Set Implicit Arguments.

#[local] Open Scope bool_scope.
Open Scope list_scope.

Section Map.

  Variables (A : Type) (B : Type).
  Variable f : A -> B.

  Fixpoint map (l:list A) : list B :=
    match l with
      | nil => nil
      | a :: l => (f a) :: (map l)
    end.

End Map.

Section NatSeq.

  Fixpoint seq (start len:nat) : list nat :=
    match len with
      | 0 => nil
      | S len => start :: seq (S start) len
    end.

End NatSeq.

Section Repeat.

  Variable A : Type.
  Fixpoint repeat (x : A) (n: nat ) :=
    match n with
      | O => nil
      | S k => x :: repeat x k
    end.

End Repeat.

Section Elts.

  Variable A : Type.

  Fixpoint nth (n:nat) (l:list A) (default:A) {struct l} : A :=
    match n, l with
      | O, x :: l' => x
      | O, nil => default
      | S m, nil => default
      | S m, x :: l' => nth m l' default
    end.

End Elts.

Section Cutting.

  Variable A : Type.

  Fixpoint firstn (n:nat) (l:list A) : list A :=
    match n with
      | 0 => nil
      | S n => match l with
                 | nil => nil
                 | a::l => a::(firstn n l)
               end
    end.

  Fixpoint skipn (n:nat) (l:list A) : list A :=
    match n with
      | 0 => l
      | S n => match l with
                 | nil => nil
                 | a::l => skipn n l
               end
    end.

End Cutting.

Section Exists_Forall.

  Variable A:Type.

  Section One_predicate.

    Variable P : A -> Prop.

    Inductive Forall : list A -> Prop :=
      | Forall_nil : Forall nil
      | Forall_cons : forall x l, P x -> Forall l -> Forall (x :: l).

  End One_predicate.

End Exists_Forall.

Section Compare.

  Variable A : Type.
  Variable cmp : A -> A -> comparison.

  Fixpoint list_compare (xs ys : list A) : comparison :=
    match xs, ys with
    | nil , nil => Eq
    | nil , _ => Lt
    | _ , nil => Gt
    | x :: xs, y :: ys =>
        match cmp x y with
        | Eq => list_compare xs ys
        | c => c
        end
    end%list.

End Compare.