Data.Wedge

This module contains the definition for the Wedge datatype. In practice, this type is isomorphic to 'Maybe (Either a b)' - the type with two possibly non-exclusive values and an empty case.

Datatypes

Categorically, the Wedge datatype represents the coproduct (like, Either) in the category Hask* of pointed Hask types, called a wedge sum. The category Hask* consists of Hask types affixed with a dedicated base point along with an object. In Hask, this is equivalent to `1 + a`, also known as 'Maybe a'. Because we can conflate basepoints of different types (there is only one Nothing type), the wedge sum is can be viewed as the type `1 + a + b`, or `Maybe (Either a b)` in Hask. Pictorially, one can visualize this as:

Wedge:
                a
                |
Nowhere +-------+
                |
                b

The fact that we can think about Wedge as a coproduct gives us some reasoning power about how a Wedge will interact with the product in Hask*, called Can. Namely, we know that a product of a type and a coproduct, `a * (b + c)`, is equivalent to `(a + b) * (a + c)`. Additioally, we may derive other facts about its associativity, distributivity, commutativity, and any more. As an exercise, think of soemthing Either can do. Now do it with Wedge!

data Wedge a b

The Wedge data type represents values with two exclusive possibilities, and an empty case. This is a coproduct of pointed types - i.e. of Maybe values. The result is a type, 'Wedge a b', which is isomorphic to 'Maybe (Either a b)'.

Constructors

Nowhere

Here a

There b

Instances

Bitraversable Wedge

Defined in Data.Wedge

Methods

bitraverse :: Applicative f => (a -> f c) -> (b -> f d) -> Wedge a b -> f (Wedge c d)

Bifoldable Wedge

Defined in Data.Wedge

Methods

bifold :: Monoid m => Wedge m m -> m

bifoldMap :: Monoid m => (a -> m) -> (b -> m) -> Wedge a b -> m

bifoldr :: (a -> c -> c) -> (b -> c -> c) -> c -> Wedge a b -> c

bifoldl :: (c -> a -> c) -> (c -> b -> c) -> c -> Wedge a b -> c

Bifunctor Wedge

Defined in Data.Wedge

Methods

bimap :: (a -> b) -> (c -> d) -> Wedge a c -> Wedge b d

first :: (a -> b) -> Wedge a c -> Wedge b c

second :: (b -> c) -> Wedge a b -> Wedge a c

Semigroup a => Monad (Wedge a)

Defined in Data.Wedge

Methods

(>>=) :: Wedge a a0 -> (a0 -> Wedge a b) -> Wedge a b

(>>) :: Wedge a a0 -> Wedge a b -> Wedge a b

return :: a0 -> Wedge a a0

fail :: String -> Wedge a a0

Functor (Wedge a)

Defined in Data.Wedge

Methods

fmap :: (a0 -> b) -> Wedge a a0 -> Wedge a b

(<$) :: a0 -> Wedge a b -> Wedge a a0

Semigroup a => Applicative (Wedge a)

Defined in Data.Wedge

Methods

pure :: a0 -> Wedge a a0

(<*>) :: Wedge a (a0 -> b) -> Wedge a a0 -> Wedge a b

liftA2 :: (a0 -> b -> c) -> Wedge a a0 -> Wedge a b -> Wedge a c

(*>) :: Wedge a a0 -> Wedge a b -> Wedge a b

(<*) :: Wedge a a0 -> Wedge a b -> Wedge a a0

Foldable (Wedge a)

Defined in Data.Wedge

Methods

fold :: Monoid m => Wedge a m -> m

foldMap :: Monoid m => (a0 -> m) -> Wedge a a0 -> m

foldr :: (a0 -> b -> b) -> b -> Wedge a a0 -> b

foldr' :: (a0 -> b -> b) -> b -> Wedge a a0 -> b

foldl :: (b -> a0 -> b) -> b -> Wedge a a0 -> b

foldl' :: (b -> a0 -> b) -> b -> Wedge a a0 -> b

foldr1 :: (a0 -> a0 -> a0) -> Wedge a a0 -> a0

foldl1 :: (a0 -> a0 -> a0) -> Wedge a a0 -> a0

toList :: Wedge a a0 -> [a0]

null :: Wedge a a0 -> Bool

length :: Wedge a a0 -> Int

elem :: Eq a0 => a0 -> Wedge a a0 -> Bool

maximum :: Ord a0 => Wedge a a0 -> a0

minimum :: Ord a0 => Wedge a a0 -> a0

sum :: Num a0 => Wedge a a0 -> a0

product :: Num a0 => Wedge a a0 -> a0

Traversable (Wedge a)

Defined in Data.Wedge

Methods

traverse :: Applicative f => (a0 -> f b) -> Wedge a a0 -> f (Wedge a b)

sequenceA :: Applicative f => Wedge a (f a0) -> f (Wedge a a0)

mapM :: Monad m => (a0 -> m b) -> Wedge a a0 -> m (Wedge a b)

sequence :: Monad m => Wedge a (m a0) -> m (Wedge a a0)

Generic1 (Wedge a :: Type -> Type)

Defined in Data.Wedge

Associated Types

type Rep1 (Wedge a) :: k -> Type

Methods

from1 :: Wedge a a0 -> Rep1 (Wedge a) a0

to1 :: Rep1 (Wedge a) a0 -> Wedge a a0

(Eq a, Eq b) => Eq (Wedge a b)

Defined in Data.Wedge

Methods

(==) :: Wedge a b -> Wedge a b -> Bool

(/=) :: Wedge a b -> Wedge a b -> Bool

(Data a, Data b) => Data (Wedge a b)

Defined in Data.Wedge

Methods

gfoldl :: (forall d b0. Data d => c (d -> b0) -> d -> c b0) -> (forall g. g -> c g) -> Wedge a b -> c (Wedge a b)

gunfold :: (forall b0 r. Data b0 => c (b0 -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Wedge a b)

toConstr :: Wedge a b -> Constr

dataTypeOf :: Wedge a b -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Wedge a b))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Wedge a b))

gmapT :: (forall b0. Data b0 => b0 -> b0) -> Wedge a b -> Wedge a b

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Wedge a b -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Wedge a b -> r

gmapQ :: (forall d. Data d => d -> u) -> Wedge a b -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Wedge a b -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Wedge a b -> m (Wedge a b)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Wedge a b -> m (Wedge a b)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Wedge a b -> m (Wedge a b)

(Ord a, Ord b) => Ord (Wedge a b)

Defined in Data.Wedge

Methods

compare :: Wedge a b -> Wedge a b -> Ordering

(<) :: Wedge a b -> Wedge a b -> Bool

(<=) :: Wedge a b -> Wedge a b -> Bool

(>) :: Wedge a b -> Wedge a b -> Bool

(>=) :: Wedge a b -> Wedge a b -> Bool

max :: Wedge a b -> Wedge a b -> Wedge a b

min :: Wedge a b -> Wedge a b -> Wedge a b

(Read a, Read b) => Read (Wedge a b)

Defined in Data.Wedge

Methods

readsPrec :: Int -> ReadS (Wedge a b)

readList :: ReadS [Wedge a b]

readPrec :: ReadPrec (Wedge a b)

readListPrec :: ReadPrec [Wedge a b]

(Show a, Show b) => Show (Wedge a b)

Defined in Data.Wedge

Methods

showsPrec :: Int -> Wedge a b -> ShowS

show :: Wedge a b -> String

showList :: [Wedge a b] -> ShowS

Generic (Wedge a b)

Defined in Data.Wedge

Associated Types

type Rep (Wedge a b) :: Type -> Type

Methods

from :: Wedge a b -> Rep (Wedge a b) x

to :: Rep (Wedge a b) x -> Wedge a b

(Semigroup a, Semigroup b) => Semigroup (Wedge a b)

Defined in Data.Wedge

Methods

(<>) :: Wedge a b -> Wedge a b -> Wedge a b

sconcat :: NonEmpty (Wedge a b) -> Wedge a b

stimes :: Integral b0 => b0 -> Wedge a b -> Wedge a b

(Semigroup a, Semigroup b) => Monoid (Wedge a b)

Defined in Data.Wedge

Methods

mempty :: Wedge a b

mappend :: Wedge a b -> Wedge a b -> Wedge a b

mconcat :: [Wedge a b] -> Wedge a b

(Binary a, Binary b) => Binary (Wedge a b)

Defined in Data.Wedge

Methods

put :: Wedge a b -> Put

get :: Get (Wedge a b)

putList :: [Wedge a b] -> Put

(NFData a, NFData b) => NFData (Wedge a b)

Defined in Data.Wedge

Methods

rnf :: Wedge a b -> ()

(Hashable a, Hashable b) => Hashable (Wedge a b)

Defined in Data.Wedge

Methods

hashWithSalt :: Int -> Wedge a b -> Int

hash :: Wedge a b -> Int

type Rep1 (Wedge a :: Type -> Type)

Defined in Data.Wedge

type Rep1 (Wedge a :: Type -> Type) = D1 (MetaData "Wedge" "Data.Wedge" "smash-0.1.1.0-8T61bCPN60yH3RLoZzb2ua" False) (C1 (MetaCons "Nowhere" PrefixI False) (U1 :: Type -> Type) :+: (C1 (MetaCons "Here" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)) :+: C1 (MetaCons "There" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1)))

type Rep (Wedge a b)

Defined in Data.Wedge

type Rep (Wedge a b) = D1 (MetaData "Wedge" "Data.Wedge" "smash-0.1.1.0-8T61bCPN60yH3RLoZzb2ua" False) (C1 (MetaCons "Nowhere" PrefixI False) (U1 :: Type -> Type) :+: (C1 (MetaCons "Here" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)) :+: C1 (MetaCons "There" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 b))))

Combinators

quotWedge :: Either (Maybe a) (Maybe b) -> Wedge a b

Given two possible pointed types, produce a Wedge by considering the left case, the right case, and mapping their Nothing cases to Nowhere. This is a pushout of pointed types `A * - B`.

wedgeLeft :: Maybe a -> Wedge a b

Inject a Maybe value into the Here case of a Wedge, or Nowhere if the empty case is given. This is analogous to the Left constructor for Either.

wedgeRight :: Maybe b -> Wedge a b

Inject a Maybe value into the There case of a Wedge, or Nowhere if the empty case is given. This is analogous to the Right constructor for Either.

fromWedge :: Wedge a b -> Maybe (Either a b)

Convert a 'Wedge a b' into a 'Maybe (Either a b)' value.

toWedge :: Maybe (Either a b) -> Wedge a b

Convert a 'Maybe (Either a b)' value into a Wedge

isHere :: Wedge a b -> Bool

Detect if a Wedge is a Here case.

isThere :: Wedge a b -> Bool

Detect if a Wedge is a There case.

isNowhere :: Wedge a b -> Bool

Detect if a Wedge is a Nowhere empty case.

Eliminators

wedge :: c -> (a -> c) -> (b -> c) -> Wedge a b -> c

Case elimination for the Wedge datatype.

Filtering

heres :: Foldable f => f (Wedge a b) -> [a]

Given a Foldable of Wedges, collect the Here cases, if any.

theres :: Foldable f => f (Wedge a b) -> [b]

Given a Foldable of Wedges, collect the There cases, if any.

filterHeres :: Foldable f => f (Wedge a b) -> [Wedge a b]

Filter the Here cases of a Foldable of Wedges.

filterTheres :: Foldable f => f (Wedge a b) -> [Wedge a b]

Filter the There cases of a Foldable of Wedges.

filterNowheres :: Foldable f => f (Wedge a b) -> [Wedge a b]

Filter the Nowhere cases of a Foldable of Wedges.

Folding

foldHeres :: Foldable f => (a -> m -> m) -> m -> f (Wedge a b) -> m

Fold over the Here cases of a Foldable of Wedges by some accumulating function.

foldTheres :: Foldable f => (b -> m -> m) -> m -> f (Wedge a b) -> m

Fold over the There cases of a Foldable of Wedges by some accumulating function.

gatherWedges :: Wedge [a] [b] -> [Wedge a b]

Given a Wedge of lists, produce a list of wedges by mapping the list of as to Here values, or the list of bs to There values.

Partitioning

partitionWedges :: forall f t a b. (Foldable t, Alternative f) => t (Wedge a b) -> (f a, f b)

Given a Foldable of Wedges, partition it into a tuple of alternatives their parts.

mapWedges :: forall f t a b c. (Alternative f, Traversable t) => (a -> Wedge b c) -> t a -> (f b, f c)

Partition a structure by mapping its contents into Wedges, and folding over '(|)'.

Distributivity

distributeWedge :: Wedge (a, b) c -> (Wedge a c, Wedge b c)

Distribute a Wedge over a product.

codistributeWedge :: Either (Wedge a c) (Wedge b c) -> Wedge (Either a b) c

Codistribute Wedges over a coproduct

Associativity

reassocLR :: Wedge (Wedge a b) c -> Wedge a (Wedge b c)

Re-associate a Wedge of Wedges from left to right.

reassocRL :: Wedge a (Wedge b c) -> Wedge (Wedge a b) c

Re-associate a Wedge of Wedges from left to right.

Symmetry

swapWedge :: Wedge a b -> Wedge b a

Swap the positions of the a's and the b's in a Wedge.