induction 1; intuition; subst; simpl in *.
specialize (div2_double x); intuition; simpl in *.
rewrite H.
omega.
destruct m.
omega.
destruct (Even.even_odd_dec m).
rewrite even_div2.
assert (div2 (S m) >= x).
eapply IHle.
trivial.
omega.
trivial.
rewrite odd_div2.
eapply IHle.
trivial.
trivial.
Qed.
Instance expnat_nz : forall k n (p : nz n),
nz (expnat n k).
intuition.
induction k; intuition; simpl in *.
econstructor.
omega.
econstructor.
edestruct IHk; eauto.
destruct p.
eapply mult_gt_0; intuition.
Qed.

(* Specifications: *)
Lemma expnat_2_ge_1 : forall n,
(1 <= expnat 2 n)%nat.