singletons 2.7

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 (rae@cs.brynmawr.edu) and with significant contributions by Jan Stolarek (jan.stolarek@p.lodz.pl) and Ryan Scott (ryan.gl.scott@gmail.com). 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".

Ryan Scott (ryan.gl.scott@gmail.com) is the active maintainer.

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 and singling functions to dependently typed equivalents. Accordingly, it exports a Prelude of promoted and singled functions, mirroring functions and datatypes found in the Prelude, Data.Bool, Data.Maybe, Data.Either, Data.Tuple and Data.List. See the promotion paper for a more thorough introduction.

This blog series, authored by Justin Le, offers a tutorial for this library that assumes no knowledge of dependent types.

Compatibility

The singletons library requires GHC 8.10.1 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:

• DataKinds
• DefaultSignatures
• EmptyCase
• ExistentialQuantification
• FlexibleContexts
• FlexibleInstances
• GADTs
• InstanceSigs
• KindSignatures
• NoCUSKs
• NoStarIsType
• PolyKinds
• RankNTypes
• ScopedTypeVariables
• StandaloneKindSignatures
• TemplateHaskell
• TypeApplications
• TypeFamilies
• TypeOperators
• UndecidableInstances

In particular, NoStarIsType is needed to use the * type family from the PNum class because with StarIsType enabled, GHC thinks * is a synonym for Type.

You may also want to consider toggling various warning flags:

• -Wno-redundant-constraints. The code that singletons generates uses redundant constraints, and there seems to be no way, without a large library redesign, to avoid this.
• -fenable-th-splice-warnings. By default, GHC does not run pattern-match coverage checker warnings on code inside of Template Haskell quotes. This is an extremely common thing to do in singletons, so you may consider opting in to these warnings.

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 promoted and singled equivalents of functions from the real Prelude. Note that not all functions from original Prelude could be promoted or singled.

Data.Singletons.Prelude.* modules provide promoted and singled equivalents of definitions found in several commonly used base library modules, including (but not limited to) Data.Bool, Data.Maybe, Data.Either, Data.List, Data.Tuple, Data.Void and GHC.Base. We also provide promoted and singled versions of common type classes, including (but not limited to) Eq, Ord, Show, Enum, and Bounded.

Data.Singletons.Decide exports type classes for propositional equality.

Data.Singletons.TypeLits exports definitions for working with GHC.TypeLits.

Functions to generate singletons

The top-level functions used to generate promoted or singled definitions 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]

This function 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.

type Sing :: k -> Type
type family Sing

The type family of singleton types. A new instance of this type family is generated for every new singleton type.

class SingI a where
  sing :: Sing a

A class used to pass singleton values implicitly. The sing method produces an explicit singleton value.

type SomeSing :: Type -> Type
data SomeSing k where
  SomeSing :: Sing (a :: k) -> SomeSing 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 Thing is isomorphic to Thing.

type SingKind :: Type -> Constraint
class SingKind k where
  type Demote k :: *
  fromSing :: Sing (a :: k) -> Demote k
  toSing   :: Demote k -> SomeSing k

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 Demote associated kind-indexed type family maps the kind Nat back to the type Nat.

type SingInstance :: k -> Type
data SingInstance a 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 (==) (in the PEq class) and the (%==) method (in the SEq class). 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 SEq, SDecide, TestEquality, and TestCoercion 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.

Show classes

Promoted and singled versions of the Show class (PShow and SShow, respectively) are provided in the Data.Singletons.Prelude.Show module. In addition, there is a ShowSing constraint synonym provided in the Data.Singletons.ShowSing module:

type ShowSing :: Type -> Constraint
type ShowSing k = (forall z. Show (Sing (z :: k)) -- Approximately

This facilitates the ability to write Show instances for Sing instances.

What distinguishes all of these Shows? Let's use the False constructor as an example. If you used the PShow Bool instance, then the output of calling Show_ on False is "False", much like the value-level Show Bool instance (similarly for the SShow Bool instance). However, the Show (Sing (z :: Bool)) instance (i.e., ShowSing Bool) is intended for printing the value of the singleton constructor SFalse, so calling show SFalse yields "SFalse".

Instance of PShow, SShow, and Show (for the singleton type) are generated when singletons is called on a datatype that has deriving Show. You can also generate these instances directly through functions exported from Data.Singletons.TH.

A promoted and singled Show instance is provided for Symbol, but it is only a crude approximation of the value-level Show instance for String. On the value level, showing Strings escapes special characters (such as double quotes), but implementing this requires pattern-matching on character literals, something which is currently impossible at the type level. As a consequence, the type-level Show instance for Symbols does not do any character escaping.

Errors

The singletons library provides two different ways to handle errors:

• The Error type family, from Data.Singletons.TypeLits:

  type Error :: a -> k
  type family Error str where {}
  

  This is simply an empty, closed type family, which means that it will fail to reduce regardless of its input. The typical use case is giving it a Symbol as an argument, so that something akin to Error "This is an error message" appears in error messages.

• The TypeError type family, from Data.Singletons.TypeError. This is a drop-in replacement for TypeError from GHC.TypeLits which can be used at both the type level and the value level (via the typeError function).

  Unlike Error, TypeError will result in an actual compile-time error message, which may be more desirable depending on the use case.

Pre-defined singletons

The singletons library defines a number of singleton types and functions by default. These include (but are not limited to):

• 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 the "Supported Haskell constructs" section of this README for a full list.

Promoted definitions are usually generated by calling the 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 defunctionalization symbols. These are required to represent partial application at the type level. For more information, refer to the "Promotion and partial application" section below.

Users also have access to Data.Singletons.Prelude and its submodules (e.g., 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.

Promotion and partial application

Promoting higher-order functions proves to be surprisingly tricky. Consider this example:

$(promote [d|
  map :: (a -> b) -> [a] -> [b]
  map _ []     = []
  map f (x:xs) = f x : map f xs
  |])

A naïve attempt to promote map would be:

type Map :: (a -> b) -> [a] -> [b]
type family Map f xs where
  Map _ '[]    = '[]
  Map f (x:xs) = f x : Map f xs

While this compiles, it is much less useful than we would like. In particular, common idioms like Map Id xs will not typecheck, since GHC requires that all invocations of type families be fully saturated. That is, the use of Id in Map Id xs is rejected since it is not applied to one argument, which the number of arguments that Id was defined with. For more information on this point, refer to the promotion paper.

Not having the ability to partially apply functions at the type level is rather painful, so we do the next best thing: we defunctionalize all promoted functions so that we can emulate partial application. For example, if one were to promote the id function:

$(promote [d|
  id :: a -> a
  id x = x
  |]

Then in addition to generating the promoted Id type family, two defunctionalization symbols will be generated:

type IdSym0 :: a ~> a
type IdSym0 x = x

type IdSym1 (x :: a) = Id a

In general, a function that accepts N arguments generates N+1 defunctionalization symbols when promoted.

IdSym1 is a fully saturated defunctionalization symbol and is usually only needed when generating code through the Template Haskell machinery. IdSym0 is more interesting: it has the kind a ~> a, which has a special arrow type (~>). Defunctionalization symbols using the (~>) kind are type-level constants that can be "applied" using a special Apply type family:

type Apply :: (a ~> b) -> a -> b
type family Apply f x

Every defunctionalization symbol comes with a corresponding Apply instance (except for fully saturated defunctionalization symbols). For instance, here is the Apply instance for IdSym0:

type instance Apply IdSym0 x = IdSym1 x

The (~>) kind is used when promoting higher-order functions so that partially applied arguments can be passed to them. For instance, here is our final attempt at promoting map:

type Map :: (a ~> b) -> [a] -> [b]
type family Map f xs where
  Map _ '[]    = '[]
  Map f (x:xs) = Apply f x : Map f xs

Now map id xs can be promoted to Map IdSym0 xs, which typechecks without issue.

Defunctionalizing existing type families

The most common way to defunctionalize functions is by promoting them with the Template Haskell machinery. One can also defunctionalize existing type families, however, by using genDefunSymbols. For example:

type MyTypeFamily :: Nat -> Bool
type family MyTypeFamily n

$(genDefunSymbols [''MyTypeFamily])

This can be especially useful if MyTypeFamily needs to be implemented by hand. Be aware of the following design limitations of genDefunSymbols:

• genDefunSymbols only works for type-level declarations. Namely, it only works when given the names of type classes, type families, type synonyms, or data types. Attempting to pass the name of a term level function, class method, data constructor, or record selector will throw an error.
• Passing the name of a data type to genDefunSymbols will cause its data constructors to be defunctionalized but not its record selectors.
• Passing the name of a type class to genDefunSymbols will cause the class itself to be defunctionalized, but /not/ its associated type families or methods.

Note that the limitations above reflect the current design of genDefunSymbols. As a result, they are subject to change in the future.

Defunctionalization and visible dependent quantification

Unlike most other parts of singletons, which disallow visible dependent quantification (VDQ), genDefunSymbols has limited support for VDQ. Consider this example:

type MyProxy :: forall (k :: Type) -> k -> Type
type family MyProxy k (a :: k) :: Type where
  MyProxy k (a :: k) = Proxy a

$(genDefunSymbols [''MyProxy])

This will generate the following defunctionalization symbols:

type MyProxySym0 ::              Type  ~> k ~> Type
type MyProxySym1 :: forall (k :: Type) -> k ~> Type
type MyProxySym2 k (a :: k) = MyProxy k a

Note that MyProxySym0 is a bit more general than it ought to be, since there is no dependency between the first kind (Type) and the second kind (k). But this would require the ability to write something like this:

type MyProxySym0 :: forall (k :: Type) ~> k ~> Type

This currently isn't possible. So for the time being, the kind of MyProxySym0 will be slightly more general, which means that under rare circumstances, you may have to provide extra type signatures if you write code which exploits the dependency in MyProxy's kind.

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 PEq a => POrd 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 SEq a => SOrd 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 singled 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 Bool where
  type Compare 'False 'False = 'EQ
  type Compare 'False 'True  = 'LT
  type Compare 'True  'False = 'GT
  type Compare 'True  'True  = 'EQ

instance SOrd 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:

1. original datatype: Nat

   promoted kind: Nat

   singleton type: SNat (which is really a synonym for Sing)

2. original datatype: /\

   promoted kind: /\

   singleton type: %/\

3. original constructor: Succ

   promoted type: 'Succ (you can use Succ when unambiguous)

   singleton constructor: SSucc

   symbols: SuccSym0, SuccSym1

4. original constructor: :+:

   promoted type: ':+:

   singleton constructor: :%+:

   symbols: :+:@#@$, :+:@#@$$, :+:@#@$$$

5. original value: pred

   promoted type: Pred

   singleton value: sPred

   symbols: PredSym0, PredSym1

6. original value: +

   promoted type: +

   singleton value: %+

   symbols: +@#@$, +@#@$$, +@#@$$$

7. original class: Num

   promoted class: PNum

   singleton class: SNum

8. original class: ~>

   promoted class: #~>

   singleton class: %~>

Special names

There are some special cases, listed below (with asterisks* denoting special treatment):

1. original datatype: []

   promoted kind: []

   singleton type*: SList

2. original constructor: []

   promoted type: '[]

   singleton constructor*: SNil

   symbols*: NilSym0

3. original constructor: :

   promoted type: ':

   singleton constructor*: SCons

   symbols: :@#@$, :@#@$$, :@#@$$$

4. original datatype: (,)

   promoted kind: (,)

   singleton type*: STuple2

5. original constructor: (,)

   promoted type: '(,)

   singleton constructor*: STuple2

   symbols*: Tuple2Sym0, Tuple2Sym1, Tuple2Sym2

   All tuples (including the 0-tuple, unit) are treated similarly.

6. original value: ___foo

   promoted type*: US___foo ("US" stands for "underscore")

   singleton value*: ___sfoo

   symbols*: US___fooSym0

   All functions that begin with leading underscores are treated similarly.

If desired, you can pick your own naming conventions by using the Data.Singletons.TH.Options module. Here is an example of how this module can be used to prefix a singled data constructor with MyS instead of S:

import Control.Monad.Trans.Class
import Data.Singletons.TH
import Data.Singletons.TH.Options
import Language.Haskell.TH (Name, mkName, nameBase)

$(let myPrefix :: Name -> Name
      myPrefix name = mkName ("MyS" ++ nameBase name) in

      withOptions defaultOptions{singledDataConName = myPrefix} $
      singletons $ lift [d| data T = MkT |])

Supported Haskell constructs

Full support

The following constructs are fully supported:

• variables
• tuples
• constructors
• if statements
• infix expressions and types
• _ patterns
• aliased patterns
• lists (including list comprehensions)
• do-notation
• sections
• undefined
• error
• 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)
• pattern guards
• case
• let
• lambda expressions
• ! and ~ patterns (silently but successfully ignored during promotion)
• class and instance declarations
• signatures (e.g., (x :: Maybe a)) in expressions
• InstanceSigs

Partial support

The following constructs are partially supported:

• deriving
• finite arithmetic sequences
• records
• signatures (e.g., (x :: Maybe a)) in patterns
• functional dependencies
• type families

See the following sections for more details.

deriving

singletons is slightly more conservative with respect to deriving than GHC is. The only classes that singletons can derive without an explicit deriving strategy are the following stock classes:

• Eq
• Ord
• Show
• Bounded
• Enum
• Functor
• Foldable
• Traversable

To do anything more exotic, one must explicitly indicate one's intentions by using the DerivingStrategies extension. singletons fully supports the anyclass strategy as well as the stock strategy (at least, for the classes listed above). singletons does not support the newtype or via strategies, as there is no equivalent of coerce at the type level.

Finite arithmetic sequences

singletons has partial support for arithmetic sequences (which desugar to methods from the Enum class under the hood). Finite sequences (e.g., [0..42]) are fully supported. However, infinite sequences (e.g., [0..]), which desugar to calls to enumFromTo or enumFromThenTo, are not supported, as these would require using infinite lists at the type level.

Records

Record selectors are promoted to top-level functions, as there is no record syntax at the type level. Record selectors are also singled to top-level functions because embedding records directly into singleton data constructors can result in surprising behavior (see this bug report for more details on this point). TH-generated code is not affected by this limitation since singletons desugars away most uses of record syntax. On the other hand, it is not possible to write out code like SIdentity { sRunIdentity = SIdentity STrue } by hand.

Signatures in patterns

singletons can promote basic pattern signatures, such as in the following examples:

f :: forall a. a -> a
f (x :: a) = (x :: a)

g :: forall a. a -> a
g (x :: b) = (x :: b) -- b is the same as a

What does /not/ work are more advanced uses of pattern signatures that take advantage of the fact that type variables in pattern signatures can alias other types. Here are some examples of functions that one cannot promote:

• h :: a -> a -> a
  h (x :: a) (_ :: b) = x
  

  This typechecks by virtue of the fact that b aliases a. However, the same trick does not work when h is promoted to a type family, as a type family would consider a and b to be distinct type variables.

• i :: Bool -> Bool
  i (x :: a) = x
  

  This typechecks by virtue of the fact that a aliases Bool. Again, this would not work at the type level, as a type family would consider a to be a separate type from Bool.

Functional dependencies

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.

Type families

Promoting functions with types that contain type families is likely to fail due to GHC#12564. Note that promoting type family declarations is fine (and often desired, since that produces defunctionalization symbols for them).

Support for promotion, but not singling

The following constructs are supported for promotion but not singleton generation:

• data constructors with contexts
• overlapping patterns
• GADTs
• instances of poly-kinded type classes

See the following sections for more details.

Data constructors with contexts

For example, the following datatype does not single:

data T a where
  MkT :: Show a => a -> T a

Constructors like these do not interact well with the current design of the SingKind class. But see this bug report, which proposes a redesign for SingKind (in a future version of GHC with certain bugfixes) which could permit constructors with equality constraints.

Overlapping patterns

Note that overlapping patterns are sometimes not obvious. For example, the filter function does not single 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, which is always true and thus overlaps with pred x guard.

Another non-obvious source of overlapping patterns comes from partial pattern matches in do-notation. For example:

f :: [()]
f = do
  Just () <- [Nothing]
  return ()

This has overlap because the partial pattern match desugars to the following:

f :: [()]
f = case [Nothing] of
      Just () -> return ()
      _ -> fail "Partial pattern match in do notation"

Here, it is more evident that the catch-all pattern _ overlaps with the one above it.

GADTs

Singling GADTs is likely to fail due to the generated SingKind instances not typechecking. (See #150). However, one can often work around the issue by suppressing the generation of SingKind instances by using custom Options. See the T150 test case for an example.

Instances of poly-kinded type classes

Singling instances of poly-kinded type classes is likely to fail due to #358. However, one can often work around the issue by using InstanceSigs. For instance, the following code will not single:

class C (f :: k -> Type) where
  method :: f a

instance C [] where
  method = []

Adding a type signature for method in the C [] is sufficient to work around the issue, though:

instance C [] where
  method :: [a]
  method = []

Little to no support

The following constructs are either unsupported or almost never work:

• scoped type variables
• datatypes that store arrows, Nat, or Symbol
• rank-n types
• promoting TypeReps
• TypeApplications

See the following sections for more details.

Scoped type variables

Promoting functions that rely on the behavior of ScopedTypeVariables is very tricky—see this GitHub issue for an extended discussion on the topic. This is not to say that promoting functions that rely on ScopedTypeVariables is guaranteed to fail, but it is rather fragile. To demonstrate how fragile this is, note that the following function will promote successfully:

f :: forall a. a -> a
f x = id x :: a

But this one will not:

g :: forall a. a -> a
g x = id (x :: a)

There are usually workarounds one can use instead of ScopedTypeVariables:

1. Use pattern signatures:

   g :: forall a. a -> a
   g (x :: a) = id (x :: a)
   

2. Use local definitions:

   g :: forall a. a -> a
   g x = id' a
     where
       id' :: a -> a
       id' x = x
   

Arrows, Nat, Symbol, and literals

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. Functions working over 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.

Rank-n types

singletons does not support type signatures that have higher-rank types. More precisely, the only types that can be promoted or singled are vanilla types, where a vanilla function type is a type that:

1. Only uses a forall at the top level, if used at all. That is to say, it does not contain any nested or higher-rank foralls.

2. Only uses a context (e.g., c => ...) at the top level, if used at all, and only after the top-level forall if one is present. That is to say, it does not contain any nested or higher-rank contexts.

3. Contains no visible dependent quantification.

Promoting TypeReps

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:

1. The module Data.Singletons.TypeRepTYPE 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:

   type instance Sing @(TYPE rep) = TypeRep
   

   (Recall that type * = TYPE LiftedRep.) Note that any datatypes that store TypeReps will not generally work as expected; the built-in promotion mechanism will not promote TypeRep to *.

2. 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.

TypeApplications

singletons currently cannot handle promoting or singling code that uses TypeApplications syntax, so singletons will simply drop any visible type applications. For example, id @Bool True will be promoted to Id True and singled to sId STrue. See #378 for a discussion of how singletons may support TypeApplications in the future.

On the other hand, singletons does make an effort to preserve the order of type variables when promoting and singling certain constructors. These include:

• Kind signatures of promoted top-level functions
• Type signatures of singled top-level functions
• Kind signatures of singled data type declarations
• Type signatures of singled data constructors
• Kind signatures of singled class declarations
• Type signatures of singled class methods

For example, consider this type signature:

const2 :: forall b a. a -> b -> a

The promoted version of const will have the following kind signature:

type Const2 :: forall b a. a -> b -> a

The singled version of const2 will have the following type signature:

sConst2 :: forall b a (x :: a) (y :: a). Sing x -> Sing y -> Sing (Const x y)

Therefore, writing const2 @T1 @T2 works just as well as writing Const2 @T1 @T2 or sConst2 @T1 @T2, since the signatures for const2, Const2, and sConst2 all begin with forall b a., in that order. Again, it is worth emphasizing that the TH machinery does not support promoting or singling const2 @T1 @T2 directly, but you can write the type applications by hand if you so choose.

singletons also has limited support for preserving the order of type variables for the following constructs:

• Kind signatures of defunctionalization symbols. The order of type variables is only guaranteed to be preserved if:

  1. The thing being defunctionalized has a standalone type (or kind) signature.
  2. The type (or kind) signature of the thing being defunctionalized is a vanilla type. (See the "Rank-n types" section above for what "vanilla" means.)

  If either of these conditions do not hold, singletons will fall back to a slightly different approach to generating defunctionalization symbols that does not guarantee the order of type variables. As an example, consider the following example:

  data T (x :: a) :: forall b. b -> Type
  $(genDefunSymbols [''T])
  

  The kind of T is forall a. a -> forall b. b -> Type, which is not vanilla. Currently, singletons will generate the following defunctionalization symbols for T:

  data TSym0 :: a ~> b ~> Type
  data TSym1 (x :: a) :: b ~> Type
  

  In both symbols, the kind starts with forall a b. rather than quantifying the b after the visible argument of kind a. These symbols can still be useful even with this flaw, so singletons permits generating them regardless. Be aware of this drawback if you try doing something similar yourself!

• Kind signatures of promoted class methods. The order of type variables will often "just work" by happy coincidence, but there are some situations where this does not happen. Consider the following class:

  class C (b :: Type) where
    m :: forall a. a -> b -> a
  

  The full type of m is forall b. C b => forall a. a -> b -> a, which binds b before a. This order is preserved when singling m, but not when promoting m. This is because the C class is promoted as follows:

  class PC (b :: Type) where
    type M (x :: a) (y :: b) :: a
  

  Due to the way GHC kind-checks associated type families, the kind of M is forall a b. a -> b -> a, which binds b after a. Moreover, the StandaloneKindSignatures extension does not provide a way to explicitly declare the full kind of an associated type family, so this limitation is not easy to work around.

  The defunctionalization symbols for M will also follow a similar order of type variables:

  type MSym0 :: forall a b. a ~> b ~> a
  type MSym1 :: forall a b. a -> b ~> a