# 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 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 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`

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.