Creating types
Products¶
You can make your own types like this:
 The choice of the names
Square
andSq
are both arbitrary. Both must be capitalized.
 This requires the GADT extension, which is included with GHC2021.
This creates a new type Square
; values of type Square
look like Sq i j
, where i
and j
are Int
s.
Sq
is referred to as a data constructor, and is a Haskell function, with type shown explicitly in the "modern" version above.
> :t Sq 5 4
Sq 5 4 :: Square
> :t Sq 5 True
"Couldn't match expected type ‘Int’ with actual type ‘Bool’"
> :t Sq
Sq :: Int > (Int > Square)  (1)!
> :t Sq 3
Sq 3 :: Int > Square  (2)!

Actually, the repl will drop the brackets, and show:
Int > Int > Square
. 
If this is unclear, see here for more info.
Tip
The type Square
contains the same information as the product type (Int, Int)
. These types are different, in the sense that code which expects one will not work with the other, but it is easy to write lossless functions between them:
Note
The number of types following Sq
can be 0 or more. For example:
If the number of types is 1
, you will see a suggestion to replace data
with newtype
. See more about newtype
here
Records¶
You can also name entries:
row
and col
are now accessing functions:
> entity = Sq 4 5
> :t entity
entity :: Entity
> row entity
4
> col entity
5
> :t row
row :: Entity > Int
> :t col
col :: Entity > Int
Sums¶
Entity
is a type, butSq
andPlayer
are values belonging to that type
The vertical bar 
indicates that an Entity
is either made with Sq
or with Piece
:
 This requires the GADT extension, which is included with GHC2021.
The newline indicates that an Entity
is either made with Sq
or with Player
:
> Sq 4 6
Sq 4 6 :: Entity
> :t Player False
Player False :: Entity
> :t Player
Player :: Bool > Entity
Tip
The type Entity
contains the same information as the type Either (Int, Int) Bool
, and one can write lossless functions between them:
Note
You can combine products and sums, using your own types:
data ChessPiece = Piece PieceType Color  SquareType Square  (1)!
data Color = Black  White
data PieceType = Bishop  Rook
data Square = Sq Int Int
 Note that Haskell is unconcerned by the order of definitions: the definition of
Color
comes after its use in the definition ofChessPiece
.
Parameterized types¶
One can also create types which take another type as a parameter:
This creates Piece Bool
, Piece Int
, and so on:
> data Piece c = Bishop c  Knight c  King c
> :t Knight True
Knight True :: Piece Bool
let four = 4 :: Int
> :t King four
King four :: Piece Int
> :t King (King True)
King (King True) :: Piece (Piece Bool)
Note
One can think of Piece
as a function on types, which gives a type c
, produces a type Piece c
.
To make this idea explicit, one can say that Piece
has kind Type > Type
, where kind is a name for the types that types themselves have.
Recursive types¶
Here, BinTree
is being used recursively in its own definition. Values of BinTree
include:
> data BinTree = Leaf Int  Branch BinTree BinTree
> :t Leaf 4
Leaf 4 :: BinTree
> :t Leaf True
> :t Branch (Leaf 4) (Leaf 5)
Branch (Leaf 4) (Leaf 5) :: BinTree
> :t Branch (Leaf 3) ((Branch (Leaf 6) (Leaf 8)))
Branch (Leaf 3) ((Branch (Leaf 6) (Leaf 8))) :: BinTree
Recursive types can also be parametrized:
data BinTree a = Leaf a  Branch (BinTree a) (BinTree a)
> :t Leaf True
Leaf True :: BinTree Bool
> :t Leaf ()
Leaf () :: BinTree ()
> :t Branch (Leaf True) (Leaf False)
Branch (Leaf True) (Leaf False) :: BinTree Bool
> :t Branch (Leaf True) (Leaf ())
"Couldn't match expected type ‘Bool’ with actual type ‘()’"  (1)!
 The definition of
Branch
requires that the left and right branch be trees of the same type, which is why this doesn't work.
Here is a more complex recursive type and a program of that type:
data Machine a b = M (a > (b, Machine a b))
machine :: Machine Int Int
machine = machine1 where
machine1 :: Machine Int Int
machine1 = M (\i > (i, if i > 10 then machine2 else machine1))
machine2 :: Machine Int Int
machine2 = M (\i > (0, machine2))
Note
The list type can be defined recursively in this way:
In fact, the [a]
type in Haskell is defined in this way, with the [1,2,3]
being extra syntax for convenience:
:
is the data constructor, analogous toHeadThenList
above, but written infix.[]
is analogous toEmptyList
.
Synonyms¶
One can also give new names to existing types:
Note
Here, Number
and Double
are not distinguished as separate types by the compiler, so replacing one by the other in a type signature will always be fine. This would not be true for:
This can be useful for readability, particularly for quite complex types:
Isomorphic types¶
Two types are isomorphic (or more colloquially, the same) if there are functions to convert between them in both directions that are lossless:
data WrappedInt = MkW {getInt :: Int}
> :t MkW
MkW :: Int > WrappedInt  one direction
> :t getInt
getInt :: WrappedInt > Int  the other direction
> :t (getInt . MkW)
(getInt . MkW) :: Int > Int
> (getInt . MkW) 4  (1)!
4
> :t (MkW . getInt)
(MkW . getInt) :: WrappedInt > WrappedInt  (2)!
(getInt . MkW)
is the identity function.(MkW . getInt)
is also the identity function.
Types which are isomorphic can be regarded as "the same" for practical purposes, since you can always convert between them, although they may have different performance characteristics. They may also have different typeclass instances.
Reference table of isomorphic types¶
A type  An isomorphic type 

data WrappedInt = MkW Int 
Int 
data Wrapped a = MkW a 
a 
data Unit = U 
() 
data Pair = P Int Bool 
(Int, Bool) 
data OneOf = I Int  B Bool 
Either Int Bool 
a > b > c 
(a, b) > c 
Bool > a 
(a, a) 
Maybe a 
Either () a 
Reader e a 
e > a 
State s a 
s > (a, s) 
Except err a 
Either err a 
ReaderT e m a 
e > m a 
StateT s m a 
s > m (a, s) 
ExceptT err m a 
m (Either err a) 
Created: January 7, 2023