simpl.
intros.
apply head_insertion.
intros.
simpl.
apply tail_insertion.
apply IHinsertion.
Qed.

Lemma is_permutation_append : forall (A:Set)(xs ys:list A),(is_permutation A xs ys)->forall (xs' ys':list A),(is_permutation A xs' ys')->(is_permutation A (app xs xs') (app ys ys')).
induction 1;intros.