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singletons

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This is the README file for the singletons, singletons-th, and singletons-base libraries. This file contains documentation for the definitions and functions in these libraries.

The singletons libraries were written by Richard Eisenberg ([email protected]) and with significant contributions by Jan Stolarek ([email protected]) and Ryan Scott ([email protected]). There are two papers that describe the libraries. 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 ([email protected]) is the active maintainer.

Purpose of the libraries

Broadly speaking, the singletons libraries define an ecosystem of singleton types, which allow programmers to use dependently typed techniques to enforce rich constraints among the types in their programs. To that end, the three libraries serve the following roles:

  • The singletons library is a small, foundational library that defines basic singleton-related types and definitions.
  • The singletons-th library defines Template Haskell functionality that allows promotion of term-level functions to type-level equivalents and singling functions to dependently typed equivalents.
  • The singletons-base library uses singletons-th to define promoted and singled functions from the base library, including the Prelude.

Besides the functionality of the libraries themselves, singletons differs from singletons-th and singletons-base by aiming to be compatible with a wider range of GHC versions. See the "Compatibility" section for further details.

Some other introductions to the ideas in these libraries include:

  • The singletons paper and promotion papers.
  • This blog series, authored by Justin Le, which offers a tutorial for these libraries that assumes no knowledge of dependent types.

Compatibility

singletons, singletons-th, and singletons-base have different support windows for requirements on the compiler version needed to build each library:

  • singletons is a minimal library, and as such, it has a relatively wide support window. singletons must be built with GHC 8.0 or later. singletons aims to support cross-compilers as well (e.g., GHC's JavaScript backend).
  • singletons-th and singletons-base require use of many bleeding-edge GHC language extensions, even more so than singletons itself. As such, it is difficult to maintain support for multiple GHC versions in any given release of either library, so they only support the latest major GHC version (currently GHC 9.10).

Any code that uses the singleton-generation functionality from singletons-th or singletons-base 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
  • NoNamedWildCards
  • NoStarIsType
  • PolyKinds
  • RankNTypes
  • ScopedTypeVariables
  • StandaloneDeriving
  • StandaloneKindSignatures
  • TemplateHaskell
  • TypeAbstractions
  • TypeApplications
  • TypeFamilies
  • TypeOperators
  • UndecidableInstances

Some notes on the use of No* extensions:

  • NoNamedWildCards is needed since singletons-th will single code like f _x = ... to sF (_sx :: Sing _x) = ..., which crucially relies on the _x in Sing _x being treated as a type variable, not a wildcard.
  • 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-th, so you may consider opting in to these warnings.

Modules for singleton types

Data.Singletons (from 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.Decide (from singletons) exports type classes for propositional equality. See the "Equality classes" section for more information.

Data.Singletons.TH (from singletons-th) exports all the definitions needed to use the Template Haskell code to generate new singletons. Data.Singletons.Base.TH (from singletons-base) re-exports Data.Singletons.TH plus any promoted or singled definitions that are likely to appear in TH-generated code. For instance, singling a deriving Eq clause will make use of SEq, the singled Eq class, so Data.Singletons.TH re-exports SEq.

Prelude.Singletons (from singletons-base) 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.

The singletons-base library provides 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, and Data.Void. We also provide promoted and singled versions of common type classes, including (but not limited to) Eq, Ord, Show, Enum, and Bounded.

GHC.TypeLits.Singletons (from singletons-base) 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 in singletons-th. The most common case is just calling the singletons function, 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

This section contains a brief overview of some of the most important types from Data.Singletons (from 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.

type SingI :: forall {k}. k -> Constraint
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.

In addition to SingI, there are also higher-order versions named SingI1 and SingI2:

type SingI1 :: forall {k1 k2}. (k1 -> k2) -> Constraint
class (forall x. SingI x => SingI (f x)) => SingI1 f where
  liftSing :: Sing x -> Sing (f x)

type SingI2 :: forall {k1 k2 k3}. (k1 -> k2 -> k3) -> Constraint
class (forall x y. (SingI x, SingI y) => SingI (f x y)) => SingI2 f where
  liftSing2 :: Sing x -> Sing y -> Sing (f x y)

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.Eq.Singletons module from singletons-base 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 from singletons 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, Eq, TestEquality, and TestCoercion are generated when singletons-th is used to single a data type that has a deriving Eq clause. The structure of the SEq and SDecide instances will match that of the derived Eq instance, while the other instances will be boilerplate code:

instance Eq (SExample a) where
  _ == _ = True

-- See Data.Singletons.Decide for how decideEquality and decideCoercion are
-- implemented

instance TestEquality (SExample a) where
  testEquality = decideEquality

instance TestCoercion (SExample a) where
  testEquality = decideCoercion

You can also generate these instances directly through functions exported from Data.Singletons.TH (from singletons-th) and Data.Singletons.Base.TH (from singletons-base).

Ord classes

Promoted and singled versions of the Ord class (POrd and SOrd, respectively) are provided in the Data.Ord.Singletons module from singletons-base.

Just like how singling a data type with a deriving Eq clause will generate a boilerplate Eq instance, singling a data type with a deriving Ord clause will generate a boilerplate Ord instance:

instance Ord (SExample a) where
  compare _ _ = EQ

Show classes

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

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 (from singletons-th) and Data.Singletons.Base.TH (from singletons-base).

Errors

The singletons-base library provides two different ways to handle errors:

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

    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.Base.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-base 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 Prelude.Singletons. 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 "Haskell constructs supported by singletons-th" 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 Prelude.Singletons and related modules (e.g., Data.Bool.Singletons, Data.Either.Singletons, Data.List.Singletons, Data.Maybe.Singletons, Data.Tuple.Singletons, etc.) in singletons-base. 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
data IdSym0 x

type IdSym1 :: a -> a
type family IdSym1 x where
  IdSym1 x = Id x

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 = Id 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-th, 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 :: forall (k :: Type) -> k -> Type

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 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-th 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. Furthermore, due to the lack of levity polymorphism at the kind level (see GHC#14180), unboxed tuple data types and data constructors are promoted and singled as if they were boxed tuples. For example, the (#,#) data constructor is promoted to (,).

  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.

  7. Any data type constructor Rep (regardless of where or how Rep is defined) is promoted to Type. This is needed to make Data.Singletons.TH.CustomStar work.

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

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 [d| data T = MkT |])

Haskell constructs supported by singletons-th

Full support

The following constructs are fully supported:

  • variables
  • tuples
  • constructors
  • if statements
  • infix expressions and types
  • fixity declarations for infix expressions and types
  • _ patterns
  • aliased patterns
  • lists (including list comprehensions)
  • do-notation
  • sections
  • undefined
  • error
  • literal expressions (for Natural, Symbol, and Char), including overloaded number literals
  • datatypes that store Natural
  • 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
  • TypeData
  • TypeAbstractions

Partial support

The following constructs are partially supported:

  • class constraints
  • scoped type variables
  • deriving
  • finite arithmetic sequences
  • records
  • signatures (e.g., (x :: Maybe a)) in patterns
  • functional dependencies
  • type families
  • TypeApplications
  • wildcard types
  • inferred type variable binders

See the following sections for more details.

Class constraints

singletons-th supports promoting and singling class constraints, but with some caveats about the particular sorts of constraints involved. Suppose you have a function that looks like this:

foo :: Eq a => a -> Bool
foo x = (x == x)

singletons-th will promote foo to the following type family:

type Foo :: a -> Bool
type family Foo x where
  Foo x = (x == x)

Note that the standalone kind signature for Foo does not mention PEq (the promoted version of Eq) anywhere. This is because GHC does not permit constraints in kinds, so it would be an error to give Foo the kind Eq a => a -> Bool. Thankfully, we don't need to mention PEq in Foo's kind anyway, as the promoted (==) type family can be used independently of a PEq constraint.

singletons-th will single foo to the following definition:

sFoo :: forall a (x :: a). SEq a => Sing x -> Sing (Foo x)
sFoo sX = sX %== sX

This time, we can (and must) mention SEq (the singled version of Eq) in the type signature for sFoo.

In general, singletons-th only supports constraints of the form C T_1 ... T_n, where C is a type class constructor and each T_i is a type that does not mention any other type classes. singletons-th does not support non-class constraints, such as equality constraint ((~)) or Coercible constraints.

singletons-th only supports constraints that appear in vanilla types. (See the "Rank-n types" section below for what "vanilla" means.) singletons-th does not support existential constraints that appear in data constructors.

Because singletons-th omits constraints when promoting types to kinds, it is possible for some promoted types to have more general kinds than what is intended. For example, consider this example:

bar :: Alternative f => f a
bar = empty

The kind of f in the type of bar should be Type -> Type, and GHC will infer this because of the Alternative f constraint. If you promote this to a type family, however, you instead get:

type Bar :: f a
type family Bar @f @a where
  Bar = Empty

Now, GHC will infer that the full kind of Bar is forall {k} (f :: k -> Type) (a :: k). f a, which is more general that the original definition! This is not desirable, as this means that Bar can be called on kinds that cannot have PAlternative instances.

In general, the only way to avoid this problem is to ensure that type variables like f are given explicit kinds. For example, singletons-th will promote this to a type family with the correct kind:

bar :: forall (f :: Type -> Type) a. Alternative f => f a
bar = empty

Be especially aware of this limitation is you are dealing with classes that are parameterized over higher-kinded type variables such as f.

Scoped type variables

singletons-th makes an effort to track scoped type variables during promotion so that they "just work". For instance, this function:

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

Will be promoted to the following type family:

type F :: forall a. a -> a
type family F x where
  F @a x = (x :: a)

Note the use of @a on the left-hand side of the type family equation, which ensures that a is in scope on the right-hand side. This also works for local definitions, so this:

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

Will promote to the following type families:

type F :: forall a. a -> a
type family F x where
  F @a x = LetG a x

type family LetG a x where
  LetG a x = (x :: a)

Note that LetG includes both a and x as arguments to ensure that they are in scope on the right-hand side.

Besides foralls in type signatures, scoped type variables can also come from pattern signatures. For example, this will also work:

f (x :: a) = g
  where
    g = (x :: a)

Not all forms of scoped type variables are currently supported:

  • If a type signature brings a type variable into scope over the body of a function with a forall, then any pattern signatures must consistently use the same type variable in argument types that mention it. For example, while this will work:

    f :: forall a. a -> a
f (x :: a) = a -- (x :: a) mentions the same type variable `a` from the type signature

    The following will not work:

    f' :: forall a. a -> a
f' (x :: b) = b -- BAD: (x :: b) mentions `b` instead of `a`

    This is because singletons-th would attempt to promote f' to the following:

    type F' :: forall a. a -> a
type family F' x where
  F' @a (x :: b) = b

    And GHC will not infer that b should stand for a.

  • There is limited support for local variables that themselves bring type variables into scope with foralls in their type signatures. For example, this example works:

    f local = g
  where
    g :: forall a. a -> a
    g x = const (x :: a) local

    This is because f and g will be promoted like so:

    type family F local where
  F local = G local

type family G local x where
  G local (x :: a) = Const (x :: a) local

    It is not straightforward to give G a standalone kind signature, and without a standalone kind signature, GHC will not allow writing @a on the left-hand side of G's type family equation. On the other hand, we can still ensure that a is brought into scope by writing (x :: a) instead of x on the left-hand side. While the :: a signature was not present in the original definition, we include it anyway during promotion to ensure that definitions like G kind-check.

    This trick only works if the scoped type variable is mentioned in one of the local definition's arguments, however. If the scoped type variable is only mentioned in the return type, then the promoted definition will not kind-check. This means that examples like this one will not work:

    konst :: a -> Maybe Bool -> a
konst x _ = x

f x = konst x y
  where
    y :: forall b. Maybe b
    y = Nothing :: Maybe b

deriving

singletons-th is slightly more conservative with respect to deriving than GHC is. The only classes that singletons-th 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-th fully supports the anyclass strategy as well as the stock strategy (at least, for the classes listed above). singletons-th does not support the newtype or via strategies, as there is no equivalent of coerce at the type level.

Finite arithmetic sequences

singletons-th 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-th 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.

Another caveat is that GHC allows defining so-called "naughty" record selectors that mention existential type variables that do not appear in the constructor's return type. Naughty record selectors can be used in pattern matching, but they cannot be used as top-level functions. Here is one example of a naughty record selector:

data Some :: (Type -> Type) -> Type where
  MkSome :: { getSome :: f a } -> Some f

Because singletons-th promotes all records to top-level functions, however, attempting to promote getSome will result in an invalid definition. (It may typecheck, but it will not behave like you would expect.) Theoretically, singletons-th could refrain from promoting naughty record selectors, but this would require detecting which type variables in a data constructor are existentially quantified. This is very challenging in general, so we stick to the dumb-but-predictable approach of always promoting record selectors, regardless of whether they are naughty or not.

Signatures in patterns

singletons-th 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).

TypeApplications

Currently, singletons-th will only promote or single code that uses TypeApplications syntax in one specific situation: when type applications are used in data constructor patterns, such as in the following example:

foo :: Maybe a -> [a]
foo (Just @a x)  = [x :: a]
foo (Nothing @a) = [] :: [a]

For all other forms of TypeApplications, singletons-th 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-th may support TypeApplications in more places in the future.

On the other hand, singletons-th 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-th 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-th 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-th 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-th 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.

    As a result, one should not depend on the order of kind variables for promoted class methods like M. The order of kind variables in defunctionalization symbols for promoted class methods (e.g. MSym0 and MSym1) is similarly unspecified.

Wildcard types

Currently, singletons-th can only handle promoting or singling wildcard types if they appear within type applications in data constructor patterns, such as in the following example:

bar :: Maybe () -> Bool
bar (Nothing @_) = False
bar (Just ())    = True

singletons-th will error if trying to promote or single a wildcard type in any other context. Ultimately, this is due to a GHC restriction, as GHC itself will forbid using wildcards in most kind-level contexts. For example, GHC will permit f :: _ but reject type F :: _.

Inferred type variable binders

singletons-th supports promoting inferred type variable binders in most circumstances. For example, singletons-th can promote this definition:

konst :: forall a {b}. a -> b -> a
konst x _ = x

To this type family:

type Konst :: forall a {b}. a -> b -> a
type family Konst @a x y where
  Konst @a (x :: a) (_ :: b) = x

There is one (somewhat obscure) corner case that singletons-th cannot handle, which requires both of the following criteria to be met:

  • A definition must not have any visible arguments.
  • A definition must have an inferred type variable binder as the last type variable in an outermost forall.

For instance, singletons-th cannot promote this definition:

bad :: forall {a}. Either a Bool
bad = Right True

This is because singletons-th will attempt to generate this type family:

type Bad :: forall {a}. Either a Bool
type family Bad where
  Bad = Right True

GHC will not kind-check Bad, however. GHC will kind-check the standalone kind signature and conclude that Bad has arity 0, i.e., that it does not bind any arguments (visible or invisible). However, the definition of Bad requires an arity of 1, as it implicitly binds an argument:

Bad @{a} = Right @{a} @Bool True

In order to make this kind-check, we would need to be able to generate something like this:

type Bad :: forall {a}. Either a Bool
type family Bad @{a} where
  Bad = Right True

However, GHC does not allow users to things like @{a}, and this is by design. (See this part of the relevant GHC proposal about invisible binders in type declarations.) As such, there is no way for singletons-th to promote this definition exactly as written. As a workaround, you can change the forall {a} to forall a, or you can remove the standalone kind signature.

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
  • literal patterns

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

Overlapping patterns pose a challenge for singling. Consider this implementation of isNothing:

isNothing :: Maybe a -> Bool
isNothing Nothing = True
isNothing _       = False

If we single this in a naïve way, we would end up with something like this:

sIsNothing :: forall a (m :: Maybe a). Sing m -> Sing (IsNothing m)
sIsNothing SNothing = STrue
sIsNothing _        = SFalse

This won't typecheck, however. The issue is that due to the way GADT pattern matches are typechecked in GHC, the second equation of sIsNothing has no way of knowing that m should be equal to Just, even though the first equation already matched m against Nothing. As a result, IsNothing m will not reduce in the second equation, which means that the SFalse, the right-hand side, will not typecheck.

Often times, one can work around the issue by "match flattening" the definition. Match flattening works by removing problematic wildcard patterns in favor of enumerating all possible constructors of the data type being matched against. Here are the match-flattened versions of isNothing and sIsNothing, for example:

isNothing :: Maybe a -> Bool
isNothing Nothing  = True
isNothing (Just _) = False

sIsNothing :: forall a (m :: Maybe a). Sing m -> Sing (IsNothing m)
sIsNothing SNothing  = STrue
sIsNothing (SJust _) = SFalse

Now each equation matches on a data constructor of Maybe, so all is well. Note that the wildcard pattern inside of Just _ is fine. The only time a wildcard pattern is problematic is when it prevents a type family in the type signature from reducing (e.g., IsNothing).

Be warned that match flattening is not a perfect workaround. If you have a function with n arguments that need to be flattened, then the flattened version will have approximately n² equations. Nevertheless, it is the only workaround available at the moment.

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. This can be workaround by either replacing the guards with an if pred x then ... else ... expression or a case pred x of { True -> ...; False -> ...} expression.

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. Again, this can be worked around by rewriting the partial pattern match to an explicit case expression that matches on both Just and Nothing.

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 = []

Literal patterns

Patterns which match on numeric, string, or character literals cannot be singled. Using this code as an example:

isZero :: Natural -> Bool
isZero 0 = True
isZero _ = False

Due to the way that the singled version of Natural works, this code would need to be singled into something like this:

sIsZero :: forall (n :: Natural). Sing n -> Sing (IsZero n)
sIsZero sn =
  case sameNat sn (SNat @0) of
    Just Refl -> STrue
    Nothing   -> SFalse

This runs into the same issues with overlapping patterns discussed earlier. Unfortunately, there is not a good workaround this time. It is impossible to match-flatten isZero since there are infinitely many Natural patterns.

If you really want to make something like this work, one can write unsafeCoerce SFalse to force it to typecheck. The singletons-base library itself uses this trick in certain places when it has no other alternatives. See the source code for the SEq Natural instance for inspiration. singletons-th will not generate such code for you, however, so if you want to reach for unsafeCoerce, you are on your own.

Support for singling, but not promotion

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

  • bang patterns

See the following sections for more details.

Bang patterns

Bang patterns (e.g., f !x = (x, x)) cannot be translated to a type-level setting as type families lack an equivalent of bang patterns. As a result, singletons-th will ignore any bang patterns and will simply promote the underyling pattern instead.

Little to no support

The following constructs are either unsupported or almost never work:

  • datatypes that store arrows or Symbol
  • rank-n types
  • embedded type expressions and patterns
  • promoting TypeReps
  • TypeApplications
  • Irrefutable patterns
  • {-# UNPACK #-} pragmas
  • partial application of the (->) type
  • invisible type patterns
  • AllowAmbiguousTypes

See the following sections for more details.

Arrows, Symbol, and literals

As described in the promotion paper, automatic promotion of datatypes that store arrows is currently impossible. So if you have a declaration such as

$(promote [d|
  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 strings can be promoted by rewriting them to use Symbol. Since Symbol does not exist at the term level, it will only be possible to use the promoted definition, but not the original, term-level one.

For now, one way to work around this issue is to define two variants of a data type: one for use at the value level, and one for use at the type level. The example below demonstrates this workaround in the context of a data type that has a Symbol field:

import Data.Kind
import Data.Singletons.TH
import Data.Singletons.TH.Options
import Data.Text (Text)
import GHC.TypeLits.Singletons
import Language.Haskell.TH (Name)

-- Term-level
newtype Message = MkMessage Text
-- Type-level
newtype PMessage = PMkMessage Symbol

$(let customPromote :: Name -> Name
      customPromote n
        | n == ''Message  = ''PMessage
        | n == 'MkMessage = 'PMkMessage
        | n == ''Text     = ''Symbol
        | otherwise       = promotedDataTypeOrConName defaultOptions n

      customDefun :: Name -> Int -> Name
      customDefun n sat = defunctionalizedName defaultOptions (customPromote n) sat in

  withOptions defaultOptions{ promotedDataTypeOrConName = customPromote
                            , defunctionalizedName      = customDefun
                            } $ do
    decs1 <- genSingletons [''Message]
    decs2 <- singletons [d|
               hello :: Message
               hello = MkMessage "hello"
               |]
    return $ decs1 ++ decs2)

Here is breakdown of what each part of this code is doing:

  • Message defines a data type with a field of type Text (from Data.Text). PMessage is what we wish to be the promoted counterpart to Message. The "P" in PMessage stands for "promoted", but this is naming convention is not strictly enforced; you may name your types however you choose.

    PMessage is identical to Message modulo names and the use of Symbol instead of Text. The choice of Symbol is intentional, since Demote Symbol = Text.

  • customPromote defines a mapping from Template Haskell Names to their promoted Name equivalents. We define special cases for the three special types in our program: Message (which will promote to PMessage), MkMessage (which will promote to PMkMessage), and Text (which will promote to Symbol). All other names will go through the default promotedDataTypeOrConName hook (from Data.Singletons.TH.Options).

  • customDefun is like customPromote, but it handles defunctionalization symbols in particular (see the "Promotion and partial application" section). This is needed to ensure that partial applications of MkMessage are promoted to PMkMessageSym0 rather than MkMessageSym0.

  • We use customPromote and customDefun to override the defaultOptions for the Template Haskell machinery. This will ensure that everything in the last argument to withOptions will recognize the names Message, MkMessage, and Symbol, promoting them according to our custom rules.

  • genSingletons [''Message] generates a Sing instance for PMessage, defunctionalization symbols for PMkMessage, etc. These are needed for the next part of the code.

  • Finally, the hello function is promoted and singled using the Template Haskell machinery. Note that the literal "hello" works as both Text and a Symbol.

Besides Text/Symbol, another common use case for this technique is higher-order functions, e.g.,

-- Term-level
newtype Function a b = MkFunction (a -> b)
-- Type-level
newtype PFunction a b = PMkFunction (a ~> b)

Rank-n types

singletons-th 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.

Embedded type expressions and patterns

As a consequence of singletons-th not supporting types with visible dependent quantification (see the "Rank-n types" section above), singletons-th will not support embedded types in expressions or patterns. This means that singletons-th will reject the following examples:

idv :: forall a -> a -> a
idv (type a) (x :: a) = x

x = idv (type Bool) True

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.Base.TypeRepTYPE (from singletons-base) 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.TH.CustomStar (from singletons-th) allows the programmer to define a subset of types with which to work. See the Haddock documentation for the function singletonStar for more info.

Irrefutable patterns

singletons-th will ignore irrefutable patterns (e.g., f ~(x, y) = (y, x)) and will simply promote or single the underlying patterns instead. singletons-th cannot promote irrefutable patterns for the same reason it cannot promote bang patterns: there is no equivalent syntax for type families. Moreover, singletons-th cannot single irrefutable patterns since singled data constructors are implemented as GADTs, as irrefutably matching on a GADT constructor will not bring the underlying type information into scope. Since essentially all singled code relies on using GADT type information in this way, it cannot reasonably be combined with irrefutable patterns, which prevent this key feature of GADT pattern matching.

{-# UNPACK #-} pragmas

singletons-th will ignore {-# UNPACK #-} pragmas on the fields of a data constructor (e.g., data T = MkT {-# UNPACK #-} !()). This is because singled data types represent their argument types using existential type variables, and any data constructor that explicitly uses existential quantification cannot be unpacked. See GHC#10016.

Partial application of the (->) type

singletons-th can only promote (->) when it is applied to exactly two arguments. Attempting to promote (->) to zero or one argument will result in an error. As a consequence, it is impossible to promote instances like the Functor ((->) r) instance, so singletons-base does not provide them.

Invisible type patterns

singletons-th currently does not support invisible type patterns, such as the use of @t in this example:

f :: a -> a
f @t x = x :: t

See this singletons issue.

AllowAmbiguousTypes

singletons-th does not currently support promoting or singling types with ambiguous type variables, which require the AllowAmbiguousTypes language extension to define. For instance, singletons-th does not support definitions such as this one:

{-# LANGUAGE AllowAmbiguousTypes #-}
class HasName a where
  name :: String -- This type is ambiguous