apply nil_is_permutation.
eapply cons_is_permutation;[apply IHl | apply head_insertion].
Qed.

(** if l' is l where x has been inserted then l' is a permutation of x::l *)
Lemma insertion_is_permutation : forall (A:Set)(l l':list A)(x:A),(insertion A x l l')->(is_permutation A (x::l) l').
induction 1.