{-# OPTIONS_GHC -Wunused-imports #-}

-- | A simple overlay over Data.Map to manage unordered sets with duplicates.

module Agda.Utils.Bag where

import Prelude hiding (null, map)

import           Text.Show.Functions    () -- instance only

import qualified Data.List              as List
import           Data.List.NonEmpty     ( NonEmpty, pattern (:|) )
import qualified Data.List.NonEmpty     as List1
  -- NB: Not importing Agda.Utils.List1 to avoid import cycles.
import           Data.Map               ( Map )
import qualified Data.Map               as Map
import           Data.Semigroup

import           Agda.Utils.Functor

-- | A set with duplicates.
--   Faithfully stores elements which are equal with regard to (==).
newtype Bag a = Bag
  { forall a. Bag a -> Map a (NonEmpty a)
bag :: Map a (NonEmpty a)
      -- ^ The list contains all occurrences of @a@ (not just the duplicates!).
      --   Hence, the invariant: the list is never empty.
  }
  deriving (Bag a -> Bag a -> Bool
(Bag a -> Bag a -> Bool) -> (Bag a -> Bag a -> Bool) -> Eq (Bag a)
forall a. Eq a => Bag a -> Bag a -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
$c== :: forall a. Eq a => Bag a -> Bag a -> Bool
== :: Bag a -> Bag a -> Bool
$c/= :: forall a. Eq a => Bag a -> Bag a -> Bool
/= :: Bag a -> Bag a -> Bool
Eq, Eq (Bag a)
Eq (Bag a) =>
(Bag a -> Bag a -> Ordering)
-> (Bag a -> Bag a -> Bool)
-> (Bag a -> Bag a -> Bool)
-> (Bag a -> Bag a -> Bool)
-> (Bag a -> Bag a -> Bool)
-> (Bag a -> Bag a -> Bag a)
-> (Bag a -> Bag a -> Bag a)
-> Ord (Bag a)
Bag a -> Bag a -> Bool
Bag a -> Bag a -> Ordering
Bag a -> Bag a -> Bag a
forall a.
Eq a =>
(a -> a -> Ordering)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> a)
-> (a -> a -> a)
-> Ord a
forall a. Ord a => Eq (Bag a)
forall a. Ord a => Bag a -> Bag a -> Bool
forall a. Ord a => Bag a -> Bag a -> Ordering
forall a. Ord a => Bag a -> Bag a -> Bag a
$ccompare :: forall a. Ord a => Bag a -> Bag a -> Ordering
compare :: Bag a -> Bag a -> Ordering
$c< :: forall a. Ord a => Bag a -> Bag a -> Bool
< :: Bag a -> Bag a -> Bool
$c<= :: forall a. Ord a => Bag a -> Bag a -> Bool
<= :: Bag a -> Bag a -> Bool
$c> :: forall a. Ord a => Bag a -> Bag a -> Bool
> :: Bag a -> Bag a -> Bool
$c>= :: forall a. Ord a => Bag a -> Bag a -> Bool
>= :: Bag a -> Bag a -> Bool
$cmax :: forall a. Ord a => Bag a -> Bag a -> Bag a
max :: Bag a -> Bag a -> Bag a
$cmin :: forall a. Ord a => Bag a -> Bag a -> Bag a
min :: Bag a -> Bag a -> Bag a
Ord)
  -- The list contains all occurrences of @a@ (not just the duplicates!).
  -- Hence the invariant: the list is never empty.
  --
  -- This is slightly wasteful, but much easier to implement
  -- in terms of @Map@ as the alternative, which is to store
  -- only the duplicates in the list.
  -- See, e.g., implementation of 'union' which would be impossible
  -- to do in the other representation.  We would need a
  -- 'Map.unionWithKey' that passes us *both* keys.
  -- But Map works under the assumption that Eq for keys is identity,
  -- it does not honor information in keys that goes beyond Ord.

------------------------------------------------------------------------
-- * Query
------------------------------------------------------------------------

-- | Is the bag empty?
null :: Bag a -> Bool
null :: forall a. Bag a -> Bool
null = Map a (NonEmpty a) -> Bool
forall k a. Map k a -> Bool
Map.null (Map a (NonEmpty a) -> Bool)
-> (Bag a -> Map a (NonEmpty a)) -> Bag a -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Bag a -> Map a (NonEmpty a)
forall a. Bag a -> Map a (NonEmpty a)
bag

-- | Number of elements in the bag.  Duplicates count. O(n).
size :: Bag a -> Int
size :: forall a. Bag a -> Int
size = Sum Int -> Int
forall a. Sum a -> a
getSum (Sum Int -> Int) -> (Bag a -> Sum Int) -> Bag a -> Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (NonEmpty a -> Sum Int) -> Map a (NonEmpty a) -> Sum Int
forall m a. Monoid m => (a -> m) -> Map a a -> m
forall (t :: * -> *) m a.
(Foldable t, Monoid m) =>
(a -> m) -> t a -> m
foldMap (Int -> Sum Int
forall a. a -> Sum a
Sum (Int -> Sum Int) -> (NonEmpty a -> Int) -> NonEmpty a -> Sum Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. NonEmpty a -> Int
forall a. NonEmpty a -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length) (Map a (NonEmpty a) -> Sum Int)
-> (Bag a -> Map a (NonEmpty a)) -> Bag a -> Sum Int
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Bag a -> Map a (NonEmpty a)
forall a. Bag a -> Map a (NonEmpty a)
bag

-- | @(bag ! a)@ finds all elements equal to @a@.  O(log n).
--   Total function, returns @[]@ if none are.
(!) :: Ord a => Bag a -> a -> [a]
! :: forall a. Ord a => Bag a -> a -> [a]
(!) (Bag Map a (NonEmpty a)
b) a
a = [a] -> (NonEmpty a -> [a]) -> Maybe (NonEmpty a) -> [a]
forall b a. b -> (a -> b) -> Maybe a -> b
maybe [] NonEmpty a -> [a]
forall a. NonEmpty a -> [a]
List1.toList (Maybe (NonEmpty a) -> [a]) -> Maybe (NonEmpty a) -> [a]
forall a b. (a -> b) -> a -> b
$ a -> Map a (NonEmpty a) -> Maybe (NonEmpty a)
forall k a. Ord k => k -> Map k a -> Maybe a
Map.lookup a
a Map a (NonEmpty a)
b
  -- Note: not defined infix because of BangPatterns.

-- | O(log n).
member :: Ord a => a -> Bag a -> Bool
member :: forall a. Ord a => a -> Bag a -> Bool
member a
a = Bool -> Bool
not (Bool -> Bool) -> (Bag a -> Bool) -> Bag a -> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. a -> Bag a -> Bool
forall a. Ord a => a -> Bag a -> Bool
notMember a
a

-- | O(log n).
notMember :: Ord a => a -> Bag a -> Bool
notMember :: forall a. Ord a => a -> Bag a -> Bool
notMember a
a Bag a
b = [a] -> Bool
forall a. [a] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
List.null (Bag a
b Bag a -> a -> [a]
forall a. Ord a => Bag a -> a -> [a]
! a
a)

-- | Return the multiplicity of the given element. O(log n + count _ _).
count :: Ord a => a -> Bag a -> Int
count :: forall a. Ord a => a -> Bag a -> Int
count a
a Bag a
b = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length (Bag a
b Bag a -> a -> [a]
forall a. Ord a => Bag a -> a -> [a]
! a
a)

------------------------------------------------------------------------
-- * Construction
------------------------------------------------------------------------

-- | O(1)
empty :: Bag a
empty :: forall a. Bag a
empty = Map a (NonEmpty a) -> Bag a
forall a. Map a (NonEmpty a) -> Bag a
Bag (Map a (NonEmpty a) -> Bag a) -> Map a (NonEmpty a) -> Bag a
forall a b. (a -> b) -> a -> b
$ Map a (NonEmpty a)
forall k a. Map k a
Map.empty

-- | O(1)
singleton :: a -> Bag a
singleton :: forall a. a -> Bag a
singleton a
a = Map a (NonEmpty a) -> Bag a
forall a. Map a (NonEmpty a) -> Bag a
Bag (Map a (NonEmpty a) -> Bag a) -> Map a (NonEmpty a) -> Bag a
forall a b. (a -> b) -> a -> b
$ a -> NonEmpty a -> Map a (NonEmpty a)
forall k a. k -> a -> Map k a
Map.singleton a
a (a
a a -> [a] -> NonEmpty a
forall a. a -> [a] -> NonEmpty a
:| [])

union :: Ord a => Bag a -> Bag a -> Bag a
union :: forall a. Ord a => Bag a -> Bag a -> Bag a
union (Bag Map a (NonEmpty a)
b) (Bag Map a (NonEmpty a)
c) = Map a (NonEmpty a) -> Bag a
forall a. Map a (NonEmpty a) -> Bag a
Bag (Map a (NonEmpty a) -> Bag a) -> Map a (NonEmpty a) -> Bag a
forall a b. (a -> b) -> a -> b
$ (NonEmpty a -> NonEmpty a -> NonEmpty a)
-> Map a (NonEmpty a) -> Map a (NonEmpty a) -> Map a (NonEmpty a)
forall k a. Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
Map.unionWith NonEmpty a -> NonEmpty a -> NonEmpty a
forall a. Semigroup a => a -> a -> a
(<>) Map a (NonEmpty a)
b Map a (NonEmpty a)
c

unions :: Ord a => [Bag a] -> Bag a
unions :: forall a. Ord a => [Bag a] -> Bag a
unions = Map a (NonEmpty a) -> Bag a
forall a. Map a (NonEmpty a) -> Bag a
Bag (Map a (NonEmpty a) -> Bag a)
-> ([Bag a] -> Map a (NonEmpty a)) -> [Bag a] -> Bag a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (NonEmpty a -> NonEmpty a -> NonEmpty a)
-> [Map a (NonEmpty a)] -> Map a (NonEmpty a)
forall (f :: * -> *) k a.
(Foldable f, Ord k) =>
(a -> a -> a) -> f (Map k a) -> Map k a
Map.unionsWith NonEmpty a -> NonEmpty a -> NonEmpty a
forall a. Semigroup a => a -> a -> a
(<>)  ([Map a (NonEmpty a)] -> Map a (NonEmpty a))
-> ([Bag a] -> [Map a (NonEmpty a)])
-> [Bag a]
-> Map a (NonEmpty a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Bag a -> Map a (NonEmpty a)) -> [Bag a] -> [Map a (NonEmpty a)]
forall a b. (a -> b) -> [a] -> [b]
List.map Bag a -> Map a (NonEmpty a)
forall a. Bag a -> Map a (NonEmpty a)
bag

-- | @insert a b = union b (singleton a)@
insert :: Ord a => a -> Bag a -> Bag a
insert :: forall a. Ord a => a -> Bag a -> Bag a
insert a
a = Map a (NonEmpty a) -> Bag a
forall a. Map a (NonEmpty a) -> Bag a
Bag (Map a (NonEmpty a) -> Bag a)
-> (Bag a -> Map a (NonEmpty a)) -> Bag a -> Bag a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (NonEmpty a -> NonEmpty a -> NonEmpty a)
-> a -> NonEmpty a -> Map a (NonEmpty a) -> Map a (NonEmpty a)
forall k a. Ord k => (a -> a -> a) -> k -> a -> Map k a -> Map k a
Map.insertWith NonEmpty a -> NonEmpty a -> NonEmpty a
forall a. Semigroup a => a -> a -> a
(<>) a
a (a
a a -> [a] -> NonEmpty a
forall a. a -> [a] -> NonEmpty a
:| []) (Map a (NonEmpty a) -> Map a (NonEmpty a))
-> (Bag a -> Map a (NonEmpty a)) -> Bag a -> Map a (NonEmpty a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Bag a -> Map a (NonEmpty a)
forall a. Bag a -> Map a (NonEmpty a)
bag

-- | @fromList = unions . map singleton@
fromList :: Ord a => [a] -> Bag a
fromList :: forall a. Ord a => [a] -> Bag a
fromList = Map a (NonEmpty a) -> Bag a
forall a. Map a (NonEmpty a) -> Bag a
Bag (Map a (NonEmpty a) -> Bag a)
-> ([a] -> Map a (NonEmpty a)) -> [a] -> Bag a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (NonEmpty a -> NonEmpty a -> NonEmpty a)
-> [(a, NonEmpty a)] -> Map a (NonEmpty a)
forall k a. Ord k => (a -> a -> a) -> [(k, a)] -> Map k a
Map.fromListWith NonEmpty a -> NonEmpty a -> NonEmpty a
forall a. Semigroup a => a -> a -> a
(<>) ([(a, NonEmpty a)] -> Map a (NonEmpty a))
-> ([a] -> [(a, NonEmpty a)]) -> [a] -> Map a (NonEmpty a)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (a -> (a, NonEmpty a)) -> [a] -> [(a, NonEmpty a)]
forall a b. (a -> b) -> [a] -> [b]
List.map (\ a
a -> (a
a, a
a a -> [a] -> NonEmpty a
forall a. a -> [a] -> NonEmpty a
:| []))

------------------------------------------------------------------------
-- * Destruction
------------------------------------------------------------------------

-- | Returns the elements of the bag, grouped by equality (==).
groups :: Bag a -> [NonEmpty a]
groups :: forall a. Bag a -> [NonEmpty a]
groups = Map a (NonEmpty a) -> [NonEmpty a]
forall k a. Map k a -> [a]
Map.elems (Map a (NonEmpty a) -> [NonEmpty a])
-> (Bag a -> Map a (NonEmpty a)) -> Bag a -> [NonEmpty a]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Bag a -> Map a (NonEmpty a)
forall a. Bag a -> Map a (NonEmpty a)
bag

-- | Returns the bag, with duplicates.
toList :: Bag a -> [a]
toList :: forall a. Bag a -> [a]
toList = (NonEmpty a -> [a]) -> [NonEmpty a] -> [a]
forall (t :: * -> *) a b. Foldable t => (a -> [b]) -> t a -> [b]
concatMap NonEmpty a -> [a]
forall a. NonEmpty a -> [a]
List1.toList ([NonEmpty a] -> [a]) -> (Bag a -> [NonEmpty a]) -> Bag a -> [a]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Bag a -> [NonEmpty a]
forall a. Bag a -> [NonEmpty a]
groups

-- | Returns the bag without duplicates.
keys :: Bag a -> [a]
keys :: forall a. Bag a -> [a]
keys = Map a (NonEmpty a) -> [a]
forall k a. Map k a -> [k]
Map.keys (Map a (NonEmpty a) -> [a])
-> (Bag a -> Map a (NonEmpty a)) -> Bag a -> [a]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Bag a -> Map a (NonEmpty a)
forall a. Bag a -> Map a (NonEmpty a)
bag
-- Works because of the invariant!
-- keys = catMaybes . map listToMaybe . Map.elems . bag
--   -- Map.keys does not work, as zero copies @(a,[])@
--   -- should count as not present in the bag.

-- | Returns the bag, with duplicates.
elems :: Bag a -> [a]
elems :: forall a. Bag a -> [a]
elems = Bag a -> [a]
forall a. Bag a -> [a]
toList

toAscList :: Bag a -> [a]
toAscList :: forall a. Bag a -> [a]
toAscList = Bag a -> [a]
forall a. Bag a -> [a]
toList

------------------------------------------------------------------------
-- * Traversal
------------------------------------------------------------------------

-- | O(n).
map :: Ord b => (a -> b) -> Bag a -> Bag b
map :: forall b a. Ord b => (a -> b) -> Bag a -> Bag b
map a -> b
f = Map b (NonEmpty b) -> Bag b
forall a. Map a (NonEmpty a) -> Bag a
Bag (Map b (NonEmpty b) -> Bag b)
-> (Bag a -> Map b (NonEmpty b)) -> Bag a -> Bag b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (NonEmpty b -> NonEmpty b -> NonEmpty b)
-> [(b, NonEmpty b)] -> Map b (NonEmpty b)
forall k a. Ord k => (a -> a -> a) -> [(k, a)] -> Map k a
Map.fromListWith NonEmpty b -> NonEmpty b -> NonEmpty b
forall a. Semigroup a => a -> a -> a
(<>) ([(b, NonEmpty b)] -> Map b (NonEmpty b))
-> (Bag a -> [(b, NonEmpty b)]) -> Bag a -> Map b (NonEmpty b)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (NonEmpty a -> (b, NonEmpty b))
-> [NonEmpty a] -> [(b, NonEmpty b)]
forall a b. (a -> b) -> [a] -> [b]
List.map NonEmpty a -> (b, NonEmpty b)
ff ([NonEmpty a] -> [(b, NonEmpty b)])
-> (Bag a -> [NonEmpty a]) -> Bag a -> [(b, NonEmpty b)]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Map a (NonEmpty a) -> [NonEmpty a]
forall k a. Map k a -> [a]
Map.elems (Map a (NonEmpty a) -> [NonEmpty a])
-> (Bag a -> Map a (NonEmpty a)) -> Bag a -> [NonEmpty a]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Bag a -> Map a (NonEmpty a)
forall a. Bag a -> Map a (NonEmpty a)
bag
  where
    ff :: NonEmpty a -> (b, NonEmpty b)
ff (a
a :| [a]
as) = (b
b, b
b b -> [b] -> NonEmpty b
forall a. a -> [a] -> NonEmpty a
:| (a -> b) -> [a] -> [b]
forall a b. (a -> b) -> [a] -> [b]
List.map a -> b
f [a]
as) where b :: b
b = a -> b
f a
a

traverse' :: forall a b m . (Applicative m, Ord b) =>
             (a -> m b) -> Bag a -> m (Bag b)
traverse' :: forall a b (m :: * -> *).
(Applicative m, Ord b) =>
(a -> m b) -> Bag a -> m (Bag b)
traverse' a -> m b
f = (Map b (NonEmpty b) -> Bag b
forall a. Map a (NonEmpty a) -> Bag a
Bag (Map b (NonEmpty b) -> Bag b)
-> ([(b, NonEmpty b)] -> Map b (NonEmpty b))
-> [(b, NonEmpty b)]
-> Bag b
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (NonEmpty b -> NonEmpty b -> NonEmpty b)
-> [(b, NonEmpty b)] -> Map b (NonEmpty b)
forall k a. Ord k => (a -> a -> a) -> [(k, a)] -> Map k a
Map.fromListWith NonEmpty b -> NonEmpty b -> NonEmpty b
forall a. Semigroup a => a -> a -> a
(<>)) ([(b, NonEmpty b)] -> Bag b)
-> (Bag a -> m [(b, NonEmpty b)]) -> Bag a -> m (Bag b)
forall (m :: * -> *) b c a.
Functor m =>
(b -> c) -> (a -> m b) -> a -> m c
<.> (NonEmpty a -> m (b, NonEmpty b))
-> [NonEmpty a] -> m [(b, NonEmpty b)]
forall (t :: * -> *) (f :: * -> *) a b.
(Traversable t, Applicative f) =>
(a -> f b) -> t a -> f (t b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> [a] -> f [b]
traverse NonEmpty a -> m (b, NonEmpty b)
trav ([NonEmpty a] -> m [(b, NonEmpty b)])
-> (Bag a -> [NonEmpty a]) -> Bag a -> m [(b, NonEmpty b)]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Map a (NonEmpty a) -> [NonEmpty a]
forall k a. Map k a -> [a]
Map.elems (Map a (NonEmpty a) -> [NonEmpty a])
-> (Bag a -> Map a (NonEmpty a)) -> Bag a -> [NonEmpty a]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Bag a -> Map a (NonEmpty a)
forall a. Bag a -> Map a (NonEmpty a)
bag
  where
    trav :: NonEmpty a -> m (b, NonEmpty b)
    trav :: NonEmpty a -> m (b, NonEmpty b)
trav (a
a :| [a]
as) = (\ b
b [b]
bs -> (b
b, b
b b -> [b] -> NonEmpty b
forall a. a -> [a] -> NonEmpty a
:| [b]
bs)) (b -> [b] -> (b, NonEmpty b)) -> m b -> m ([b] -> (b, NonEmpty b))
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> m b
f a
a m ([b] -> (b, NonEmpty b)) -> m [b] -> m (b, NonEmpty b)
forall a b. m (a -> b) -> m a -> m b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> (a -> m b) -> [a] -> m [b]
forall (t :: * -> *) (f :: * -> *) a b.
(Traversable t, Applicative f) =>
(a -> f b) -> t a -> f (t b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> [a] -> f [b]
traverse a -> m b
f [a]
as


------------------------------------------------------------------------
-- Instances
------------------------------------------------------------------------

instance Show a => Show (Bag a) where
  showsPrec :: Int -> Bag a -> ShowS
showsPrec Int
_ (Bag Map a (NonEmpty a)
b) = (String
"Agda.Utils.Bag.Bag (" String -> ShowS
forall a. [a] -> [a] -> [a]
++) ShowS -> ShowS -> ShowS
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Map a (NonEmpty a) -> ShowS
forall a. Show a => a -> ShowS
shows Map a (NonEmpty a)
b ShowS -> ShowS -> ShowS
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Char
')'Char -> ShowS
forall a. a -> [a] -> [a]
:)

instance Ord a => Semigroup (Bag a) where
  <> :: Bag a -> Bag a -> Bag a
(<>) = Bag a -> Bag a -> Bag a
forall a. Ord a => Bag a -> Bag a -> Bag a
union

instance Ord a => Monoid (Bag a) where
  mempty :: Bag a
mempty  = Bag a
forall a. Bag a
empty
  mappend :: Bag a -> Bag a -> Bag a
mappend = Bag a -> Bag a -> Bag a
forall a. Semigroup a => a -> a -> a
(<>)
  mconcat :: [Bag a] -> Bag a
mconcat = [Bag a] -> Bag a
forall a. Ord a => [Bag a] -> Bag a
unions

instance Foldable Bag where
  foldMap :: forall m a. Monoid m => (a -> m) -> Bag a -> m
foldMap a -> m
f = (a -> m) -> [a] -> m
forall m a. Monoid m => (a -> m) -> [a] -> m
forall (t :: * -> *) m a.
(Foldable t, Monoid m) =>
(a -> m) -> t a -> m
foldMap a -> m
f ([a] -> m) -> (Bag a -> [a]) -> Bag a -> m
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Bag a -> [a]
forall a. Bag a -> [a]
toList

-- not a Functor (only works for 'Ord'ered types)
-- not Traversable (only works for 'Ord'ered types)