(** 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 permutation_insertion_permutation : forall (A:Set)(l l':list A),(is_permutation A l l')->forall (x:A)(l'':list A),(insertion A x l' l'')->forall (l''':list A),(is_permutation A l'' l''')->(is_permutation A (x::l) l''').