universe-1.1: A class for finite and recursively enumerable types.

Safe HaskellSafe
LanguageHaskell2010

Data.Universe

Description

A convenience module that imports the submodules Instances.Base, Instances.Containers, Instances.Extended, and Instances.Trans to provide instances of Universe and Finite for a wide variety of types.

Synopsis

Documentation

class Universe a where #

Creating an instance of this class is a declaration that your type is recursively enumerable (and that universe is that enumeration). In particular, you promise that any finite inhabitant has a finite index in universe, and that no inhabitant appears at two different finite indices.

Well-behaved instance should produce elements lazily.

Laws:

elem x universe                    -- any inhabitant has a finite index
let pfx = take n universe          -- any finite prefix of universe has unique elements
in length pfx = length (nub pfx)

Minimal complete definition

Nothing

Methods

universe :: [a] #

Instances
Universe Bool 
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universe :: [Bool] #

Universe Char 
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universe :: [Char] #

Universe Int 
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universe :: [Int] #

Universe Int8 
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universe :: [Int8] #

Universe Int16 
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universe :: [Int16] #

Universe Int32 
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universe :: [Int32] #

Universe Int64 
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universe :: [Int64] #

Universe Integer 
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universe :: [Integer] #

Universe Natural 
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universe :: [Natural] #

Universe Ordering 
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universe :: [Ordering] #

Universe Word 
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universe :: [Word] #

Universe Word8 
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universe :: [Word8] #

Universe Word16 
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universe :: [Word16] #

Universe Word32 
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universe :: [Word32] #

Universe Word64 
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universe :: [Word64] #

Universe () 
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universe :: [()] #

Universe Void 
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universe :: [Void] #

Universe All 
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universe :: [All] #

Universe Any 
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universe :: [Any] #

Universe a => Universe [a] 
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universe :: [[a]] #

Universe a => Universe (Maybe a) 
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universe :: [Maybe a] #

RationalUniverse a => Universe (Ratio a) 
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universe :: [Ratio a] #

Universe a => Universe (Min a) 
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universe :: [Min a] #

Universe a => Universe (Max a) 
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universe :: [Max a] #

Universe a => Universe (First a) 
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universe :: [First a] #

Universe a => Universe (Last a) 
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universe :: [Last a] #

Universe a => Universe (Identity a) 
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universe :: [Identity a] #

Universe a => Universe (First a) 
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universe :: [First a] #

Universe a => Universe (Last a) 
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universe :: [Last a] #

Universe a => Universe (Dual a) 
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universe :: [Dual a] #

Universe a => Universe (Sum a) 
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universe :: [Sum a] #

Universe a => Universe (Product a) 
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universe :: [Product a] #

Universe a => Universe (NonEmpty a) 
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universe :: [NonEmpty a] #

(Ord a, Universe a) => Universe (Set a) 
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universe :: [Set a] #

(Finite a, Ord a, Universe b) => Universe (a -> b) 
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universe :: [a -> b] #

(Universe a, Universe b) => Universe (Either a b) 
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universe :: [Either a b] #

(Universe a, Universe b) => Universe (a, b) 
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universe :: [(a, b)] #

Universe (Proxy a) 
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universe :: [Proxy a] #

(Ord k, Finite k, Universe v) => Universe (Map k v) 
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universe :: [Map k v] #

(Universe a, Universe b, Universe c) => Universe (a, b, c) 
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universe :: [(a, b, c)] #

Universe a => Universe (Const a b) 
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universe :: [Const a b] #

Universe (f a) => Universe (IdentityT f a) 
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universe :: [IdentityT f a] #

Universe a => Universe (Tagged b a) 
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universe :: [Tagged b a] #

(Universe a, Universe b, Universe c, Universe d) => Universe (a, b, c, d) 
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universe :: [(a, b, c, d)] #

(Universe (f a), Universe (g a)) => Universe (Product f g a) 
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universe :: [Product f g a] #

(Universe (f a), Universe (g a)) => Universe (Sum f g a) 
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universe :: [Sum f g a] #

(Finite e, Ord e, Universe (m a)) => Universe (ReaderT e m a) 
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universe :: [ReaderT e m a] #

(Universe a, Universe b, Universe c, Universe d, Universe e) => Universe (a, b, c, d, e) 
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universe :: [(a, b, c, d, e)] #

Universe (f (g a)) => Universe (Compose f g a) 
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universe :: [Compose f g a] #

class Universe a => Finite a where #

Creating an instance of this class is a declaration that your universe eventually ends. Minimal definition: no methods defined. By default, universeF = universe, but for some types (like Either) the universeF method may have a more intuitive ordering.

Laws:

elem x universeF                       -- any inhabitant has a finite index
length (filter (== x) universeF) == 1  -- should terminate
(xs -> cardinality xs == genericLength xs) universeF

Note: elemIndex x universe == elemIndex x universeF may not hold for all types, though the laws imply that universe is a permutation of universeF.

>>> elemIndex (Left True :: Either Bool Bool) universe
Just 2
>>> elemIndex (Left True :: Either Bool Bool) universeF
Just 1

Minimal complete definition

Nothing

Instances
Finite Bool 
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Finite Char 
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Finite Int 
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Finite Int8 
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Finite Int16 
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Finite Int32 
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Finite Int64 
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Finite Ordering 
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Finite Word 
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Finite Word8 
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Finite Word16 
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Finite Word32 
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Finite Word64 
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Finite () 
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universeF :: [()] #

cardinality :: Tagged () Natural #

Finite Void 
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Finite All 
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Finite Any 
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Finite a => Finite (Maybe a) 
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Finite a => Finite (Min a) 
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Finite a => Finite (Max a) 
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Finite a => Finite (First a) 
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Finite a => Finite (Last a) 
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Finite a => Finite (Identity a) 
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Finite a => Finite (First a) 
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Finite a => Finite (Last a) 
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Finite a => Finite (Dual a) 
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Finite a => Finite (Sum a) 
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Finite a => Finite (Product a) 
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(Ord a, Finite a) => Finite (Set a) 
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(Ord a, Finite a, Finite b) => Finite (a -> b) 
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universeF :: [a -> b] #

cardinality :: Tagged (a -> b) Natural #

(Finite a, Finite b) => Finite (Either a b) 
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(Finite a, Finite b) => Finite (a, b) 
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universeF :: [(a, b)] #

cardinality :: Tagged (a, b) Natural #

Finite (Proxy a) 
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(Ord k, Finite k, Finite v) => Finite (Map k v) 
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universeF :: [Map k v] #

cardinality :: Tagged (Map k v) Natural #

(Finite a, Finite b, Finite c) => Finite (a, b, c) 
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universeF :: [(a, b, c)] #

cardinality :: Tagged (a, b, c) Natural #

Finite a => Finite (Const a b) 
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universeF :: [Const a b] #

cardinality :: Tagged (Const a b) Natural #

Finite (f a) => Finite (IdentityT f a) 
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Finite a => Finite (Tagged b a) 
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(Finite a, Finite b, Finite c, Finite d) => Finite (a, b, c, d) 
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universeF :: [(a, b, c, d)] #

cardinality :: Tagged (a, b, c, d) Natural #

(Finite (f a), Finite (g a)) => Finite (Product f g a) 
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universeF :: [Product f g a] #

cardinality :: Tagged (Product f g a) Natural #

(Finite (f a), Finite (g a)) => Finite (Sum f g a) 
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universeF :: [Sum f g a] #

cardinality :: Tagged (Sum f g a) Natural #

(Finite e, Ord e, Finite (m a)) => Finite (ReaderT e m a) 
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Defined in Data.Universe.Class

Methods

universeF :: [ReaderT e m a] #

cardinality :: Tagged (ReaderT e m a) Natural #

(Finite a, Finite b, Finite c, Finite d, Finite e) => Finite (a, b, c, d, e) 
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Defined in Data.Universe.Class

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universeF :: [(a, b, c, d, e)] #

cardinality :: Tagged (a, b, c, d, e) Natural #

Finite (f (g a)) => Finite (Compose f g a) 
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Defined in Data.Universe.Class

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universeF :: [Compose f g a] #

cardinality :: Tagged (Compose f g a) Natural #

universeGeneric :: (Generic a, GUniverse (Rep a)) => [a] #

>>> data Zero deriving (Show, Generic)
>>> universeGeneric :: [Zero]
[]
>>> data One = One deriving (Show, Generic)
>>> universeGeneric :: [One]
[One]
>>> data Big = B0 Bool Bool | B1 Bool deriving (Show, Generic)
>>> universeGeneric :: [Big]
[B0 False False,B1 False,B0 False True,B1 True,B0 True False,B0 True True]
>>> universeGeneric :: [Maybe Ordering]
[Nothing,Just LT,Just EQ,Just GT]
>>> take 10 (universeGeneric :: [Either Integer Integer])
[Left 0,Right 0,Left 1,Right 1,Left (-1),Right (-1),Left 2,Right 2,Left (-2),Right (-2)]
>>> take 10 (universeGeneric :: [(Integer, Integer, Integer)])
[(0,0,0),(0,0,1),(1,0,0),(0,1,0),(1,0,1),(-1,0,0),(0,0,-1),(1,1,0),(-1,0,1),(2,0,0)]