.66	   sqr(0)
3.36/1.66	   sqr(s(x0))
3.36/1.66	   dbl(0)
3.36/1.66	   dbl(s(x0))
3.36/1.66	   add(0, x0)
3.36/1.66	   add(s(x0), x1)
3.36/1.66	   first(0, x0)
3.36/1.66	   first(s(x0), cons(x1))
3.36/1.66	
3.36/1.66	
3.36/1.66	----------------------------------------
3.36/1.66	
3.36/1.66	(1) QTRSRRRProof (EQUIVALENT)
3.36/1.66	Used ordering:
3.36/1.66	Quasi precedence:
3.36/1.66	terms_1 > [cons_1, first_2]
3.36/1.66	terms_1 > recip_1
3.36/1.66	terms_1 > sqr_1 > 0 > nil
3.36/1.66	terms_1 > sqr_1 > add_2 > s_1
3.36/1.66	terms_1 > sqr_1 > dbl_1 > s_1
3.36/1.66	
3.36/1.66	
3.36/1.66	Status:
3.36/1.66	terms_1: [1]
3.36/1.66	cons_1: [1]
3.36/1.66	recip_1: multiset status
3.36/1.66	sqr_1: multiset status
3.36/1.66	0: multiset status
3.36/1.66	s_1: multiset status
3.36/1.66	add_2: multiset status
3.36/1.66	dbl_1: multiset status
3.36/1.66	first_2: multiset status
3.36/1.66	nil: multiset status
3.36/1.66	
3.36/1.66	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.36/1.66	
3.36/1.66	   terms(N) -> cons(recip(sqr(N)))
3.36/1.66	   sqr(0) -> 0
3.36/1.66	   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
3.36/1.66	   dbl(0) -> 0
3.36/1.66	   dbl(s(X)) -> s(s(dbl(X)))
3.36/1.66	   add(0, X) -> X
3.36/1.66	   add(s(X), Y) -> s(add(X, Y))
3.36/1.66	   first(0, X) -> nil
3.36/1.66	   first(s(X), cons(Y)) -> cons(Y)
3.36/1.66	
3.36/1.66	
3.36/1.66	
3.36/1.66	
3.36/1.66	----------------------------------------
3.36/1.66	
3.36/1.66	(2)
3.36/1.66	Obligation:
3.36/1.66	Q restricted rewrite system:
3.36/1.66	R is empty.
3.36/1.66	The set Q consists of the following terms:
3.36/1.66	
3.36/1.66	   terms(x0)
3.36/1.66	   sqr(0)
3.36/1.66	   sqr(s(x0))
3.36/1.66	   dbl(0)
3.36/1.66	   dbl(s(x0))
3.36/1.66	   add(0, x0)
3.36/1.66	   add(s(x0), x1)
3.36/1.66	   first(0, x0)
3.36/1.66	   first(s(x0), cons(x1))
3.36/1.66	
3.36/1.66	
3.36/1.66	----------------------------------------
3.36/1.66	
3.36/1.66	(3) RisEmptyProof (EQUIVALENT)
3.36/1.67	The TRS R is empty. Hence, termination is trivially proven.
3.36/1.67	----------------------------------------
3.36/1.67	
3.36/1.67	(4)
3.36/1.67	YES
3.36/1.69	EOF