(** 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 fibonacci_pos : forall n, 0 <= fibonacci n.