Fixpoint ge_bool (n m  : nat) : bool :=
 match n with
   | O   => match m with
            |  O => true
            | S _ => false
          end
   | S n  => match m with
               | O => true
               | S m => ge_bool n m
             end
   end.

Lemma ge_bool_cases : forall n m, (if ge_bool n m then n >= m else n < m)%nat.