Existential witnesses, singletons, and classes for operations on GHC TypeLits


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MIT licensed by Justin Le
Maintained by [email protected]
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Module documentation for

Used by 2 packages in lts-13.19(full list with versions):


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Provides witnesses for KnownNat and KnownSymbol instances for various operations on GHC TypeLits — in particular, the arithmetic operations defined in GHC.TypeLits, and also for type-level lists of KnownNat and KnownSymbol instances.

This is useful for situations where you have KnownNat n, and you want to prove to GHC KnownNat (n + 3), or KnownNat (2*n + 4).

It’s also useful for when you want to work with type level lists of KnownNat/KnownSymbol instances and singletons for traversing them, and be able to apply analogies of natVal/symbolVal to lists with analogies for SomeNat and SomeSymbol.

Note that most of the functionality in this library can be reproduced in a more generic way using the great singletons library. The versions here are provided as a “plumbing included” alternative that makes some commonly found design patterns involving GHC’s TypeLits functionality a little smoother, especially when working with external libraries or GHC TypeLit’s Nat comparison API.


Provides tools for refining upper and lower bounds on KnownNats and proving inequalities involving GHC.TypeLits’s comparison API. (Both with <=? and CmpNat).

If a library function requires 1 <= n constraint, but only KnownNat n is available:

foo :: (KnownNat n, 1 <= n) => Proxy n -> Int

bar :: KnownNat n => Proxy n -> Int
bar n = case Proxy @1 %<=? n of
          LE  Refl -> foo n
          NLE _    -> 0

foo requires that 1 <= n, but bar has to handle all cases of n. %<=? lets you compare the KnownNats in two Proxys and returns a :<=?, which has two constructors, LE and NLE.

If you pattern match on the result, in the LE branch, the constraint 1 <= n will be satisfied according to GHC, so bar can safely call foo, and GHC will recognize that 1 <= n.

In the NLE branch, the constraint that 1 > n is satisfied, so any functions that require that constraint would be callable.

For convenience, isLE and isNLE are also offered:

bar :: KnownNat n => Proxy n -> Int
bar n = case Proxy @1 `isLE` n of
          Just Refl -> foo n
          Nothing   -> 0

Similarly, if a library function requires something involving CmpNat, you can use cmpNat and the SCmpNat type:

foo1 :: (KnownNat n, CmpNat 5 n ~ LT) => Proxy n -> Int
foo2 :: (KnownNat n, CmpNat 5 n ~ GT) => Proxy n -> Int

bar :: KnownNat n => Proxy n -> Int
bar n = case Proxy @5 `cmpNat` n of
          CLT Refl -> foo1 n
          CEQ _    -> 0
          CGT Refl -> foo2 n

You can use the Refl that cmpNat gives you with flipCmpNat and cmpNatLE to “flip” the inequality or turn it into something compatible with <=? (useful for when you have to work with libraries that mix the two methods) or cmpNatEq and eqCmpNat to get to/from witnesses for equality of the two Nats.


SOFT DEPRECATED: Use singletons library instead! However, this module is still here in case people want the functionality of singletons without requiring the entire library.

Provides witnesses for instances arising from the arithmetic operations defined in GHC.TypeLits.

In general, if you have KnownNat n, GHC can’t infer KnownNat (n + 1); and if you have KnownNat m, as well, GHC can’t infer KnownNat (n + m).

This can be extremely annoying when dealing with libraries and applications where one regularly adds and subtracts type-level nats and expects KnownNat instances to follow. For example, vector concatenation of length-encoded vector types can be:

concat :: (KnownNat n, KnownNat m)
       => Vector n       a
       -> Vector m       a
       -> Vector (n + m) a

But, n + m now does not have a KnownNat instance, which severely hinders what you can do with this!

Consider this concrete (but silly) example:

getDoubled :: KnownNat n => Proxy n -> Integer
getDoubled p = natVal (Proxy @(n * 2))

Which is supposed to call natVal with n * 2. However, this fails, because while n is a KnownNat, n * 2 is not necessarily so. This module lets you re-assure GHC that this is okay.

The most straightforward/high-level usage is with withNatOp:

getDoubled :: forall n. KnownNat n => Proxy n -> Integer
getDoubled p = withNatOp (%*) p (Proxy @2) $
    natVal (Proxy @(n * 2))

Within the scope of the argument of withNatOp (%*) (Proxy @n) (Proxy @m), n * m is an instance of KnownNat, so you can use natVal on it, and get the expected result:

> getDoubled (Proxy @12)

There are four “nat operations” defined here, corresponding to the four type-level operations on Nat provided in GHC.TypeLits: (%+), (%-), (%*), and (%^), corresponding to addition, subtraction, multiplication, and exponentiation, respectively.

Note that (%-) is implemented in a way that allows for the result to be a negative Nat.

There are more advanced operations dealing with low-level machinery, as well, in the module. See module documentation for more detail.

Singletons replacement

This module is deprecated, and it is recommended you use the functionality from the singletons package instead. A direct translation using Proxy would be:

getDoubled :: forall n. KnownNat n => Proxy n -> Integer
getDoubled p = withKnownNat (SNat @n %:* SNat @2) $
    natVal (Proxy @(n * 2))

But one using singletons throughout the whole process would be:

getDoubled :: forall n. KnownNat n => Sing n -> Integer
getDoubled s = withKnownNat (s %:* SNat @2) $
    natVal (Proxy @(n * 2))


**HARD DEPRECATED: Use singletons library instead! This module is extremely unweildy, and using singletons is much, much smoother on many levels, and integrates everything together in a nice way. This module will likely be removed in a future version.

Provides analogies of KnownNat, SomeNat, natVal, etc., to type-level lists of KnownNat instances, and also singletons for iterating over type-level lists of Nats and Symbols.

If you had KnownNats ns, then you have two things you can do with it; first, natsVal, which is like natVal but for type-level lists of KnownNats:

> natsVal (Proxy @[1,2,3])

And more importantly, natsList, which provides singletons that you can pattern match on to “reify” the structure of the list, getting a Proxy n for every item in the list with a KnownNat/KnownSymbol instance in scope for you to use:

printNats :: NatList ns -> IO ()
printNats nl = case nl of
                 ØNL       ->
                   return ()
                 p :># nl' -> do
                   print $ natVal p
                   printNats nl'
> printNats (natsList :: @[1,2,3])

Without this, there is no way to “iterate over” and “access” every Nat in a list of KnownNats. You can’t “iterate” over [1,2,3] in Proxy [1,2,3], but you can iterate over them in NatList [1,2,3].

This module also lets you “reify” lists of Integers or Strings into NatLists and SymbolLists, so you can access them at the type level for some dependent types fun.

> reifyNats [1,2,3] $ \nl -> do
    print nl
    printNats nl
Proxy :<# Proxy :<# Proxy :<# ØNL

Another thing you can do is provide witneses that two [Nat]s or [Symbol]s are the same/were instantiated with the same numbers/symbols.

> reifyNats [1,2,3] $ \ns -> do
  reifyNats [1,2,3] $ \ms -> do
    case sameNats ns ms of
      Just Refl -> -- in this branch, ns and ms are the same.
      Nothing   -> -- in this branch, they aren't

The above would match on the Just Refl branch.

See module documentation for more details and variations.

Singletons replacement

This module is deprecated, and it is recommended you use the functionality from the singletons package instead. natsVal is fromSing, reifyNats is toSing/withSomeSing, sameNats is simply %~, and you can traverse/reify singletons of lists too:

printNats :: forall (ns :: [Nat]). Sing ns -> IO ()
printNats ss = case ss of
                 SNil             ->
                   return ()
                 s `SCons` ss' -> do
                   print $ fromSing s
                   printNats ss'




October 1, 2018

  • Fix build for GHC 8.6 and TypeIsStar changes.
  • Break compatibility with GHC < 8.0



March 30, 2018

  • Functions taking multiple p m in GHC.TypeLits.Compare now are able to take different “proxy” types (p and q).



  • Documentation fixes because deprecation warnings were not showing up on haddocks for GHC.TypeLits.List.



  • Added extra witnesses inside the constructors of (:<=?).
  • Allowed functions to polymorphically expect p n instead of Proxy n whenever possible.
  • Soft deprecation of GHC.TypeLits.Witnesses, in case people want to use the functionality of singletons without the full library.
  • Formal deprecation of GHC.TypeLits.List, with migration information, because it’s just so much more unweidly than using singletons.



  • Added the GHC.TypeLits.Compare module for refining bounds and proving inequalities on KnownNats and associated utility functions.



  • Removed redundant KnownNats and KnownSymbols constraints for sameNats and sameSymbols.



  • Added “eliminators”, a staple of dependently typed programming, for NatList and SymbolList.



  • Breaking: Changed the name of someNatsVal' to someNatsValPos, to break away from the “just add '” anti-pattern and to make the function name a bit more meaningful.

  • Added reifyNats', a “safe” version of reifyNats. Ideally, reifyNats should be the safe one, but its connection to reifyNat from the reflection package is very strong and worth preserving, I think.



  • Added mapNatList' and mapSymbolList' companions to mapNatList and mapSymbolList; they use NatList and SymbolList instead of Rank-2 types, so they can work better with function composition with (.) and other things that Rank-2 types would have trouble with.

  • Added sameNats and sameSymbols, modeled after sameNat and sameSymbol. They provide witnesses to GHC that KnownNats passed in are both the same.



  • Added strict fields to NatList, SomeNats, SymbolList, and SomeSymbols. It really doesn’t make any sense for them to be lazy.



  • Added README to the cabal package as an extra source file, for viewing on Hackage.



  • Initial version.