GHC.TypeNats

This module is an internal GHC module. It declares the constants used in the implementation of type-level natural numbers. The programmer interface for working with type-level naturals should be defined in a separate library.

Since: base-4.10.0.0

Nat Kind

data Natural

Natural number

Invariant: numbers <= 0xffffffffffffffff use the NS constructor

Instances

Data Natural

Since: base-4.8.0.0

Defined in Data.Data

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Natural -> c Natural

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Natural

toConstr :: Natural -> Constr

dataTypeOf :: Natural -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c Natural)

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Natural)

gmapT :: (forall b. Data b => b -> b) -> Natural -> Natural

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Natural -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Natural -> r

gmapQ :: (forall d. Data d => d -> u) -> Natural -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Natural -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Natural -> m Natural

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Natural -> m Natural

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Natural -> m Natural

Bits Natural

Since: base-4.8.0

Defined in GHC.Bits

Methods

(.&.) :: Natural -> Natural -> Natural

(.|.) :: Natural -> Natural -> Natural

xor :: Natural -> Natural -> Natural

complement :: Natural -> Natural

shift :: Natural -> Int -> Natural

rotate :: Natural -> Int -> Natural

zeroBits :: Natural

bit :: Int -> Natural

setBit :: Natural -> Int -> Natural

clearBit :: Natural -> Int -> Natural

complementBit :: Natural -> Int -> Natural

testBit :: Natural -> Int -> Bool

bitSizeMaybe :: Natural -> Maybe Int

bitSize :: Natural -> Int

isSigned :: Natural -> Bool

shiftL :: Natural -> Int -> Natural

unsafeShiftL :: Natural -> Int -> Natural

shiftR :: Natural -> Int -> Natural

unsafeShiftR :: Natural -> Int -> Natural

rotateL :: Natural -> Int -> Natural

rotateR :: Natural -> Int -> Natural

popCount :: Natural -> Int

Enum Natural

Since: base-4.8.0.0

Defined in GHC.Enum

Methods

succ :: Natural -> Natural

pred :: Natural -> Natural

toEnum :: Int -> Natural

fromEnum :: Natural -> Int

enumFrom :: Natural -> [Natural]

enumFromThen :: Natural -> Natural -> [Natural]

enumFromTo :: Natural -> Natural -> [Natural]

enumFromThenTo :: Natural -> Natural -> Natural -> [Natural]

Ix Natural

Since: base-4.8.0.0

Defined in GHC.Ix

Methods

range :: (Natural, Natural) -> [Natural]

index :: (Natural, Natural) -> Natural -> Int

unsafeIndex :: (Natural, Natural) -> Natural -> Int

inRange :: (Natural, Natural) -> Natural -> Bool

rangeSize :: (Natural, Natural) -> Int

unsafeRangeSize :: (Natural, Natural) -> Int

Num Natural

Note that Natural's Num instance isn't a ring: no element but 0 has an additive inverse. It is a semiring though.

Since: base-4.8.0.0

Defined in GHC.Num

Methods

(+) :: Natural -> Natural -> Natural

(-) :: Natural -> Natural -> Natural

(*) :: Natural -> Natural -> Natural

negate :: Natural -> Natural

abs :: Natural -> Natural

signum :: Natural -> Natural

fromInteger :: Integer -> Natural

Read Natural

Since: base-4.8.0.0

Defined in GHC.Read

Methods

readsPrec :: Int -> ReadS Natural

readList :: ReadS [Natural]

readPrec :: ReadPrec Natural

readListPrec :: ReadPrec [Natural]

Integral Natural

Since: base-4.8.0.0

Defined in GHC.Real

Methods

quot :: Natural -> Natural -> Natural

rem :: Natural -> Natural -> Natural

div :: Natural -> Natural -> Natural

mod :: Natural -> Natural -> Natural

quotRem :: Natural -> Natural -> (Natural, Natural)

divMod :: Natural -> Natural -> (Natural, Natural)

toInteger :: Natural -> Integer

Real Natural

Since: base-4.8.0.0

Defined in GHC.Real

Methods

toRational :: Natural -> Rational

Show Natural

Since: base-4.8.0.0

Defined in GHC.Show

Methods

showsPrec :: Int -> Natural -> ShowS

show :: Natural -> String

showList :: [Natural] -> ShowS

PrintfArg Natural

Since: base-4.8.0.0

Defined in Text.Printf

Methods

formatArg :: Natural -> FieldFormatter

parseFormat :: Natural -> ModifierParser

Eq Natural

Defined in GHC.Num.Natural

Methods

(==) :: Natural -> Natural -> Bool

(/=) :: Natural -> Natural -> Bool

Ord Natural

Defined in GHC.Num.Natural

Methods

compare :: Natural -> Natural -> Ordering

(<) :: Natural -> Natural -> Bool

(<=) :: Natural -> Natural -> Bool

(>) :: Natural -> Natural -> Bool

(>=) :: Natural -> Natural -> Bool

max :: Natural -> Natural -> Natural

min :: Natural -> Natural -> Natural

KnownNat n => HasResolution (n :: Nat)

For example, Fixed 1000 will give you a Fixed with a resolution of 1000.

Defined in Data.Fixed

Methods

resolution :: p n -> Integer

TestCoercion SNat

Since: base-4.18.0.0

Defined in GHC.TypeNats

Methods

testCoercion :: forall (a :: Nat) (b :: Nat). SNat a -> SNat b -> Maybe (Coercion a b)

TestEquality SNat

Since: base-4.18.0.0

Defined in GHC.TypeNats

Methods

testEquality :: forall (a :: Nat) (b :: Nat). SNat a -> SNat b -> Maybe (a :~: b)

type Compare (a :: Natural) (b :: Natural)

Defined in Data.Type.Ord

type Compare (a :: Natural) (b :: Natural) = CmpNat a b

type Nat = Natural

A type synonym for Natural.

Previously, this was an opaque data type, but it was changed to a type synonym.

Since: base-4.16.0.0

Linking type and value level

class KnownNat (n :: Nat) where

This class gives the integer associated with a type-level natural. There are instances of the class for every concrete literal: 0, 1, 2, etc.

Since: base-4.7.0.0

Methods

natSing :: SNat n

natVal :: forall (n :: Nat) proxy. KnownNat n => proxy n -> Natural

Since: base-4.10.0.0

natVal' :: forall (n :: Nat). KnownNat n => Proxy# n -> Natural

Since: base-4.10.0.0

data SomeNat

This type represents unknown type-level natural numbers.

Since: base-4.10.0.0

Constructors

KnownNat n => SomeNat (Proxy n)

Instances

Read SomeNat

Since: base-4.7.0.0

Defined in GHC.TypeNats

Methods

readsPrec :: Int -> ReadS SomeNat

readList :: ReadS [SomeNat]

readPrec :: ReadPrec SomeNat

readListPrec :: ReadPrec [SomeNat]

Show SomeNat

Since: base-4.7.0.0

Defined in GHC.TypeNats

Methods

showsPrec :: Int -> SomeNat -> ShowS

show :: SomeNat -> String

showList :: [SomeNat] -> ShowS

Eq SomeNat

Since: base-4.7.0.0

Defined in GHC.TypeNats

Methods

(==) :: SomeNat -> SomeNat -> Bool

(/=) :: SomeNat -> SomeNat -> Bool

Ord SomeNat

Since: base-4.7.0.0

Defined in GHC.TypeNats

Methods

compare :: SomeNat -> SomeNat -> Ordering

(<) :: SomeNat -> SomeNat -> Bool

(<=) :: SomeNat -> SomeNat -> Bool

(>) :: SomeNat -> SomeNat -> Bool

(>=) :: SomeNat -> SomeNat -> Bool

max :: SomeNat -> SomeNat -> SomeNat

min :: SomeNat -> SomeNat -> SomeNat

someNatVal :: Natural -> SomeNat

Convert an integer into an unknown type-level natural.

Since: base-4.10.0.0

sameNat :: forall (a :: Nat) (b :: Nat) proxy1 proxy2. (KnownNat a, KnownNat b) => proxy1 a -> proxy2 b -> Maybe (a :~: b)

We either get evidence that this function was instantiated with the same type-level numbers, or Nothing.

Since: base-4.7.0.0

decideNat :: forall (a :: Nat) (b :: Nat) proxy1 proxy2. (KnownNat a, KnownNat b) => proxy1 a -> proxy2 b -> Either ((a :~: b) -> Void) (a :~: b)

We either get evidence that this function was instantiated with the same type-level numbers, or that the type-level numbers are distinct.

Since: base-4.19.0.0

Singleton values

data SNat (n :: Nat)

A value-level witness for a type-level natural number. This is commonly referred to as a singleton type, as for each n, there is a single value that inhabits the type SNat n (aside from bottom).

The definition of SNat is intentionally left abstract. To obtain an SNat value, use one of the following:

1. The natSing method of KnownNat.
2. The SNat pattern synonym.
3. The withSomeSNat function, which creates an SNat from a Natural number.

Since: base-4.18.0.0

Instances

TestCoercion SNat

Since: base-4.18.0.0

Defined in GHC.TypeNats

Methods

testCoercion :: forall (a :: Nat) (b :: Nat). SNat a -> SNat b -> Maybe (Coercion a b)

TestEquality SNat

Since: base-4.18.0.0

Defined in GHC.TypeNats

Methods

testEquality :: forall (a :: Nat) (b :: Nat). SNat a -> SNat b -> Maybe (a :~: b)

Show (SNat n)

Since: base-4.18.0.0

Defined in GHC.TypeNats

Methods

showsPrec :: Int -> SNat n -> ShowS

show :: SNat n -> String

showList :: [SNat n] -> ShowS

Eq (SNat n)

Since: base-4.19.0.0

Defined in GHC.TypeNats

Methods

(==) :: SNat n -> SNat n -> Bool

(/=) :: SNat n -> SNat n -> Bool

Ord (SNat n)

Since: base-4.19.0.0

Defined in GHC.TypeNats

Methods

compare :: SNat n -> SNat n -> Ordering

(<) :: SNat n -> SNat n -> Bool

(<=) :: SNat n -> SNat n -> Bool

(>) :: SNat n -> SNat n -> Bool

(>=) :: SNat n -> SNat n -> Bool

max :: SNat n -> SNat n -> SNat n

min :: SNat n -> SNat n -> SNat n

pattern SNat :: () => KnownNat n => SNat n

A explicitly bidirectional pattern synonym relating an SNat to a KnownNat constraint.

As an expression: Constructs an explicit SNat n value from an implicit KnownNat n constraint:

SNat @n :: KnownNat n => SNat n

As a pattern: Matches on an explicit SNat n value bringing an implicit KnownNat n constraint into scope:

f :: SNat n -> ..
f SNat = {- SNat n in scope -}

Since: base-4.18.0.0

fromSNat :: forall (n :: Nat). SNat n -> Natural

Return the Natural number corresponding to n in an SNat n value.

Since: base-4.18.0.0

withSomeSNat :: Natural -> (forall (n :: Nat). SNat n -> r) -> r

Convert a Natural number into an SNat n value, where n is a fresh type-level natural number.

Since: base-4.18.0.0

withKnownNat :: forall (n :: Nat) r. SNat n -> (KnownNat n => r) -> r

Convert an explicit SNat n value into an implicit KnownNat n constraint.

Since: base-4.18.0.0

Functions on type literals

type (<=) (x :: t) (y :: t) = Assert (x <=? y) (LeErrMsg x y :: Constraint) infix 4

Comparison (<=) of comparable types, as a constraint.

Since: base-4.16.0.0

type (<=?) (m :: k) (n :: k) = OrdCond (Compare m n) 'True 'True 'False infix 4

Comparison (<=) of comparable types, as a function.

Since: base-4.16.0.0

type family (a :: Natural) + (b :: Natural) :: Natural where ... infixl 6

Addition of type-level naturals.

Since: base-4.7.0.0

type family (a :: Natural) * (b :: Natural) :: Natural where ... infixl 7

Multiplication of type-level naturals.

Since: base-4.7.0.0

type family (a :: Natural) ^ (b :: Natural) :: Natural where ... infixr 8

Exponentiation of type-level naturals.

Since: base-4.7.0.0

type family (a :: Natural) - (b :: Natural) :: Natural where ... infixl 6

Subtraction of type-level naturals.

Since: base-4.7.0.0

type family CmpNat (a :: Natural) (b :: Natural) :: Ordering where ...

Comparison of type-level naturals, as a function.

Since: base-4.7.0.0

cmpNat :: forall (a :: Nat) (b :: Nat) proxy1 proxy2. (KnownNat a, KnownNat b) => proxy1 a -> proxy2 b -> OrderingI a b

Like sameNat, but if the numbers aren't equal, this additionally provides proof of LT or GT.

Since: base-4.16.0.0

type family Div (a :: Natural) (b :: Natural) :: Natural where ... infixl 7

Division (round down) of natural numbers. Div x 0 is undefined (i.e., it cannot be reduced).

Since: base-4.11.0.0

type family Mod (a :: Natural) (b :: Natural) :: Natural where ... infixl 7

Modulus of natural numbers. Mod x 0 is undefined (i.e., it cannot be reduced).

Since: base-4.11.0.0

type family Log2 (a :: Natural) :: Natural where ...

Log base 2 (round down) of natural numbers. Log 0 is undefined (i.e., it cannot be reduced).

Since: base-4.11.0.0