3.49/1.64	WORST_CASE(?, O(1))
3.49/1.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.49/1.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.49/1.64	
3.49/1.64	
3.49/1.64	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.49/1.64	
3.49/1.64	(0) CpxTRS
3.49/1.64	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
3.49/1.64	(2) CpxTRS
3.49/1.64	(3) NarrowingOnBasicTermsTerminatesProof [FINISHED, 13 ms]
3.49/1.64	(4) BOUNDS(1, 1)
3.49/1.64	
3.49/1.64	
3.49/1.64	----------------------------------------
3.49/1.64	
3.49/1.64	(0)
3.49/1.64	Obligation:
3.49/1.64	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.49/1.64	
3.49/1.64	
3.49/1.64	The TRS R consists of the following rules:
3.49/1.64	
3.49/1.64	   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
3.49/1.64	   sqr(0) -> 0
3.49/1.64	   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
3.49/1.64	   dbl(0) -> 0
3.49/1.64	   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.49/1.64	   add(0, X) -> X
3.49/1.64	   add(s(X), Y) -> s(n__add(activate(X), Y))
3.49/1.64	   first(0, X) -> nil
3.49/1.64	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
3.49/1.64	   terms(X) -> n__terms(X)
3.49/1.64	   add(X1, X2) -> n__add(X1, X2)
3.49/1.64	   s(X) -> n__s(X)
3.49/1.64	   dbl(X) -> n__dbl(X)
3.49/1.64	   first(X1, X2) -> n__first(X1, X2)
3.49/1.64	   activate(n__terms(X)) -> terms(X)
3.49/1.64	   activate(n__add(X1, X2)) -> add(X1, X2)
3.49/1.64	   activate(n__s(X)) -> s(X)
3.49/1.64	   activate(n__dbl(X)) -> dbl(X)
3.49/1.64	   activate(n__first(X1, X2)) -> first(X1, X2)
3.49/1.64	   activate(X) -> X
3.49/1.64	
3.49/1.64	S is empty.
3.49/1.64	Rewrite Strategy: FULL
3.49/1.64	----------------------------------------
3.49/1.64	
3.49/1.64	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
3.49/1.64	The TRS does not nest defined symbols.
3.49/1.64	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
3.49/1.64	sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
3.49/1.64	dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
3.49/1.64	add(s(X), Y) -> s(n__add(activate(X), Y))
3.49/1.64	first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
3.49/1.64	
3.49/1.64	----------------------------------------
3.49/1.64	
3.49/1.64	(2)
3.49/1.64	Obligation:
3.49/1.64	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, 1).
3.49/1.64	
3.49/1.64	
3.49/1.64	The TRS R consists of the following rules:
3.49/1.64	
3.49/1.64	   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
3.49/1.64	   sqr(0) -> 0
3.49/1.64	   dbl(0) -> 0
3.49/1.64	   add(0, X) -> X
3.49/1.64	   first(0, X) -> nil
3.49/1.64	   terms(X) -> n__terms(X)
3.49/1.64	   add(X1, X2) -> n__add(X1, X2)
3.49/1.64	   s(X) -> n__s(X)
3.49/1.64	   dbl(X) -> n__dbl(X)
3.49/1.64	   first(X1, X2) -> n__first(X1, X2)
3.49/1.64	   activate(n__terms(X)) -> terms(X)
3.49/1.64	   activate(n__add(X1, X2)) -> add(X1, X2)
3.49/1.64	   activate(n__s(X)) -> s(X)
3.49/1.64	   activate(n__dbl(X)) -> dbl(X)
3.49/1.64	   activate(n__first(X1, X2)) -> first(X1, X2)
3.49/1.64	   activate(X) -> X
3.49/1.64	
3.49/1.64	S is empty.
3.49/1.64	Rewrite Strategy: FULL
3.49/1.64	----------------------------------------
3.49/1.64	
3.49/1.64	(3) NarrowingOnBasicTermsTerminatesProof (FINISHED)
3.49/1.64	Constant runtime complexity proven by termination of constructor-based narrowing.
3.49/1.64	
3.49/1.64	The maximal most general narrowing sequences give rise to the following rewrite sequences:
3.49/1.64	
3.49/1.64	activate(n__first(x0, x1)) ->^* n__first(x0, x1)
3.49/1.64	
3.49/1.64	activate(n__first(0, x0)) ->^* nil
3.49/1.64	
3.49/1.64	activate(n__dbl(x0)) ->^* n__dbl(x0)
3.49/1.64	
3.49/1.64	activate(n__dbl(0)) ->^* 0
3.49/1.64	
3.49/1.64	activate(n__s(x0)) ->^* n__s(x0)
3.49/1.64	
3.49/1.64	activate(n__add(x0, x1)) ->^* n__add(x0, x1)
3.49/1.64	
3.49/1.64	activate(n__terms(x0)) ->^* n__terms(x0)
3.49/1.64	
3.49/1.64	activate(n__terms(0)) ->^* cons(recip(0), n__terms(n__s(0)))
3.49/1.64	
3.49/1.64	first(x0, x1) ->^* n__first(x0, x1)
3.49/1.64	
3.49/1.64	first(0, x0) ->^* nil
3.49/1.64	
3.49/1.64	dbl(x0) ->^* n__dbl(x0)
3.49/1.64	
3.49/1.64	dbl(0) ->^* 0
3.49/1.64	
3.49/1.64	add(x0, x1) ->^* n__add(x0, x1)
3.49/1.64	
3.49/1.64	terms(x0) ->^* n__terms(x0)
3.49/1.64	
3.49/1.64	terms(0) ->^* cons(recip(0), n__terms(n__s(0)))
3.49/1.64	
3.49/1.64	sqr(0) ->^* 0
3.49/1.64	
3.49/1.64	s(x0) ->^* n__s(x0)
3.49/1.64	
3.49/1.64	
3.49/1.64	
3.49/1.64	
3.49/1.64	----------------------------------------
3.49/1.64	
3.49/1.64	(4)
3.49/1.64	BOUNDS(1, 1)
3.49/1.67	EOF