base-4.19.0.0: Basic libraries
Safe HaskellTrustworthy
LanguageHaskell2010

GHC.TypeNats

Description

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

Synopsis

Nat Kind

data Natural Source #

Natural number

Invariant: numbers <= 0xffffffffffffffff use the NS constructor

Instances

Instances details
Data Natural Source #

Since: base-4.8.0.0

Instance details

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 Source #

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

toConstr :: Natural -> Constr Source #

dataTypeOf :: Natural -> DataType Source #

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

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

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

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

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

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

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

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

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

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

Bits Natural Source #

Since: base-4.8.0

Instance details

Defined in GHC.Bits

Methods

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

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

xor :: Natural -> Natural -> Natural Source #

complement :: Natural -> Natural Source #

shift :: Natural -> Int -> Natural Source #

rotate :: Natural -> Int -> Natural Source #

zeroBits :: Natural Source #

bit :: Int -> Natural Source #

setBit :: Natural -> Int -> Natural Source #

clearBit :: Natural -> Int -> Natural Source #

complementBit :: Natural -> Int -> Natural Source #

testBit :: Natural -> Int -> Bool Source #

bitSizeMaybe :: Natural -> Maybe Int Source #

bitSize :: Natural -> Int Source #

isSigned :: Natural -> Bool Source #

shiftL :: Natural -> Int -> Natural Source #

unsafeShiftL :: Natural -> Int -> Natural Source #

shiftR :: Natural -> Int -> Natural Source #

unsafeShiftR :: Natural -> Int -> Natural Source #

rotateL :: Natural -> Int -> Natural Source #

rotateR :: Natural -> Int -> Natural Source #

popCount :: Natural -> Int Source #

Enum Natural Source #

Since: base-4.8.0.0

Instance details

Defined in GHC.Enum

Methods

succ :: Natural -> Natural Source #

pred :: Natural -> Natural Source #

toEnum :: Int -> Natural Source #

fromEnum :: Natural -> Int Source #

enumFrom :: Natural -> [Natural] Source #

enumFromThen :: Natural -> Natural -> [Natural] Source #

enumFromTo :: Natural -> Natural -> [Natural] Source #

enumFromThenTo :: Natural -> Natural -> Natural -> [Natural] Source #

Ix Natural Source #

Since: base-4.8.0.0

Instance details

Defined in GHC.Ix

Methods

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

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

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

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

rangeSize :: (Natural, Natural) -> Int Source #

unsafeRangeSize :: (Natural, Natural) -> Int Source #

Num Natural Source #

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

Instance details

Defined in GHC.Num

Methods

(+) :: Natural -> Natural -> Natural Source #

(-) :: Natural -> Natural -> Natural Source #

(*) :: Natural -> Natural -> Natural Source #

negate :: Natural -> Natural Source #

abs :: Natural -> Natural Source #

signum :: Natural -> Natural Source #

fromInteger :: Integer -> Natural Source #

Read Natural Source #

Since: base-4.8.0.0

Instance details

Defined in GHC.Read

Methods

readsPrec :: Int -> ReadS Natural Source #

readList :: ReadS [Natural] Source #

readPrec :: ReadPrec Natural Source #

readListPrec :: ReadPrec [Natural] Source #

Integral Natural Source #

Since: base-4.8.0.0

Instance details

Defined in GHC.Real

Methods

quot :: Natural -> Natural -> Natural Source #

rem :: Natural -> Natural -> Natural Source #

div :: Natural -> Natural -> Natural Source #

mod :: Natural -> Natural -> Natural Source #

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

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

toInteger :: Natural -> Integer Source #

Real Natural Source #

Since: base-4.8.0.0

Instance details

Defined in GHC.Real

Methods

toRational :: Natural -> Rational Source #

Show Natural Source #

Since: base-4.8.0.0

Instance details

Defined in GHC.Show

Methods

showsPrec :: Int -> Natural -> ShowS Source #

show :: Natural -> String Source #

showList :: [Natural] -> ShowS Source #

PrintfArg Natural Source #

Since: base-4.8.0.0

Instance details

Defined in Text.Printf

Methods

formatArg :: Natural -> FieldFormatter Source #

parseFormat :: Natural -> ModifierParser Source #

Eq Natural 
Instance details

Defined in GHC.Num.Natural

Methods

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

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

Ord Natural 
Instance details

Defined in GHC.Num.Natural

Methods

compare :: Natural -> Natural -> Ordering Source #

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

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

(>) :: Natural -> Natural -> Bool Source #

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

max :: Natural -> Natural -> Natural Source #

min :: Natural -> Natural -> Natural Source #

KnownNat n => HasResolution (n :: Nat) Source #

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

Instance details

Defined in Data.Fixed

Methods

resolution :: p n -> Integer Source #

TestCoercion SNat Source #

Since: base-4.18.0.0

Instance details

Defined in GHC.TypeNats

Methods

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

TestEquality SNat Source #

Since: base-4.18.0.0

Instance details

Defined in GHC.TypeNats

Methods

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

type Compare (a :: Natural) (b :: Natural) Source # 
Instance details

Defined in Data.Type.Ord

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

type Nat = Natural Source #

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 Source #

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 Source #

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

Since: base-4.10.0.0

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

Since: base-4.10.0.0

data SomeNat Source #

This type represents unknown type-level natural numbers.

Since: base-4.10.0.0

Constructors

KnownNat n => SomeNat (Proxy n) 

Instances

Instances details
Read SomeNat Source #

Since: base-4.7.0.0

Instance details

Defined in GHC.TypeNats

Methods

readsPrec :: Int -> ReadS SomeNat Source #

readList :: ReadS [SomeNat] Source #

readPrec :: ReadPrec SomeNat Source #

readListPrec :: ReadPrec [SomeNat] Source #

Show SomeNat Source #

Since: base-4.7.0.0

Instance details

Defined in GHC.TypeNats

Methods

showsPrec :: Int -> SomeNat -> ShowS Source #

show :: SomeNat -> String Source #

showList :: [SomeNat] -> ShowS Source #

Eq SomeNat Source #

Since: base-4.7.0.0

Instance details

Defined in GHC.TypeNats

Methods

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

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

Ord SomeNat Source #

Since: base-4.7.0.0

Instance details

Defined in GHC.TypeNats

Methods

compare :: SomeNat -> SomeNat -> Ordering Source #

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

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

(>) :: SomeNat -> SomeNat -> Bool Source #

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

max :: SomeNat -> SomeNat -> SomeNat Source #

min :: SomeNat -> SomeNat -> SomeNat Source #

someNatVal :: Natural -> SomeNat Source #

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) Source #

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) Source #

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) Source #

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

Instances details
TestCoercion SNat Source #

Since: base-4.18.0.0

Instance details

Defined in GHC.TypeNats

Methods

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

TestEquality SNat Source #

Since: base-4.18.0.0

Instance details

Defined in GHC.TypeNats

Methods

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

Show (SNat n) Source #

Since: base-4.18.0.0

Instance details

Defined in GHC.TypeNats

Methods

showsPrec :: Int -> SNat n -> ShowS Source #

show :: SNat n -> String Source #

showList :: [SNat n] -> ShowS Source #

Eq (SNat n) Source #

Since: base-4.19.0.0

Instance details

Defined in GHC.TypeNats

Methods

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

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

Ord (SNat n) Source #

Since: base-4.19.0.0

Instance details

Defined in GHC.TypeNats

Methods

compare :: SNat n -> SNat n -> Ordering Source #

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

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

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

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

max :: SNat n -> SNat n -> SNat n Source #

min :: SNat n -> SNat n -> SNat n Source #

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

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 Source #

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 Source #

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 Source #

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 Source #

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 Source #

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

Since: base-4.16.0.0

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

Addition of type-level naturals.

Since: base-4.7.0.0

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

Multiplication of type-level naturals.

Since: base-4.7.0.0

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

Exponentiation of type-level naturals.

Since: base-4.7.0.0

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

Subtraction of type-level naturals.

Since: base-4.7.0.0

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

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 Source #

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 Source #

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 Source #

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 ... Source #

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

Since: base-4.11.0.0