inversion H.
rewrite <- H3 in H0.
trivial.
elim (permutation_insertion_comm A l'' l''0 x0 H1 l''' H2).
intro l1;intro.
elim H3;clear H3;intros.
eapply cons_is_permutation.
eapply IHis_permutation.
apply H0.
apply H3.
trivial.
Qed.

(* Specifications: *)
(** is_permutation is transitive *)
Lemma is_permutation_trans : forall (A:Set)(l l':list A),(is_permutation A l l')->forall (l'':list A),(is_permutation A l' l'')->(is_permutation A l l'').
induction l.