- apply aux1 in IHp.
inversion IHp as [f' Hyp]; clear IHp.
rewrite <- app_assoc; simpl.
pose proof Hyp [Left; Right; Right] as H; clear Hyp.
inversion H as [H0 H1]; clear H.
rewrite H0; clear H0.
simpl; subst; auto.
- apply aux1 in IHp.
inversion IHp as [f' Hyp]; clear IHp.
rewrite <- app_assoc; simpl.
pose proof Hyp [Left; Left; Right] as H; clear Hyp.
inversion H as [H0 H1]; clear H.
rewrite H0; clear H0.
simpl; subst; auto.
- auto.
Qed.
Fixpoint lengthSplit {A : Type} (l l' : list A) : option (list A * list A) :=
match l, l' with
| [], x => Some ([], x)
| _::xs, y::ys =>
option_map (fun (p : list A * list A) =>
let (l1,l2) := p in (y::l1, l2)) (lengthSplit xs ys)
| _, _ => None
end.

(* Specifications: *)
Lemma lengthSplit1 : forall {A : Type} (l l' : list A),
le (length l) (length l') ->
exists p, lengthSplit l l' = Some p.