induction ls; intuition; simpl in *.
intuition; subst.
rewrite H0.
rewrite orb_true_r.
eapply fold_left_orb_true_init.
eauto.
Qed.

Theorem hd_error_Some_In :
forall (A : Type)(ls : list A) a,
hd_error ls = Some a ->
In a ls.