(** Dropping the first N elements of a list. *)

Fixpoint list_drop (A: Type) (n: nat) (x: list A) {struct n} : list A :=
  match n with
  | O => x
  | S n' => match x with nil => nil | hd :: tl => list_drop n' tl end
  end.

(* Specifications: *)
Lemma list_drop_incl: forall (A: Type) (x: A) n (l: list A), In x (list_drop n l) -> In x l.
Lemma list_drop_norepet: forall (A: Type) n (l: list A), list_norepet l -> list_norepet (list_drop n l).
Lemma list_map_drop: forall (A B: Type) (f: A -> B) n (l: list A), list_drop n (map f l) = map f (list_drop n l).
Lemma list_drop_incl: forall (A: Type) (x: A) n (l: list A), In x (list_drop n l) -> In x l.
Lemma list_drop_norepet: forall (A: Type) n (l: list A), list_norepet l -> list_norepet (list_drop n l).
Lemma list_map_drop: forall (A B: Type) (f: A -> B) n (l: list A), list_drop n (map f l) = map f (list_drop n l).