Data.Smash

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

Datatypes

Categorically, the Smash datatype represents a special type of product, a smash product, in the category Hask* of pointed Hask types. The category Hask* consists of Hask types affixed with a dedicated base point - i.e. all objects look like Maybe a. The smash product is a symmetric, monoidal tensor in Hask* that plays nicely with the product, Can, and coproduct, Wedge. Pictorially, these datatypes look like this:

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

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


Smash:


Nada +--------+ (a,b)

The fact that smash products form a closed, symmetric monoidal tensor for Hask* means that we can speak in terms of the language of linear logic for this category. Namely, we can understand how Smash, Wedge, and Can interact. Can and Wedge distribute nicely over each other, and Smash distributes well over Wedge, but is only semi-distributable over Wedge's linear counterpart, which is left out of the api. In this library, we focus on the fragment of this pointed linear logic that makes sense to use, and that will be useful to us as Haskell developers.

data Smash a b

The Smash data type represents A value which has either an empty case, or two values. The result is a type, 'Smash a b', which is isomorphic to Maybe (a,b).

Categorically, the smash product (the quotient of a pointed product by a wedge sum) has interesting properties. It forms a closed symmetric-monoidal tensor in the category Hask* of pointed haskell types (i.e. Maybe values).

Constructors

Nada

Smash a b

Instances

Bitraversable Smash

Defined in Data.Smash

Methods

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

Bifoldable Smash

Defined in Data.Smash

Methods

bifold :: Monoid m => Smash m m -> m

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

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

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

Bifunctor Smash

Defined in Data.Smash

Methods

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

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

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

Eq2 Smash

Defined in Data.Smash

Methods

liftEq2 :: (a -> b -> Bool) -> (c -> d -> Bool) -> Smash a c -> Smash b d -> Bool

Ord2 Smash

Defined in Data.Smash

Methods

liftCompare2 :: (a -> b -> Ordering) -> (c -> d -> Ordering) -> Smash a c -> Smash b d -> Ordering

Read2 Smash

Defined in Data.Smash

Methods

liftReadsPrec2 :: (Int -> ReadS a) -> ReadS [a] -> (Int -> ReadS b) -> ReadS [b] -> Int -> ReadS (Smash a b)

liftReadList2 :: (Int -> ReadS a) -> ReadS [a] -> (Int -> ReadS b) -> ReadS [b] -> ReadS [Smash a b]

liftReadPrec2 :: ReadPrec a -> ReadPrec [a] -> ReadPrec b -> ReadPrec [b] -> ReadPrec (Smash a b)

liftReadListPrec2 :: ReadPrec a -> ReadPrec [a] -> ReadPrec b -> ReadPrec [b] -> ReadPrec [Smash a b]

Show2 Smash

Defined in Data.Smash

Methods

liftShowsPrec2 :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> (Int -> b -> ShowS) -> ([b] -> ShowS) -> Int -> Smash a b -> ShowS

liftShowList2 :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> (Int -> b -> ShowS) -> ([b] -> ShowS) -> [Smash a b] -> ShowS

NFData2 Smash

Defined in Data.Smash

Methods

liftRnf2 :: (a -> ()) -> (b -> ()) -> Smash a b -> ()

Biapplicative Smash

Defined in Data.Smash

Methods

bipure :: a -> b -> Smash a b

(<<*>>) :: Smash (a -> b) (c -> d) -> Smash a c -> Smash b d

biliftA2 :: (a -> b -> c) -> (d -> e -> f) -> Smash a d -> Smash b e -> Smash c f

(*>>) :: Smash a b -> Smash c d -> Smash c d

(<<*) :: Smash a b -> Smash c d -> Smash a b

Hashable2 Smash

Defined in Data.Smash

Methods

liftHashWithSalt2 :: (Int -> a -> Int) -> (Int -> b -> Int) -> Int -> Smash a b -> Int

Monoid a => Monad (Smash a)

Defined in Data.Smash

Methods

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

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

return :: a0 -> Smash a a0

Functor (Smash a)

Defined in Data.Smash

Methods

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

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

Monoid a => Applicative (Smash a)

Defined in Data.Smash

Methods

pure :: a0 -> Smash a a0

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

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

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

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

Eq a => Eq1 (Smash a)

Defined in Data.Smash

Methods

liftEq :: (a0 -> b -> Bool) -> Smash a a0 -> Smash a b -> Bool

Ord a => Ord1 (Smash a)

Defined in Data.Smash

Methods

liftCompare :: (a0 -> b -> Ordering) -> Smash a a0 -> Smash a b -> Ordering

Read a => Read1 (Smash a)

Defined in Data.Smash

Methods

liftReadsPrec :: (Int -> ReadS a0) -> ReadS [a0] -> Int -> ReadS (Smash a a0)

liftReadList :: (Int -> ReadS a0) -> ReadS [a0] -> ReadS [Smash a a0]

liftReadPrec :: ReadPrec a0 -> ReadPrec [a0] -> ReadPrec (Smash a a0)

liftReadListPrec :: ReadPrec a0 -> ReadPrec [a0] -> ReadPrec [Smash a a0]

Show a => Show1 (Smash a)

Defined in Data.Smash

Methods

liftShowsPrec :: (Int -> a0 -> ShowS) -> ([a0] -> ShowS) -> Int -> Smash a a0 -> ShowS

liftShowList :: (Int -> a0 -> ShowS) -> ([a0] -> ShowS) -> [Smash a a0] -> ShowS

Monoid a => MonadZip (Smash a)

Defined in Data.Smash

Methods

mzip :: Smash a a0 -> Smash a b -> Smash a (a0, b)

mzipWith :: (a0 -> b -> c) -> Smash a a0 -> Smash a b -> Smash a c

munzip :: Smash a (a0, b) -> (Smash a a0, Smash a b)

Monoid a => Alternative (Smash a)

Defined in Data.Smash

Methods

empty :: Smash a a0

(<|>) :: Smash a a0 -> Smash a a0 -> Smash a a0

some :: Smash a a0 -> Smash a [a0]

many :: Smash a a0 -> Smash a [a0]

Monoid a => MonadPlus (Smash a)

Defined in Data.Smash

Methods

mzero :: Smash a a0

mplus :: Smash a a0 -> Smash a a0 -> Smash a a0

NFData a => NFData1 (Smash a)

Defined in Data.Smash

Methods

liftRnf :: (a0 -> ()) -> Smash a a0 -> ()

Hashable a => Hashable1 (Smash a)

Defined in Data.Smash

Methods

liftHashWithSalt :: (Int -> a0 -> Int) -> Int -> Smash a a0 -> Int

Generic1 (Smash a :: Type -> Type)

Defined in Data.Smash

Associated Types

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

Methods

from1 :: forall (a0 :: k). Smash a a0 -> Rep1 (Smash a) a0

to1 :: forall (a0 :: k). Rep1 (Smash a) a0 -> Smash a a0

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

Defined in Data.Smash

Methods

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

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

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

Defined in Data.Smash

Methods

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

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

toConstr :: Smash a b -> Constr

dataTypeOf :: Smash a b -> DataType

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

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

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

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

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

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

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

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

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

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

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

Defined in Data.Smash

Methods

compare :: Smash a b -> Smash a b -> Ordering

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

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

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

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

max :: Smash a b -> Smash a b -> Smash a b

min :: Smash a b -> Smash a b -> Smash a b

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

Defined in Data.Smash

Methods

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

readList :: ReadS [Smash a b]

readPrec :: ReadPrec (Smash a b)

readListPrec :: ReadPrec [Smash a b]

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

Defined in Data.Smash

Methods

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

show :: Smash a b -> String

showList :: [Smash a b] -> ShowS

Generic (Smash a b)

Defined in Data.Smash

Associated Types

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

Methods

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

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

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

Defined in Data.Smash

Methods

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

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

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

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

Defined in Data.Smash

Methods

mempty :: Smash a b

mappend :: Smash a b -> Smash a b -> Smash a b

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

(Lift a, Lift b) => Lift (Smash a b)

Defined in Data.Smash

Methods

lift :: Smash a b -> Q Exp

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

Defined in Data.Smash

Methods

put :: Smash a b -> Put

get :: Get (Smash a b)

putList :: [Smash a b] -> Put

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

Defined in Data.Smash

Methods

rnf :: Smash a b -> ()

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

Defined in Data.Smash

Methods

hashWithSalt :: Int -> Smash a b -> Int

hash :: Smash a b -> Int

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

Defined in Data.Smash

type Rep1 (Smash a :: Type -> Type) = D1 ('MetaData "Smash" "Data.Smash" "smash-0.1.2-inplace" 'False) (C1 ('MetaCons "Nada" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "Smash" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1))

type Rep (Smash a b)

Defined in Data.Smash

type Rep (Smash a b) = D1 ('MetaData "Smash" "Data.Smash" "smash-0.1.2-inplace" 'False) (C1 ('MetaCons "Nada" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "Smash" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 b)))

Type synonyms

type (⨳) a b = Smash a b

A type operator synonym for Smash

Combinators

toSmash :: Maybe (a, b) -> Smash a b

Convert a Maybe value into a Smash value

fromSmash :: Smash a b -> Maybe (a, b)

Convert a Smash value into a Maybe value

smashFst :: Smash a b -> Maybe a

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

smashSnd :: Smash a b -> Maybe b

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

quotSmash :: Can a b -> Smash a b

Smash product of pointed type modulo its wedge

hulkSmash :: a -> b -> Wedge a b -> Smash a b

Take the smash product of a wedge and two default values to place in either the left or right side of the final product

isSmash :: Smash a b -> Bool

Detect whether a Smash value is not empty

isNada :: Smash a b -> Bool

Detect whether a Smash value is empty

smashDiag :: Maybe a -> Smash a a

Create a smash product of self-similar values from a pointed object.

This is the diagonal morphism in Hask*.

smashDiag' :: a -> Smash a a

See: smashDiag. This is always a Smash value.

Eliminators

smash :: c -> (a -> b -> c) -> Smash a b -> c

Case elimination for the Smash datatype

Filtering

smashes :: Foldable f => f (Smash a b) -> [(a, b)]

Given a Foldable of Smashs, collect the values of the Smash cases, if any.

filterNadas :: Foldable f => f (Smash a b) -> [Smash a b]

Filter the Nada cases of a Foldable of Smash values.

Folding and Unfolding

foldSmashes :: Foldable f => (a -> b -> m -> m) -> m -> f (Smash a b) -> m

Fold over the Smash case of a Foldable of Smash products by some accumulating function.

gatherSmashes :: Smash [a] [b] -> [Smash a b]

Gather a Smash product of two lists and product a list of Smash values, mapping the Nada case to the empty list and zipping the two lists together with the Smash constructor otherwise.

unfoldr :: Alternative f => (b -> Smash a b) -> b -> f a

Unfold from right to left into a smash product

unfoldrM :: (Monad m, Alternative f) => (b -> m (Smash a b)) -> b -> m (f a)

Unfold from right to left into a monadic computation over a smash product

iterateUntil :: Alternative f => (b -> Smash a b) -> b -> f a

Iterate on a seed, accumulating a result. See iterateUntilM for more details.

iterateUntilM :: Monad m => Alternative f => (b -> m (Smash a b)) -> b -> m (f a)

Iterate on a seed, which may result in one of two scenarios:

1. The function yields a Nada value, which terminates the iteration.
2. The function yields a Smash value.

accumUntil :: Alternative f => Monoid b => (b -> Smash a b) -> f a

Iterate on a seed, accumulating values and monoidally updating the seed with each update.

accumUntilM :: Monad m => Alternative f => Monoid b => (b -> m (Smash a b)) -> m (f a)

Iterate on a seed, accumulating values and monoidally updating a seed within a monad.

Partitioning

partitionSmashes :: Foldable t => Alternative f => t (Smash a b) -> (f a, f b)

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

mapSmashes :: Alternative f => Traversable t => (a -> Smash b c) -> t a -> (f b, f c)

Partition a structure by mapping its contents into Smashs, and folding over (<|>).

Currying & Uncurrying

smashCurry :: (Smash a b -> Maybe c) -> Maybe a -> Maybe b -> Maybe c

Curry a map from a smash product to a pointed type. This is analogous to curry for ('->').

smashUncurry :: (Maybe a -> Maybe b -> Maybe c) -> Smash a b -> Maybe c

Uncurry a map of pointed types to a map of a smash product to a pointed type. This is analogous to uncurry for ('->').

Distributivity

distributeSmash :: Smash (Wedge a b) c -> Wedge (Smash a c) (Smash b c)

A smash product of wedges is a wedge of smash products. Smash products distribute over coproducts (Wedges) in pointed Hask

undistributeSmash :: Wedge (Smash a c) (Smash b c) -> Smash (Wedge a b) c

A wedge of smash products is a smash product of wedges. Smash products distribute over coproducts (Wedges) in pointed Hask

pairSmash :: Smash (a, b) c -> (Smash a c, Smash b c)

Distribute a Smash of a pair into a pair of Smashs

unpairSmash :: (Smash a c, Smash b c) -> Smash (a, b) c

Distribute a Smash of a pair into a pair of Smashs

pairSmashCan :: Smash (Can a b) c -> Can (Smash a c) (Smash b c)

Distribute a Smash of a Can into a Can of Smashs

unpairSmashCan :: Can (Smash a c) (Smash b c) -> Smash (Can a b) c

Undistribute a Can of Smashs into a Smash of Cans.

Associativity

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

Reassociate a Smash product from left to right.

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

Reassociate a Smash product from right to left.

Symmetry

swapSmash :: Smash a b -> Smash b a

Swap the positions of values in a Smash a b to form a Smash b a.