intros a b xs neq.
simple induction 1.
eauto.
simple induction 3.
eauto.
intros.
apply bar2.
intro.
case w.
apply prop1.
intros.
elim (letter_eq_dec l a).
intro; rewrite a0; eauto.
intro; rewrite (letter_neq a b l neq b2); eauto.
Defined.
Hint Resolve prop2.

Theorem prop3 :
forall (a : letter) (xs : list word),
bar xs -> forall zs : list word, xs <> nil -> R a xs zs -> bar zs.