simpl in |- *.
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 le_n_max2 : forall n m : nat, n <= max m n.
intros n; elim n; clear n.