    -- | 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.Map (Map)
    import qualified Data.Map as Map
    import Data.Semigroup

    import Agda.Utils.Functor

    import Agda.Utils.Impossible

    -- | A set with duplicates.
    --   Faithfully stores elements which are equal with regard to (==).
    newtype Bag a = Bag
      { forall a. Bag a -> Map a [a]
    bag :: Map a [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
    /= :: Bag a -> Bag a -> Bool
    $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
    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
    min :: Bag a -> Bag a -> Bag a
    $cmin :: forall a. Ord a => Bag a -> Bag a -> Bag a
    max :: Bag a -> Bag a -> Bag a
    $cmax :: forall a. Ord a => Bag a -> Bag a -> Bag a
    >= :: 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
    $c< :: forall a. Ord a => Bag a -> Bag a -> Bool
    compare :: Bag a -> Bag a -> Ordering
    $ccompare :: forall a. Ord a => Bag a -> Bag a -> Ordering
    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 [a] -> Bool
    forall k a. Map k a -> Bool
    Map.null (Map a [a] -> Bool) -> (Bag a -> Map a [a]) -> Bag a -> Bool
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . Bag a -> Map a [a]
    forall a. Bag a -> Map a [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
    . ([a] -> Sum Int) -> Map a [a] -> Sum Int
    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) -> ([a] -> Int) -> [a] -> Sum Int
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . [a] -> Int
    forall (t :: * -> *) a. Foldable t => t a -> Int
    length) (Map a [a] -> Sum Int) -> (Bag a -> Map a [a]) -> Bag a -> Sum Int
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . Bag a -> Map a [a]
    forall a. Bag a -> Map a [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 [a]
    b) a
    a = [a] -> a -> Map a [a] -> [a]
    forall k a. Ord k => a -> k -> Map k a -> a
    Map.findWithDefault [] a
    a Map a [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 (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 (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 [a] -> Bag a
    forall a. Map a [a] -> Bag a
    Bag (Map a [a] -> Bag a) -> Map a [a] -> Bag a
    forall a b. (a -> b) -> a -> b
    $ Map a [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 [a] -> Bag a
    forall a. Map a [a] -> Bag a
    Bag (Map a [a] -> Bag a) -> Map a [a] -> Bag a
    forall a b. (a -> b) -> a -> b
    $ a -> [a] -> Map a [a]
    forall k a. k -> a -> Map k a
    Map.singleton a
    a [a
    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 [a]
    b) (Bag Map a [a]
    c) = Map a [a] -> Bag a
    forall a. Map a [a] -> Bag a
    Bag (Map a [a] -> Bag a) -> Map a [a] -> Bag a
    forall a b. (a -> b) -> a -> b
    $ ([a] -> [a] -> [a]) -> Map a [a] -> Map a [a] -> Map a [a]
    forall k a. Ord k => (a -> a -> a) -> Map k a -> Map k a -> Map k a
    Map.unionWith [a] -> [a] -> [a]
    forall a. [a] -> [a] -> [a]
    (++) Map a [a]
    b Map a [a]
    c

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

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

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

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

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

    -- | Returns the bag without duplicates.
    keys :: Bag a -> [a]
    keys :: forall a. Bag a -> [a]
    keys = Map a [a] -> [a]
    forall k a. Map k a -> [k]
    Map.keys (Map a [a] -> [a]) -> (Bag a -> Map a [a]) -> Bag a -> [a]
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . Bag a -> Map a [a]
    forall a. Bag a -> Map a [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
    ------------------------------------------------------------------------

    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 [b] -> Bag b
    forall a. Map a [a] -> Bag a
    Bag (Map b [b] -> Bag b) -> (Bag a -> Map b [b]) -> Bag a -> Bag b
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . ([b] -> [b] -> [b]) -> [(b, [b])] -> Map b [b]
    forall k a. Ord k => (a -> a -> a) -> [(k, a)] -> Map k a
    Map.fromListWith [b] -> [b] -> [b]
    forall a. [a] -> [a] -> [a]
    (++) ([(b, [b])] -> Map b [b])
    -> (Bag a -> [(b, [b])]) -> Bag a -> Map b [b]
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . ([a] -> (b, [b])) -> [[a]] -> [(b, [b])]
    forall a b. (a -> b) -> [a] -> [b]
    List.map [a] -> (b, [b])
    ff ([[a]] -> [(b, [b])]) -> (Bag a -> [[a]]) -> Bag a -> [(b, [b])]
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . Map a [a] -> [[a]]
    forall k a. Map k a -> [a]
    Map.elems (Map a [a] -> [[a]]) -> (Bag a -> Map a [a]) -> Bag a -> [[a]]
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . Bag a -> Map a [a]
    forall a. Bag a -> Map a [a]
    bag
      where
        ff :: [a] -> (b, [b])
    ff (a
    a : [a]
    as) = (b
    b, b
    b b -> [b] -> [b]
    forall a. a -> [a] -> [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
        ff []       = (b, [b])
    forall a. HasCallStack => a
    __IMPOSSIBLE__

    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 [b] -> Bag b
    forall a. Map a [a] -> Bag a
    Bag (Map b [b] -> Bag b)
    -> ([(b, [b])] -> Map b [b]) -> [(b, [b])] -> Bag b
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . ([b] -> [b] -> [b]) -> [(b, [b])] -> Map b [b]
    forall k a. Ord k => (a -> a -> a) -> [(k, a)] -> Map k a
    Map.fromListWith [b] -> [b] -> [b]
    forall a. [a] -> [a] -> [a]
    (++)) ([(b, [b])] -> Bag b)
    -> (Bag a -> m [(b, [b])]) -> Bag a -> m (Bag b)
    forall (m :: * -> *) b c a.
    Functor m =>
    (b -> c) -> (a -> m b) -> a -> m c
    <.> ([a] -> m (b, [b])) -> [[a]] -> m [(b, [b])]
    forall (t :: * -> *) (f :: * -> *) a b.
    (Traversable t, Applicative f) =>
    (a -> f b) -> t a -> f (t b)
    traverse [a] -> m (b, [b])
    trav ([[a]] -> m [(b, [b])])
    -> (Bag a -> [[a]]) -> Bag a -> m [(b, [b])]
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . Map a [a] -> [[a]]
    forall k a. Map k a -> [a]
    Map.elems (Map a [a] -> [[a]]) -> (Bag a -> Map a [a]) -> Bag a -> [[a]]
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . Bag a -> Map a [a]
    forall a. Bag a -> Map a [a]
    bag
      where
        trav :: [a] -> m (b, [b])
        trav :: [a] -> m (b, [b])
    trav (a
    a : [a]
    as) = (\ b
    b [b]
    bs -> (b
    b, b
    bb -> [b] -> [b]
    forall a. a -> [a] -> [a]
    :[b]
    bs)) (b -> [b] -> (b, [b])) -> m b -> m ([b] -> (b, [b]))
    forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
    <$> a -> m b
    f a
    a m ([b] -> (b, [b])) -> m [b] -> m (b, [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)
    traverse a -> m b
    f [a]
    as
        trav []       = m (b, [b])
    forall a. HasCallStack => a
    __IMPOSSIBLE__

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

    instance Show a => Show (Bag a) where
      showsPrec :: Int -> Bag a -> ShowS
    showsPrec Int
    _ (Bag Map a [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 [a] -> ShowS
    forall a. Show a => a -> ShowS
    shows Map a [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 (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)