# singletons

A framework for generating singleton types http://www.github.com/goldfirere/singletons

Version on this page: | 2.2 |

LTS Haskell 8.5: | 2.2 |

Stackage Nightly 2017-03-24: | 2.2 |

Latest on Hackage: | 2.2 |

# singletons 2.2

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 follow-up
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 term-level functions to type-level
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 8.0.1 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`

`TypeOperators`

`FlexibleContexts`

`RankNTypes`

`UndecidableInstances`

`FlexibleInstances`

`InstanceSigs`

`DefaultSignatures`

`TypeInType`

You may also want

`-Wno-redundant-constraints`

as the code that `singletons`

generates uses redundant constraints, and there
seems to be no way, without a large library redesign, to avoid this.

## 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`

re-exports `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 for function promotion

Modules in `Data.Promotion`

namespace provide functionality required for
function promotion. They mostly re-export 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 re-export 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 top-level 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 in-depth 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 *existentially-quantified* 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 existentially-quantified singleton, wrapped up in a `SomeSing`

.
The `DemoteRep`

associated
kind-indexed 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`

.

## Pre-defined 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`

.

## Promoting functions

Function promotion allows to generate type-level equivalents of term-level 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
so-called "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
type-checking 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 type-checker.

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 0-tuple, 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`

catch-all 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
higher-kinded type variable. GHC does not support higher-sorted 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 built-in 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, term-level 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 built-in 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
built-in 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 lambda-expression. 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.

## Changes

# Changelog for singletons project

## 2.2

With

`TypeInType`

, we no longer kind`KProxy`

. @int-index has very helpfully removed the use of`KProxy`

from`singletons`

.Drop support for GHC 7.x.

Remove

`bugInGHC`

. That function was intended to work around GHC's difficulty in detecting exhaustiveness of GADT pattern matches. GHC 8 comes with a much better exhaustiveness checker, and so this function is no longer necessary.

## 2.1

Require

`th-desugar`

>= 1.6Work with GHC 8. GHC 8 gives the opportunity to simplify some pieces of singletons, but these opportunities are not yet fully realized. For example, injective type families means that we no longer need

`Sing`

to be a data family; it could be a type family. This might drastically simplify the way functions are singletonized. But not yet!`singletons`

now outputs a few more type/kind annotations to help GHC do type inference. There may be a few more programs accepted than before. (This is the fix for #136.)

## 2.0.1

Lots more functions in

`Data.Singletons.Prelude.List`

:`filter`

,`find`

,`elemIndex`

,`elemIndices`

,`findIndex`

,`findIndices`

,`intersect`

,`intersectBy`

,`takeWhile`

,`dropWhile`

,`dropWhileEnd`

,`span`

,`break`

,`take`

,`drop`

,`splitAt`

,`group`

,`maximum`

,`minimum`

,`insert`

,`sort`

,`groupBy`

,`lookup`

,`partition`

,`sum`

,`product`

,`length`

,`replicate`

,`transpose`

,`(!!)`

,`nub`

,`nubBy`

,`unionBy`

,`union`

,`genericLength`

## 2.0.0.2

Fix fixity of

`*`

.

## 2.0.0.1

Make haddock work.

## 2.0

Instance promotion now works properly -- it was quite buggy in 1.0.

Classes and instances can now be singletonized.

Limited support for functional dependencies.

We now have promoted and singletonized versions of

`Enum`

, as well as`Bounded`

.Deriving

`Enum`

is also now supported.Ditto for

`Num`

, which includes an instance for`Nat`

, naturally.Promoting a literal number now uses overloaded literals at the type level, using a type-level

`FromInteger`

in the type-level`Num`

class.Better support for dealing with constraints. Some previously-unsingletonizable functions that have constrained parameters now work.

No more orphan

`Quasi`

instances!Support for functions of arity 8 (instead of the old limit, 7).

Full support for fixity declarations.

A raft of bugfixes.

Drop support for GHC 7.8. You must have GHC 7.10.2.

## 1.1.2.1

Fix bug #116, thus allowing locally-declared symbols to be used in GHC 7.10.

## 1.1.2

No more GHC 7.8.2 support -- you must have GHC 7.8.3.

## 1.1.1

Update testsuite to work with th-desugar-1.5.2. No functional changes.

## 1.1

This is a maintenance release to support building (but *not* testing, due to
GHC bug #10058) with 7.10. This release also targets th-desugar-1.5. Some
types changed (using th-desugar's new `DsMonad`

instead of `Quasi`

), but
clients generally won't need to make any changes, unless they, too, generalize
over `Quasi`

.

## 1.0

This is a complete rewrite of the package.

A much wider array of surface syntax is now accepted for promotion and singletonization, including

`let`

,`case`

, partially-applied functions, and anonymous functions,`where`

, sections, among others.Classes and instances can be promoted (but not singletonized).

Derivation of promoted instances for

`Ord`

and`Bounded`

.

This release can be seen as a "technology preview". More features are coming soon.

This version drops GHC 7.6 support.

## 0.10.0

Template Haskell names are now more hygienic. In other words, `singletons`

won't try to gobble up something happened to be named `Sing`

in your project.
(Note that the Template Haskell names are not *completely* hygienic; names
generated during singleton generation can still cause conflicts.)

If a function to be promoted or singletonized is missing a type signature,
that is now an *error*, not a warning.

Added a new external module Data.Singletons.TypeLits, which contain the singletons for GHC.TypeLits. Some convenience functions are also provided.

The extension `EmptyCase`

is no longer needed. This caused pain when trying
to support both GHC 7.6.3 and 7.8.

## 0.9.3

Fix export list of Data.Singletons.TH, again again.

Add `SEq`

instances for `Nat`

and `Symbol`

.

## 0.9.2

Fix export list of Data.Singletons.TH, again.

## 0.9.1

Fix export list of Data.Singletons.TH.

## 0.9.0

Make compatible with GHC HEAD, but HEAD reports core lint errors sometimes.

Change module structure significantly. If you want to derive your own
singletons, you should import `Data.Singletons.TH`

. The module
`Data.Singletons`

now exports functions only for the *use* of singletons.

New modules `Data.Singletons.Bool`

, `...Maybe`

, `...Either`

, and `...List`

are just like their equivalents from `Data.`

, except for `List`

, which is
quite lacking in features.

For singleton equality, use `Data.Singletons.Eq`

.

For propositional singleton equality, use `Data.Singletons.Decide`

.

New module `Data.Singletons.Prelude`

is meant to mirror the Haskell Prelude,
but with singleton definitions.

Streamline representation of singletons, resulting in *exponential* speedup
at execution. (This has not been rigorously measured, but the data structures
are now *exponentially* smaller.)

Add internal support for TypeLits, because the TypeLits module no longer exports singleton definitions.

Add support for existential singletons, through the `toSing`

method of
`SingKind`

.

Remove the `SingE`

class, bundling its functionality into `SingKind`

.
Thus, the `SingRep`

synonym has also been removed.

Name change: `KindIs`

becomes `KProxy`

.

Add support for singletonizing calls to `error`

.

Add support for singletonizing empty data definitions.

## 0.8.6

Make compatible with GHC HEAD, but HEAD reports core lint errors sometimes.

## 0.8.5

Bug fix to make singletons compatible with GHC 7.6.1.

Added git info to cabal file.

## 0.8.4

Update to work with latest version of GHC (7.7.20130114).

Now use branched type family instances to allow for promotion of functions with overlapping patterns.

Permit promotion of functions with constraints by omitting constraints.

## 0.8.3

Update to work with latest version of GHC (7.7.20121031).

Removed use of Any to simulate kind classes; now using KindOf and OfKind from GHC.TypeLits.

Made compatible with GHC.TypeLits.

## 0.8.2

Added this changelog

Update to work with latest version of GHC (7.6.1). (There was a change to Template Haskell).

Moved library into Data.Singletons.

## 0.8.1

Update to work with latest version of GHC. (There was a change to Template Haskell).

Updated dependencies in cabal to include the newer version of TH.

## 0.8

Initial public release