Proof with auto.
induction l...
intros.
simpl in H.
intuition.
subst...
intros.
simpl in H.
destruct H.
subst...
apply in_map_iff in H. destruct H as [x0 [A B]].
subst.
apply IHl in B.
eauto.
Qed.

Lemma length_insert_everywhere x l:
length (insert_everywhere x l) = S (length l).