(** l' is a permutation of l *)
Inductive is_permutation (A:Set) : list A->list A->Prop :=
  nil_is_permutation : (is_permutation A nil nil)
|cons_is_permutation : forall (l l':list A),(is_permutation A l l')->forall (x:A)(l'':(list A)),(insertion A x l' l'')->(is_permutation A (x::l) l'').(** l' is a permutation of l *)
Inductive is_permutation (A:Set) : list A->list A->Prop :=
  nil_is_permutation : (is_permutation A nil nil)
|cons_is_permutation : forall (l l':list A),(is_permutation A l l')->forall (x:A)(l'':(list A)),(insertion A x l' l'')->(is_permutation A (x::l) l'').(** l' is a permutation of l *)
Inductive is_permutation (A:Set) : list A->list A->Prop :=
  nil_is_permutation : (is_permutation A nil nil)
|cons_is_permutation : forall (l l':list A),(is_permutation A l l')->forall (x:A)(l'':(list A)),(insertion A x l' l'')->(is_permutation A (x::l) l'').

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.