induction n; intuition; simpl in *.
Qed.

(* Specifications: *)
Lemma firstn_ge_all : forall n (A : Set) (ls : list A),
n >= length ls ->
firstn n ls = ls.
Theorem nth_In_exists :
forall (A : Type)(ls : list A) a def,
In a ls ->
exists n, nth n ls def = a.