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.