simpl.
auto.
simpl.
eapply cons_is_permutation.
apply IHis_permutation.
apply H1.
apply insertion_append.
auto.
Qed.

Lemma insertion_map : forall (B:Set)(y:B)(ys yss:list B),(insertion _ y ys yss)->forall (A:Set)(f:A->B)(x:A),y=f x->forall (xs:list A),ys = map f xs->exists xss:list A,yss = map f xss /\ insertion _ x xs xss.
induction 1;intros.