(** {1 Booleans} *)

(** {2 Basic theory of Booleans} *)

module Bool

  let function andb (x y : bool) : bool =
    match x with
    | True -> y
    | False -> False
    end

  let function orb (x y : bool) : bool =
    match x with
    | False -> y
    | True -> True
    end

  let function notb (x : bool) : bool =
    match x with
    | False -> True
    | True  -> False
    end

  let function xorb (x y : bool) : bool =
    match x with
    | False -> y
    | True -> notb y
    end

  let function implb (x y : bool) : bool =
    match x with
    | False -> True
    | True -> y
    end

  val (=) (x y : bool) : bool ensures { result <-> x = y }

end

(** {2 Operator if-then-else} *)

module Ite

  let function ite (b:bool) (x y : 'a) : 'a =
    match b with
    | True  -> x
    | False -> y
    end

end