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(.using
 [library
  [lux (.except)
   ["_" test (.only Test)]
   [abstract
    [monad (.only do)]
    [order (.only Order)]
    [\\specification
     ["$[0]" equivalence]]]
   [data
    ["[0]" bit ("[1]#[0]" equivalence)]
    [collection
     ["[0]" list]]]
   [math
    ["[0]" random (.only Random) ("[1]#[0]" monad)]
    [number
     ["n" nat]]]]]
 [\\library
  ["[0]" / (.only Set)
   ["[0]" //]]])

(def: size
  (random.Random Nat)
  (# random.monad each (n.% 100) random.nat))

(def: .public (random size order gen_value)
  (All (_ a) (-> Nat (Order a) (Random a) (Random (Set a))))
  (case size
    0
    (random#in (/.empty order))

    _
    (do random.monad
      [partial (random (-- size) order gen_value)
       value (random.only (|>> (/.member? partial) not)
                          gen_value)]
      (in (/.has value partial)))))

(def: .public test
  Test
  (<| (_.covering /._)
      (_.for [/.Set])
      (do [! random.monad]
        [sizeL ..size
         sizeR ..size
         usetL (random.set n.hash sizeL random.nat)
         non_memberL (random.only (|>> (//.member? usetL) not)
                                  random.nat)
         .let [listL (//.list usetL)]
         listR (|> (random.set n.hash sizeR random.nat) (# ! each //.list))
         .let [(open "/#[0]") /.equivalence
               setL (/.of_list n.order listL)
               setR (/.of_list n.order listR)
               empty (/.empty n.order)]]
        (`` (all _.and
                 (_.for [/.equivalence]
                        ($equivalence.spec /.equivalence (..random sizeL n.order random.nat)))

                 (_.coverage [/.size]
                   (n.= sizeL (/.size setL)))
                 (_.coverage [/.empty?]
                   (bit#= (n.= 0 (/.size setL))
                          (/.empty? setL)))
                 (_.coverage [/.empty]
                   (/.empty? (/.empty n.order)))
                 (_.coverage [/.list]
                   (# (list.equivalence n.equivalence) =
                      (/.list (/.of_list n.order listL))
                      (list.sorted (# n.order <) listL)))
                 (_.coverage [/.of_list]
                   (|> setL
                       /.list (/.of_list n.order)
                       (/#= setL)))
                 (~~ (template [<coverage> <comparison>]
                       [(_.coverage [<coverage>]
                          (case (<coverage> setL)
                            {.#Some value}
                            (|> setL /.list (list.every? (<comparison> value)))

                            {.#None}
                            (/.empty? setL)))]

                       [/.min n.>=]
                       [/.max n.<=]
                       ))
                 (_.coverage [/.member?]
                   (let [members_are_identified!
                         (list.every? (/.member? setL) (/.list setL))

                         non_members_are_not_identified!
                         (not (/.member? setL non_memberL))]
                     (and members_are_identified!
                          non_members_are_not_identified!)))
                 (_.coverage [/.has]
                   (let [setL+ (/.has non_memberL setL)]
                     (and (not (/.member? setL non_memberL))
                          (/.member? setL+ non_memberL)
                          (n.= (++ (/.size setL))
                               (/.size setL+)))))
                 (_.coverage [/.lacks]
                   (|> setL
                       (/.has non_memberL)
                       (/.lacks non_memberL)
                       (# /.equivalence = setL)))
                 (_.coverage [/.sub?]
                   (let [self!
                         (/.sub? setL setL)

                         empty!
                         (/.sub? setL empty)]
                     (and self!
                          empty!)))
                 (_.coverage [/.super?]
                   (let [self!
                         (/.super? setL setL)

                         empty!
                         (/.super? empty setL)

                         symmetry!
                         (bit#= (/.super? setL setR)
                                (/.sub? setR setL))]
                     (and self!
                          empty!
                          symmetry!)))
                 (~~ (template [<coverage> <relation> <empty?>]
                       [(_.coverage [<coverage>]
                          (let [self!
                                (# /.equivalence =
                                   setL
                                   (<coverage> setL setL))

                                super!
                                (and (<relation> (<coverage> setL setR) setL)
                                     (<relation> (<coverage> setL setR) setR))

                                empty!
                                (# /.equivalence =
                                   (if <empty?> empty setL)
                                   (<coverage> setL empty))

                                idempotence!
                                (# /.equivalence =
                                   (<coverage> setL (<coverage> setL setR))
                                   (<coverage> setR (<coverage> setL setR)))]
                            (and self!
                                 super!
                                 empty!
                                 idempotence!)))]

                       [/.union /.sub? false]
                       [/.intersection /.super? true]
                       ))
                 (_.coverage [/.difference]
                   (let [self!
                         (|> setL
                             (/.difference setL)
                             (# /.equivalence = empty))

                         empty!
                         (|> setL
                             (/.difference empty)
                             (# /.equivalence = setL))

                         difference!
                         (not (list.any? (/.member? (/.difference setL setR))
                                         (/.list setL)))

                         idempotence!
                         (# /.equivalence =
                            (/.difference setL setR)
                            (/.difference setL (/.difference setL setR)))]
                     (and self!
                          empty!
                          difference!
                          idempotence!)))
                 )))))