(** 2) For Euclid's algorithm, the worst-case situation corresponds
    to Fibonacci numbers. Let's define them: *)

 Fixpoint fibonacci (n:nat) : Z :=
   match n with
     | O => 1
     | S O => 1
     | S (S n as p) => fibonacci p + fibonacci n
   end.

Lemma Zgcdn_worst_is_fibonacci : forall n a b, 0 < a < b -> Zis_gcd a b (Zgcdn (S n) a b) -> Zgcdn n a b <> Zgcdn (S n) a b -> fibonacci (S n) <= a /\ fibonacci (S (S n)) <= b.