Copyright	(c) 2020 Emily Pillmore
License	BSD-3-Clause
Maintainer	Emily Pillmore <emilypi@cohomolo.gy>
Stability	Experimental
Portability	portable
Safe Haskell	Safe
Language	Haskell2010

Data.Can

Contents

Datatypes
Combinators
- Eliminators
Folding
Filtering
Curry & Uncurry
Partitioning
Distributivity
Associativity
Symmetry

Description

This module contains the definition for the Can datatype. In practice, this type is isomorphic to Maybe These - the type with two possibly non-exclusive values and an empty case.

Synopsis

data Can a b
- = Non
- | One a
- | Eno b
- | Two a b
canFst :: Can a b -> Maybe a
canSnd :: Can a b -> Maybe b
isOne :: Can a b -> Bool
isEno :: Can a b -> Bool
isTwo :: Can a b -> Bool
isNon :: Can a b -> Bool
can :: c -> (a -> c) -> (b -> c) -> (a -> b -> c) -> Can a b -> c
foldOnes :: Foldable f => (a -> m -> m) -> m -> f (Can a b) -> m
foldEnos :: Foldable f => (b -> m -> m) -> m -> f (Can a b) -> m
foldTwos :: Foldable f => (a -> b -> m -> m) -> m -> f (Can a b) -> m
gatherCans :: Can [a] [b] -> [Can a b]
ones :: Foldable f => f (Can a b) -> [a]
enos :: Foldable f => f (Can a b) -> [b]
twos :: Foldable f => f (Can a b) -> [(a, b)]
filterOnes :: Foldable f => f (Can a b) -> [Can a b]
filterEnos :: Foldable f => f (Can a b) -> [Can a b]
filterTwos :: Foldable f => f (Can a b) -> [Can a b]
filterNons :: Foldable f => f (Can a b) -> [Can a b]
canCurry :: (Can a b -> Maybe c) -> Maybe a -> Maybe b -> Maybe c
canUncurry :: (Maybe a -> Maybe b -> Maybe c) -> Can a b -> Maybe c
partitionCans :: forall f t a b. (Foldable t, Alternative f) => t (Can a b) -> (f a, f b)
partitionAll :: Foldable f => f (Can a b) -> ([a], [b], [(a, b)])
partitionEithers :: Foldable f => f (Either a b) -> Can [a] [b]
mapCans :: forall f t a b c. (Alternative f, Traversable t) => (a -> Can b c) -> t a -> (f b, f c)
distributeCan :: Can (a, b) c -> (Can a c, Can b c)
codistributeCan :: Either (Can a c) (Can b c) -> Can (Either a b) c
reassocLR :: Can (Can a b) c -> Can a (Can b c)
reassocRL :: Can a (Can b c) -> Can (Can a b) c
swapCan :: Can a b -> Can b a

Datatypes

Categorically, the Can datatype represents the pointed product in the category Hask* of pointed Hask types. The category Hask* consists of Hask types affixed with a dedicated base point of an object along with the object. In Hask*, this is equivalent to `1 + a`, or 'Maybe a' in Hask. Hence, the product is `(1 + a) * (1 + b) ~ 1 + a + b + a*b`, or `Maybe (Either (Either a b) (a,b))` in Hask. Pictorially, you can visualize this as:

Can:
        a
        |
Non +---+---+ (a,b)
        |
        b

The fact that we can think about Can as your average product gives us some reasoning power about how this thing will be able to interact with the coproduct in Hask*, called Wedge. Namely, facts about currying 'Can a b -> c ~ a -> b -> c' and distributivity over Wedge along with other facts about its associativity, commutativity, and any other analogy with `(,)` that you can think of.

data Can a b Source #

The Can data type represents values with two non-exclusive possibilities, as well as an empty case. This is a product of pointed types - i.e. of Maybe values. The result is a type, 'Can a b', which is isomorphic to 'Maybe (These a b)'.

Constructors

Non
One a
Eno b
Two a b

Instances

Bitraversable Can Source #
Instance details Defined in Data.Can Methods bitraverse :: Applicative f => (a -> f c) -> (b -> f d) -> Can a b -> f (Can c d) #
Bifoldable Can Source #
Instance details Defined in Data.Can Methods bifold :: Monoid m => Can m m -> m # bifoldMap :: Monoid m => (a -> m) -> (b -> m) -> Can a b -> m # bifoldr :: (a -> c -> c) -> (b -> c -> c) -> c -> Can a b -> c # bifoldl :: (c -> a -> c) -> (c -> b -> c) -> c -> Can a b -> c #
Bifunctor Can Source #
Instance details Defined in Data.Can Methods bimap :: (a -> b) -> (c -> d) -> Can a c -> Can b d # first :: (a -> b) -> Can a c -> Can b c # second :: (b -> c) -> Can a b -> Can a c #
Semigroup a => Monad (Can a) Source #
Instance details Defined in Data.Can Methods (>>=) :: Can a a0 -> (a0 -> Can a b) -> Can a b # (>>) :: Can a a0 -> Can a b -> Can a b # return :: a0 -> Can a a0 # fail :: String -> Can a a0 #
Functor (Can a) Source #
Instance details Defined in Data.Can Methods fmap :: (a0 -> b) -> Can a a0 -> Can a b # (<$) :: a0 -> Can a b -> Can a a0 #
Semigroup a => Applicative (Can a) Source #
Instance details Defined in Data.Can Methods pure :: a0 -> Can a a0 # (<>) :: Can a (a0 -> b) -> Can a a0 -> Can a b # liftA2 :: (a0 -> b -> c) -> Can a a0 -> Can a b -> Can a c # (>) :: Can a a0 -> Can a b -> Can a b # (<*) :: Can a a0 -> Can a b -> Can a a0 #
Foldable (Can a) Source #
Instance details Defined in Data.Can Methods fold :: Monoid m => Can a m -> m # foldMap :: Monoid m => (a0 -> m) -> Can a a0 -> m # foldr :: (a0 -> b -> b) -> b -> Can a a0 -> b # foldr' :: (a0 -> b -> b) -> b -> Can a a0 -> b # foldl :: (b -> a0 -> b) -> b -> Can a a0 -> b # foldl' :: (b -> a0 -> b) -> b -> Can a a0 -> b # foldr1 :: (a0 -> a0 -> a0) -> Can a a0 -> a0 # foldl1 :: (a0 -> a0 -> a0) -> Can a a0 -> a0 # toList :: Can a a0 -> [a0] # null :: Can a a0 -> Bool # length :: Can a a0 -> Int # elem :: Eq a0 => a0 -> Can a a0 -> Bool # maximum :: Ord a0 => Can a a0 -> a0 # minimum :: Ord a0 => Can a a0 -> a0 # sum :: Num a0 => Can a a0 -> a0 # product :: Num a0 => Can a a0 -> a0 #
Traversable (Can a) Source #
Instance details Defined in Data.Can Methods traverse :: Applicative f => (a0 -> f b) -> Can a a0 -> f (Can a b) # sequenceA :: Applicative f => Can a (f a0) -> f (Can a a0) # mapM :: Monad m => (a0 -> m b) -> Can a a0 -> m (Can a b) # sequence :: Monad m => Can a (m a0) -> m (Can a a0) #
Generic1 (Can a :: Type -> Type) Source #
Instance details Defined in Data.Can Associated Types type Rep1 (Can a) :: k -> Type # Methods from1 :: Can a a0 -> Rep1 (Can a) a0 # to1 :: Rep1 (Can a) a0 -> Can a a0 #
(Eq a, Eq b) => Eq (Can a b) Source #
Instance details Defined in Data.Can Methods (==) :: Can a b -> Can a b -> Bool # (/=) :: Can a b -> Can a b -> Bool #
(Data a, Data b) => Data (Can a b) Source #
Instance details Defined in Data.Can Methods gfoldl :: (forall d b0. Data d => c (d -> b0) -> d -> c b0) -> (forall g. g -> c g) -> Can a b -> c (Can a b) # gunfold :: (forall b0 r. Data b0 => c (b0 -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Can a b) # toConstr :: Can a b -> Constr # dataTypeOf :: Can a b -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Can a b)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Can a b)) # gmapT :: (forall b0. Data b0 => b0 -> b0) -> Can a b -> Can a b # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Can a b -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Can a b -> r # gmapQ :: (forall d. Data d => d -> u) -> Can a b -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Can a b -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Can a b -> m (Can a b) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Can a b -> m (Can a b) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Can a b -> m (Can a b) #
(Ord a, Ord b) => Ord (Can a b) Source #
Instance details Defined in Data.Can Methods compare :: Can a b -> Can a b -> Ordering # (<) :: Can a b -> Can a b -> Bool # (<=) :: Can a b -> Can a b -> Bool # (>) :: Can a b -> Can a b -> Bool # (>=) :: Can a b -> Can a b -> Bool # max :: Can a b -> Can a b -> Can a b # min :: Can a b -> Can a b -> Can a b #
(Read a, Read b) => Read (Can a b) Source #
Instance details Defined in Data.Can Methods readsPrec :: Int -> ReadS (Can a b) # readList :: ReadS [Can a b] # readPrec :: ReadPrec (Can a b) # readListPrec :: ReadPrec [Can a b] #
(Show a, Show b) => Show (Can a b) Source #
Instance details Defined in Data.Can Methods showsPrec :: Int -> Can a b -> ShowS # show :: Can a b -> String # showList :: [Can a b] -> ShowS #
Generic (Can a b) Source #
Instance details Defined in Data.Can Associated Types type Rep (Can a b) :: Type -> Type # Methods from :: Can a b -> Rep (Can a b) x # to :: Rep (Can a b) x -> Can a b #
(Semigroup a, Semigroup b) => Semigroup (Can a b) Source #
Instance details Defined in Data.Can Methods (<>) :: Can a b -> Can a b -> Can a b # sconcat :: NonEmpty (Can a b) -> Can a b # stimes :: Integral b0 => b0 -> Can a b -> Can a b #
(Semigroup a, Semigroup b) => Monoid (Can a b) Source #
Instance details Defined in Data.Can Methods mempty :: Can a b # mappend :: Can a b -> Can a b -> Can a b # mconcat :: [Can a b] -> Can a b #
(Hashable a, Hashable b) => Hashable (Can a b) Source #
Instance details Defined in Data.Can Methods hashWithSalt :: Int -> Can a b -> Int hash :: Can a b -> Int
type Rep1 (Can a :: Type -> Type) Source #
Instance details Defined in Data.Can type Rep1 (Can a :: Type -> Type) = D1 (MetaData "Can" "Data.Can" "smash-0.1.0.0-inplace" False) ((C1 (MetaCons "Non" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "One" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a))) :+: (C1 (MetaCons "Eno" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1) :+: C1 (MetaCons "Two" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1)))
type Rep (Can a b) Source #
Instance details Defined in Data.Can type Rep (Can a b) = D1 (MetaData "Can" "Data.Can" "smash-0.1.0.0-inplace" False) ((C1 (MetaCons "Non" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "One" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a))) :+: (C1 (MetaCons "Eno" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 b)) :+: C1 (MetaCons "Two" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 b))))

Combinators

canFst :: Can a b -> Maybe a Source #

Project the left value of a Can datatype. This is analogous to fst for '(,)'.

canSnd :: Can a b -> Maybe b Source #

Project the right value of a Can datatype. This is analogous to snd for '(,)'.

isOne :: Can a b -> Bool Source #

Detect if a Can is a One case.

isEno :: Can a b -> Bool Source #

Detect if a Can is a Eno case.

isTwo :: Can a b -> Bool Source #

Detect if a Can is a Two case.

isNon :: Can a b -> Bool Source #

Detect if a Can is a Non case.

Eliminators

can Source #

Arguments

:: c	default value to supply for the `Non` case
-> (a -> c)	eliminator for the `One` case
-> (b -> c)	eliminator for the `Eno` case
-> (a -> b -> c)	eliminator for the `Two` case
-> Can a b
-> c

Case elimination for the Can datatype

Folding

foldOnes :: Foldable f => (a -> m -> m) -> m -> f (Can a b) -> m Source #

Fold over the One cases of a Foldable of Cans by some accumulating function.

foldEnos :: Foldable f => (b -> m -> m) -> m -> f (Can a b) -> m Source #

Fold over the Eno cases of a Foldable of Cans by some accumulating function.

foldTwos :: Foldable f => (a -> b -> m -> m) -> m -> f (Can a b) -> m Source #

Fold over the Two cases of a Foldable of Cans by some accumulating function.

gatherCans :: Can [a] [b] -> [Can a b] Source #

Gather a Can of two lists and produce a list of Can values, mapping the Non case to the empty list, One' case to a list of Ones, the Eno case to a list of Enos, or zipping Two along both lists.

Filtering

ones :: Foldable f => f (Can a b) -> [a] Source #

Given a Foldable of Cans, collect the values of the One cases, if any.

enos :: Foldable f => f (Can a b) -> [b] Source #

Given a Foldable of Cans, collect the values of the Eno cases, if any.

twos :: Foldable f => f (Can a b) -> [(a, b)] Source #

Given a Foldable of Cans, collect the values of the Two cases, if any.

filterOnes :: Foldable f => f (Can a b) -> [Can a b] Source #

Filter the One cases of a Foldable of Can values.

filterEnos :: Foldable f => f (Can a b) -> [Can a b] Source #

Filter the Eno cases of a Foldable of Can values.

filterTwos :: Foldable f => f (Can a b) -> [Can a b] Source #

Filter the Two cases of a Foldable of Can values.

filterNons :: Foldable f => f (Can a b) -> [Can a b] Source #

Filter the Non cases of a Foldable of Can values.

Curry & Uncurry

canCurry :: (Can a b -> Maybe c) -> Maybe a -> Maybe b -> Maybe c Source #

Curry a function from a Can to a Maybe value, resulting in a function of curried Maybe values. This is analogous to currying for '(->)'.

canUncurry :: (Maybe a -> Maybe b -> Maybe c) -> Can a b -> Maybe c Source #

Uncurry a function from a Can to a Maybe value, resulting in a function of curried Maybe values. This is analogous to uncurrying for '(->)'.

Partitioning

partitionCans :: forall f t a b. (Foldable t, Alternative f) => t (Can a b) -> (f a, f b) Source #

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

partitionAll :: Foldable f => f (Can a b) -> ([a], [b], [(a, b)]) Source #

Partition a list of Can values into a triple of lists of all of their constituent parts

partitionEithers :: Foldable f => f (Either a b) -> Can [a] [b] Source #

Partition a list of Either values, separating them into a Can value of lists of left and right values, or Non in the case of an empty list.

mapCans :: forall f t a b c. (Alternative f, Traversable t) => (a -> Can b c) -> t a -> (f b, f c) Source #

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

Distributivity

distributeCan :: Can (a, b) c -> (Can a c, Can b c) Source #

Distribute a Can value over a product.

codistributeCan :: Either (Can a c) (Can b c) -> Can (Either a b) c Source #

Codistribute a coproduct over a Can value.

Associativity

reassocLR :: Can (Can a b) c -> Can a (Can b c) Source #

Re-associate a Can of cans from left to right.

reassocRL :: Can a (Can b c) -> Can (Can a b) c Source #

Re-associate a Can of cans from right to left.

Symmetry

swapCan :: Can a b -> Can b a Source #

Swap the positions of values in a Can.