simple induction 1.
eauto.
intros.
apply bar2.
simple induction w.
auto.
intros.
elim (letter_eq_dec a0 a).
intro.
rewrite a1; eauto.
intro.
apply (prop2 a0 a (l :: zs) b0 H3 ws); eauto.
Defined.
Hint Resolve prop3.

Theorem higman : bar nil.
apply bar2.