Section strong.
Variable P : nat -> Prop.
Hypothesis PH : forall n, (forall m, m < n -> P m) -> P n.
induction n; simpl; intuition; apply PH; intuition.
elimtype False; omega.
Qed.
intros; eapply strong'; eauto.
Qed.
End strong.

Theorem div2_odd : forall n,
mod2 n = true
-> n = S (2 * div2 n).