singletons 2.0
This is the README file for the singletons library. This file contains all the
documentation for the definitions and functions in the library.
The singletons library was written by Richard Eisenberg, eir@cis.upenn.edu, and
with significant contributions by Jan Stolarek, jan.stolarek@p.lodz.pl. There
are two papers that describe the library. Original one, Dependently typed
programming with singletons, is available
here and will
be referenced in this documentation as the "singletons paper". A followup
paper, Promoting Functions to Type Families in Haskell, is available
here
and will be referenced in this documentation as the
"promotion paper".
Purpose of the singletons library
The library contains a definition of singleton types, which allow programmers
to use dependently typed techniques to enforce rich constraints among the types
in their programs. See the singletons paper for a more thorough introduction.
The package also allows promotion of termlevel functions to typelevel
equivalents. Accordingly, it exports a Prelude of promoted and singletonized
functions, mirroring functions and datatypes found in Prelude, Data.Bool
,
Data.Maybe
, Data.Either
, Data.Tuple
and Data.List
. See the promotion
paper for a more thorough introduction.
Compatibility
The singletons library requires GHC 7.10.2 or greater. Any code that uses the
singleton generation primitives needs to enable a long list of GHC
extensions. This list includes, but is not necessarily limited to, the
following:
ScopedTypeVariables
TemplateHaskell
TypeFamilies
GADTs
KindSignatures
DataKinds
PolyKinds
TypeOperators
FlexibleContexts
RankNTypes
UndecidableInstances
FlexibleInstances
InstanceSigs
DefaultSignatures
Modules for singleton types
Data.Singletons
exports all the basic singletons definitions. Import this
module if you are not using Template Haskell and wish only to define your
own singletons.
Data.Singletons.TH
exports all the definitions needed to use the Template
Haskell code to generate new singletons.
Data.Singletons.Prelude
reexports Data.Singletons
along with singleton
definitions for various Prelude types. This module provides a singletonized
equivalent of the real Prelude
. Note that not all functions from original
Prelude
could be turned into singletons.
Data.Singletons.Prelude.*
modules provide singletonized equivalents of
definitions found in the following base
library modules: Data.Bool
,
Data.Maybe
, Data.Either
, Data.List
, Data.Tuple
and GHC.Base
. We also
provide singletonized Eq
and Ord
typeclasses
Data.Singletons.Decide
exports type classes for propositional equality.
Data.Singletons.TypeLits
exports definitions for working with GHC.TypeLits
.
Data.Singletons.Void
exports a Void
type, shamelessly copied from
Edward Kmett's void
package, but without the great many package dependencies
in void
.
Modules in Data.Promotion
namespace provide functionality required for
function promotion. They mostly reexport a subset of definitions from
respective Data.Singletons
modules.
Data.Promotion.TH
exports all the definitions needed to use the Template
Haskell code to generate promoted definitions.
Data.Promotion.Prelude
and Data.Promotion.Prelude.*
modules reexport all
promoted definitions from respective Data.Singletons.Prelude
modules. Data.Promotion.Prelude.List
adds a significant amount of functions
that couldn't be singletonized but can be promoted. Some functions still don't
promote  these are documented in the source code of the module. There is also
Data.Promotion.Prelude.Bounded
module that provides promoted PBounded
typeclass.
Functions to generate singletons
The toplevel functions used to generate singletons are documented in the
Data.Singletons.TH
module. The most common case is just calling singletons
,
which I'll describe here:
singletons :: Q [Dec] > Q [Dec]
Generates singletons from the definitions given. Because singleton generation
requires promotion, this also promotes all of the definitions given to the
type level.
Usage example:
$(singletons [d
data Nat = Zero  Succ Nat
pred :: Nat > Nat
pred Zero = Zero
pred (Succ n) = n
])
Definitions used to support singletons
Please refer to the singletons paper for a more indepth explanation of these
definitions. Many of the definitions were developed in tandem with Iavor Diatchki.
data family Sing (a :: k)
The data family of singleton types. A new instance of this data family is
generated for every new singleton type.
class SingI (a :: k) where
sing :: Sing a
A class used to pass singleton values implicitly. The sing
method produces
an explicit singleton value.
data SomeSing (kproxy :: KProxy k) where
SomeSing :: Sing (a :: k) > SomeSing ('KProxy :: KProxy k)
The SomeSing
type wraps up an existentiallyquantified singleton. Note that
the type parameter a
does not appear in the SomeSing
type. Thus, this type
can be used when you have a singleton, but you don't know at compile time what
it will be. SomeSing ('KProxy :: KProxy Thing)
is isomorphic to Thing
.
class (kparam ~ 'KProxy) => SingKind (kparam :: KProxy k) where
type DemoteRep kparam :: *
fromSing :: Sing (a :: k) > DemoteRep kparam
toSing :: DemoteRep kparam > SomeSing kparam
This class is used to convert a singleton value back to a value in the
original, unrefined ADT. The fromSing
method converts, say, a
singleton Nat
back to an ordinary Nat
. The toSing
method produces
an existentiallyquantified singleton, wrapped up in a SomeSing
.
The DemoteRep
associated
kindindexed type family maps a proxy of the kind Nat
back to the type Nat
.
data SingInstance (a :: k) where
SingInstance :: SingI a => SingInstance a
singInstance :: Sing a > SingInstance a
Sometimes you have an explicit singleton (a Sing
) where you need an implicit
one (a dictionary for SingI
). The SingInstance
type simply wraps a SingI
dictionary, and the singInstance
function produces this dictionary from an
explicit singleton. The singInstance
function runs in constant time, using
a little magic.
Equality classes
There are two different notions of equality applicable to singletons: Boolean
equality and propositional equality.

Boolean equality is implemented in the type family (:==)
(which is actually
a synonym for the type family (==)
from Data.Type.Equality
) and the class
SEq
. See the Data.Singletons.Prelude.Eq
module for more information.

Propositional equality is implemented through the constraint (~)
, the type
(:~:)
, and the class SDecide
. See modules Data.Type.Equality
and
Data.Singletons.Decide
for more information.
Which one do you need? That depends on your application. Boolean equality has
the advantage that your program can take action when two types do not equal,
while propositional equality has the advantage that GHC can use the equality
of types during type inference.
Instances of both SEq
and SDecide
are generated when singletons
is called
on a datatype that has deriving Eq
. You can also generate these instances
directly through functions exported from Data.Singletons.TH
.
Predefined singletons
The singletons library defines a number of singleton types and functions
by default:
Bool
Maybe
Either
Ordering
()
 tuples up to length 7
 lists
These are all available through Data.Singletons.Prelude
. Functions that
operate on these singletons are available from modules such as Data.Singletons.Bool
and Data.Singletons.Maybe
.
Function promotion allows to generate typelevel equivalents of termlevel
definitions. Almost all Haskell source constructs are supported  see last
section of this README for a full list.
Promoted definitions are usually generated by calling promote
function:
$(promote [d
data Nat = Zero  Succ Nat
pred :: Nat > Nat
pred Zero = Zero
pred (Succ n) = n
])
Every promoted function and data constructor definition comes with a set of
socalled "symbols". These are required to represent partial application at the
type level. Each function gets N+1 symbols, where N is the arity. Symbols
represent application of between 0 to N arguments. When calling any of the
promoted definitions it is important refer to it using their symbol
name. Moreover, there is new function application at the type level represented
by Apply
type family. Symbol representing arity X can have X arguments passed
in using normal function application. All other parameters must be passed by
calling Apply
.
Users also have access to Data.Promotion.Prelude
and its submodules (Base
,
Bool
, Either
, List
, Maybe
and Tuple
). These provide promoted versions
of function found in GHC's base library.
Note that GHC resolves variable names in Template Haskell quotes. You cannot
then use an undefined identifier in a quote, making idioms like this not
work:
type family Foo a where ...
$(promote [d ... foo x ... ])
In this example, foo
would be out of scope.
Refer to the promotion paper for more details on function promotion.
Classes and instances
This is best understood by example. Let's look at a stripped down Ord
:
class Eq a => Ord a where
compare :: a > a > Ordering
(<) :: a > a > Bool
x < y = case x `compare` y of
LT > True
EQ > False
GT > False
This class gets promoted to a "kind class" thus:
class (kproxy ~ 'KProxy, PEq kproxy) => POrd (kproxy :: KProxy a) where
type Compare (x :: a) (y :: a) :: Ordering
type (:<) (x :: a) (y :: a) :: Bool
type x :< y = ...  promoting `case` is yucky.
Note that default method definitions become default associated type family
instances. This works out quite nicely.
We also get this singleton class:
class (kproxy ~ 'KProxy, SEq kproxy) => SOrd (kproxy :: KProxy a) where
sCompare :: forall (x :: a) (y :: a). Sing x > Sing y > Sing (Compare x y)
(%:<) :: forall (x :: a) (y :: a). Sing x > Sing y > Sing (x :< y)
default (%:<) :: forall (x :: a) (y :: a).
((x :< y) ~ { RHS from (:<) above })
=> Sing x > Sing y > Sing (x :< y)
x %:< y = ...  this is a bit yucky too
Note that a singletonized class needs to use default
signatures, because
typechecking the default body requires that the default associated type
family instance was used in the promoted class. The extra equality constraint
on the default signature asserts this fact to the typechecker.
Instances work roughly similarly.
instance Ord Bool where
compare False False = EQ
compare False True = LT
compare True False = GT
compare True True = EQ
instance POrd ('KProxy :: KProxy Bool) where
type Compare 'False 'False = 'EQ
type Compare 'False 'True = 'LT
type Compare 'True 'False = 'GT
type Compare 'True 'True = 'EQ
instance SOrd ('KProxy :: KProxy Bool) where
sCompare :: forall (x :: a) (y :: a). Sing x > Sing y > Sing (Compare x y)
sCompare SFalse SFalse = SEQ
sCompare SFalse STrue = SLT
sCompare STrue SFalse = SGT
sCompare STrue STrue = SEQ
The only interesting bit here is the instance signature. It's not necessary
in such a simple scenario, but more complicated functions need to refer to
scoped type variables, which the instance signature can bring into scope.
The defaults all just work.
On names
The singletons library has to produce new names for the new constructs it
generates. Here are some examples showing how this is done:

original datatype: Nat
promoted kind: Nat
singleton type: SNat
(which is really a synonym for Sing
)

original datatype: :/\:
promoted kind: :/\:
singleton type: :%/\:

original constructor: Succ
promoted type: 'Succ
(you can use Succ
when unambiguous)
singleton constructor: SSucc
symbols: SuccSym0
, SuccSym1

original constructor: :+:
promoted type: ':+:
singleton constructor: :%+:
symbols: :+:$
, :+:$$
, :+:$$$

original value: pred
promoted type: Pred
singleton value: sPred
symbols: PredSym0
, PredSym1

original value: +
promoted type: :+
singleton value: %:+
symbols: :+$
, :+$$
, :+$$$

original class: Num
promoted class: PNum
singleton class: SNum

original class: ~>
promoted class: #~>
singleton class: :%~>
Special names
There are some special cases:

original datatype: []
singleton type: SList

original constructor: []
promoted type: '[]
singleton constructor: SNil
symbols: NilSym0

original constructor: :
promoted type: ':
singleton constructr: SCons
symbols: ConsSym0
, ConsSym1

original datatype: (,)
singleton type: STuple2

original constructor: (,)
promoted type: '(,)
singleton constructor: STuple2
symbols: Tuple2Sym0
, Tuple2Sym1
, Tuple2Sym2
All tuples (including the 0tuple, unit) are treated similarly.

original value: undefined
promoted type: Any
singleton value: undefined
Supported Haskell constructs
The following constructs are fully supported:
 variables
 tuples
 constructors
 if statements
 infix expressions
_
patterns
 aliased patterns
 lists
 sections
 undefined
 error
 deriving
Eq
, Ord
, Bounded
, and Enum
 class constraints (though these sometimes fail with
let
, lambda
, and case
)
 literals (for
Nat
and Symbol
), including overloaded number literals
 unboxed tuples (which are treated as normal tuples)
 records
 pattern guards
 case
 let
 lambda expressions
!
and ~
patterns (silently but successfully ignored during promotion)
 class and instance declarations
 functional dependencies (with limitations  see below)
The following constructs are supported for promotion but not singleton generation:
 scoped type variables
 overlapping patterns. Note that overlapping patterns are
sometimes not obvious. For example, the
filter
function does not
singletonize due
to overlapping patterns:
filter :: (a > Bool) > [a] > [a]
filter _pred [] = []
filter pred (x:xs)
 pred x = x : filter pred xs
 otherwise = filter pred xs
Overlap is caused by otherwise
catchall guard, that is always true and this
overlaps with pred x
guard.
The following constructs are not supported:
 list comprehensions
 do
 arithmetic sequences
 datatypes that store arrows,
Nat
, or Symbol
 literals (limited support)
Why are these out of reach? First two depend on monads, which mention a
higherkinded type variable. GHC does not support highersorted kind variables,
which would be necessary to promote/singletonize monads. There are other tricks
possible, too, but none are likely to work. See the bug report
here for more info.
Arithmetic sequences are defined using Enum
typeclass, which uses infinite
lists.
As described in the promotion paper, promotion of datatypes that store arrows is
currently impossible. So if you have a declaration such as
data Foo = Bar (Bool > Maybe Bool)
you will quickly run into errors.
Literals are problematic because we rely on GHC's builtin support, which
currently is limited. Functions that operate on strings will not work because
type level strings are no longer considered lists of characters. Function
working on integer literals can be promoted by rewriting them to use
Nat
. Since Nat
does not exist at the term level it will only be possible to
use the promoted definition, but not the original, termlevel one.
This is the same line of reasoning that forbids the use of Nat
or Symbol
in datatype definitions. But, see this bug
report for a workaround.
Support for *
The builtin Haskell promotion mechanism does not yet have a full story around
the kind *
(the kind of types that have values). Ideally, promoting some form
of TypeRep
would yield *
, but the implementation of TypeRep would have to be
updated for this to really work out. In the meantime, users who wish to
experiment with this feature have two options:

The module Data.Singletons.TypeRepStar
has all the definitions possible for
making *
the promoted version of TypeRep
, as TypeRep
is currently implemented.
The singleton associated with TypeRep
has one constructor:
data instance Sing (a :: *) where
STypeRep :: Typeable a => Sing a
Thus, an implicit TypeRep
is stored in the singleton constructor. However,
any datatypes that store TypeRep
s will not generally work as expected; the
builtin promotion mechanism will not promote TypeRep
to *
.
 The module
Data.Singletons.CustomStar
allows the programmer to define a subset
of types with which to work. See the Haddock documentation for the function
singletonStar
for more info.
Known bugs
 Record updates don't singletonize
 In obscure scenarios, GHC "forgets" constraints on functions. This should
happen only with certain uses where the constraint is needed inside of a
case
or lambdaexpression. Having type inference on result types nearby
makes this more likely to bite.
 Inference dependent on functional dependencies is unpredictably bad. The
problem is that a use of an associated type family tied to a class with
fundeps doesn't provoke the fundep to kick in. This is GHC's problem, in
the end.