Open Scope Z_scope.
Fixpoint fib (n: nat) : Z :=
match n with
| O => 1
| S O => 1
| S (S n as p) => fib p + fib n
end.
Open Scope R_scope.
Fixpoint fibr (n: nat) : R :=
match n with
| O => 1
| S O => 1
| S (S n as p) => fibr p + fibr n
end.

Lemma fibr_bound':
fibr 47 < IZR (Z.pow_pos 2  32).