induction k; intuition; simpl in *.
omega.
rewrite plus_0_r.
rewrite <- mult_plus_distr_l.
eapply mult_le_compat.
trivial.
rewrite <- plus_0_r at 1.
rewrite <- plus_assoc.
eapply IHk.
trivial.
Qed.

Theorem nat_half_plus :
forall x,
x > 1 ->
exists a b,
a > 0 /\ b <= 1 /\ x = 2 * a + b.