induction l in p |- *...
simpl.
intuition.
simpl.
intros.
destruct H.
subst...
destruct (fst (conj_prod (in_map_iff _ _ _)) H).
destruct H0.
apply IHl in H1.
subst.
simpl...
apply perm_trans with (a :: fst x :: snd x).
apply perm_swap.
apply perm_skip...
Qed.

Lemma insert_everywhere_are_perms x l:
forall y, In y (insert_everywhere x l) -> Permutation y (x :: l).