intros m.
apply le_O_n.
intros n ih m.
case m; clear m.
simpl in |- *.
trivial.
intros m0; simpl in |- *.
apply le_n_S.
apply ih.
Qed.

Lemma max_n_n : forall n : nat, max n n = n.
simple induction n; clear n; simpl in |- *.