Math.Combinat.TypeLevel

Type-level hackery.

This module is used for groups whose parameters are encoded as type-level natural numbers, for example finite cyclic groups, free groups, symmetric groups and braid groups.

Proxy

data Proxy (t :: k)

Proxy is a type that holds no data, but has a phantom parameter of arbitrary type (or even kind). Its use is to provide type information, even though there is no value available of that type (or it may be too costly to create one).

Historically, Proxy :: Proxy a is a safer alternative to the undefined :: a idiom.

>>> Proxy :: Proxy (Void, Int -> Int)
Proxy

Proxy can even hold types of higher kinds,

>>> Proxy :: Proxy Either
Proxy

>>> Proxy :: Proxy Functor
Proxy

>>> Proxy :: Proxy complicatedStructure
Proxy

Constructors

Proxy

Instances

Generic1 (Proxy :: k -> Type)

Defined in GHC.Generics

Associated Types

type Rep1 Proxy :: k -> Type

Methods

from1 :: forall (a :: k0). Proxy a -> Rep1 Proxy a

to1 :: forall (a :: k0). Rep1 Proxy a -> Proxy a

Foldable (Proxy :: TYPE LiftedRep -> Type)

Since: base-4.7.0.0

Defined in Data.Foldable

Methods

fold :: Monoid m => Proxy m -> m

foldMap :: Monoid m => (a -> m) -> Proxy a -> m

foldMap' :: Monoid m => (a -> m) -> Proxy a -> m

foldr :: (a -> b -> b) -> b -> Proxy a -> b

foldr' :: (a -> b -> b) -> b -> Proxy a -> b

foldl :: (b -> a -> b) -> b -> Proxy a -> b

foldl' :: (b -> a -> b) -> b -> Proxy a -> b

foldr1 :: (a -> a -> a) -> Proxy a -> a

foldl1 :: (a -> a -> a) -> Proxy a -> a

toList :: Proxy a -> [a]

null :: Proxy a -> Bool

length :: Proxy a -> Int

elem :: Eq a => a -> Proxy a -> Bool

maximum :: Ord a => Proxy a -> a

minimum :: Ord a => Proxy a -> a

sum :: Num a => Proxy a -> a

product :: Num a => Proxy a -> a

Eq1 (Proxy :: Type -> Type)

Since: base-4.9.0.0

Defined in Data.Functor.Classes

Methods

liftEq :: (a -> b -> Bool) -> Proxy a -> Proxy b -> Bool

Ord1 (Proxy :: Type -> Type)

Since: base-4.9.0.0

Defined in Data.Functor.Classes

Methods

liftCompare :: (a -> b -> Ordering) -> Proxy a -> Proxy b -> Ordering

Read1 (Proxy :: Type -> Type)

Since: base-4.9.0.0

Defined in Data.Functor.Classes

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Proxy a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Proxy a]

liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Proxy a)

liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Proxy a]

Show1 (Proxy :: TYPE LiftedRep -> Type)

Since: base-4.9.0.0

Defined in Data.Functor.Classes

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Proxy a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Proxy a] -> ShowS

Traversable (Proxy :: Type -> Type)

Since: base-4.7.0.0

Defined in Data.Traversable

Methods

traverse :: Applicative f => (a -> f b) -> Proxy a -> f (Proxy b)

sequenceA :: Applicative f => Proxy (f a) -> f (Proxy a)

mapM :: Monad m => (a -> m b) -> Proxy a -> m (Proxy b)

sequence :: Monad m => Proxy (m a) -> m (Proxy a)

Alternative (Proxy :: Type -> Type)

Since: base-4.9.0.0

Defined in Data.Proxy

Methods

empty :: Proxy a

(<|>) :: Proxy a -> Proxy a -> Proxy a

some :: Proxy a -> Proxy [a]

many :: Proxy a -> Proxy [a]

Applicative (Proxy :: Type -> Type)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

pure :: a -> Proxy a

(<*>) :: Proxy (a -> b) -> Proxy a -> Proxy b

liftA2 :: (a -> b -> c) -> Proxy a -> Proxy b -> Proxy c

(*>) :: Proxy a -> Proxy b -> Proxy b

(<*) :: Proxy a -> Proxy b -> Proxy a

Functor (Proxy :: Type -> Type)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

fmap :: (a -> b) -> Proxy a -> Proxy b

(<$) :: a -> Proxy b -> Proxy a

Monad (Proxy :: Type -> Type)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

(>>=) :: Proxy a -> (a -> Proxy b) -> Proxy b

(>>) :: Proxy a -> Proxy b -> Proxy b

return :: a -> Proxy a

MonadPlus (Proxy :: Type -> Type)

Since: base-4.9.0.0

Defined in Data.Proxy

Methods

mzero :: Proxy a

mplus :: Proxy a -> Proxy a -> Proxy a

Monoid (Proxy s)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

mempty :: Proxy s

mappend :: Proxy s -> Proxy s -> Proxy s

mconcat :: [Proxy s] -> Proxy s

Semigroup (Proxy s)

Since: base-4.9.0.0

Defined in Data.Proxy

Methods

(<>) :: Proxy s -> Proxy s -> Proxy s

sconcat :: NonEmpty (Proxy s) -> Proxy s

stimes :: Integral b => b -> Proxy s -> Proxy s

Bounded (Proxy t)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

minBound :: Proxy t

maxBound :: Proxy t

Enum (Proxy s)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

succ :: Proxy s -> Proxy s

pred :: Proxy s -> Proxy s

toEnum :: Int -> Proxy s

fromEnum :: Proxy s -> Int

enumFrom :: Proxy s -> [Proxy s]

enumFromThen :: Proxy s -> Proxy s -> [Proxy s]

enumFromTo :: Proxy s -> Proxy s -> [Proxy s]

enumFromThenTo :: Proxy s -> Proxy s -> Proxy s -> [Proxy s]

Generic (Proxy t)

Defined in GHC.Generics

Associated Types

type Rep (Proxy t) :: Type -> Type

Methods

from :: Proxy t -> Rep (Proxy t) x

to :: Rep (Proxy t) x -> Proxy t

Ix (Proxy s)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

range :: (Proxy s, Proxy s) -> [Proxy s]

index :: (Proxy s, Proxy s) -> Proxy s -> Int

unsafeIndex :: (Proxy s, Proxy s) -> Proxy s -> Int

inRange :: (Proxy s, Proxy s) -> Proxy s -> Bool

rangeSize :: (Proxy s, Proxy s) -> Int

unsafeRangeSize :: (Proxy s, Proxy s) -> Int

Read (Proxy t)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

readsPrec :: Int -> ReadS (Proxy t)

readList :: ReadS [Proxy t]

readPrec :: ReadPrec (Proxy t)

readListPrec :: ReadPrec [Proxy t]

Show (Proxy s)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

showsPrec :: Int -> Proxy s -> ShowS

show :: Proxy s -> String

showList :: [Proxy s] -> ShowS

Eq (Proxy s)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

(==) :: Proxy s -> Proxy s -> Bool

(/=) :: Proxy s -> Proxy s -> Bool

Ord (Proxy s)

Since: base-4.7.0.0

Defined in Data.Proxy

Methods

compare :: Proxy s -> Proxy s -> Ordering

(<) :: Proxy s -> Proxy s -> Bool

(<=) :: Proxy s -> Proxy s -> Bool

(>) :: Proxy s -> Proxy s -> Bool

(>=) :: Proxy s -> Proxy s -> Bool

max :: Proxy s -> Proxy s -> Proxy s

min :: Proxy s -> Proxy s -> Proxy s

type Rep1 (Proxy :: k -> Type)

Since: base-4.6.0.0

Defined in GHC.Generics

type Rep1 (Proxy :: k -> Type) = D1 ('MetaData "Proxy" "Data.Proxy" "base" 'False) (C1 ('MetaCons "Proxy" 'PrefixI 'False) (U1 :: k -> Type))

type Rep (Proxy t)

Since: base-4.6.0.0

Defined in GHC.Generics

type Rep (Proxy t) = D1 ('MetaData "Proxy" "Data.Proxy" "base" 'False) (C1 ('MetaCons "Proxy" 'PrefixI 'False) (U1 :: Type -> Type))

proxyUndef :: Proxy a -> a

proxyOf :: a -> Proxy a

proxyOf1 :: f a -> Proxy a

proxyOf2 :: g (f a) -> Proxy a

asProxyTypeOf :: a -> proxy a -> a

asProxyTypeOf is a type-restricted version of const. It is usually used as an infix operator, and its typing forces its first argument (which is usually overloaded) to have the same type as the tag of the second.

>>> import Data.Word
>>> :type asProxyTypeOf 123 (Proxy :: Proxy Word8)
asProxyTypeOf 123 (Proxy :: Proxy Word8) :: Word8

Note the lower-case proxy in the definition. This allows any type constructor with just one argument to be passed to the function, for example we could also write

>>> import Data.Word
>>> :type asProxyTypeOf 123 (Just (undefined :: Word8))
asProxyTypeOf 123 (Just (undefined :: Word8)) :: Word8

asProxyTypeOf1 :: f a -> Proxy a -> f a

Type-level naturals as type arguments

typeArg :: KnownNat n => f (n :: Nat) -> Integer

iTypeArg :: KnownNat n => f (n :: Nat) -> Int

Hiding the type parameter

data Some f

Hide the type parameter of a functor. Example: Some Braid

Constructors

forall n.KnownNat n => Some (f n)

withSome :: Some f -> (forall n. KnownNat n => f n -> a) -> a

Uses the value inside a Some

withSomeM :: Monad m => Some f -> (forall n. KnownNat n => f n -> m a) -> m a

Monadic version of withSome

selectSome :: Integral int => (forall n. KnownNat n => f n) -> int -> Some f

Given a polymorphic value, we select at run time the one specified by the second argument

selectSomeM :: forall m f int. (Integral int, Monad m) => (forall n. KnownNat n => m (f n)) -> int -> m (Some f)

Monadic version of selectSome

withSelected :: Integral int => (forall n. KnownNat n => f n -> a) -> (forall n. KnownNat n => f n) -> int -> a

Combination of selectSome and withSome: we make a temporary structure of the given size, but we immediately consume it.

withSelectedM :: forall m f int a. (Integral int, Monad m) => (forall n. KnownNat n => f n -> a) -> (forall n. KnownNat n => m (f n)) -> int -> m a

(Half-)monadic version of withSelected