base-4.20.2.0: Core data structures and operations

Safe Haskell | Safe
Language | Haskell2010

GHC.TypeNats

Contents

- Nat Kind
- Linking type and value level
  - Singleton values
- Functions on type literals

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 module.

Since: base-4.10.0.0

Synopsis

- data Natural
- type Nat = Natural
- class KnownNat (n :: Nat) where
  - natSing :: SNat n
- natVal :: forall (n :: Nat) proxy. KnownNat n => proxy n -> Natural
- natVal' :: forall (n :: Nat). KnownNat n => Proxy# n -> Natural
- data SomeNat = KnownNat n => SomeNat (Proxy n)
- someNatVal :: Natural -> SomeNat
- sameNat :: forall (a :: Nat) (b :: Nat) proxy1 proxy2. (KnownNat a, KnownNat b) => proxy1 a -> proxy2 b -> Maybe (a :~: b)
- decideNat :: forall (a :: Nat) (b :: Nat) proxy1 proxy2. (KnownNat a, KnownNat b) => proxy1 a -> proxy2 b -> Either ((a :~: b) -> Void) (a :~: b)
- data SNat (n :: Nat)
- pattern SNat :: () => KnownNat n => SNat n
- fromSNat :: forall (n :: Nat). SNat n -> Natural
- withSomeSNat :: Natural -> (forall (n :: Nat). SNat n -> r) -> r
- withKnownNat :: forall (n :: Nat) r. SNat n -> (KnownNat n => r) -> r
- type (<=) (x :: t) (y :: t) = Assert (x <=? y) (LeErrMsg x y :: Constraint)
- type (<=?) (m :: k) (n :: k) = OrdCond (Compare m n) 'True 'True 'False
- type family (a :: Natural) + (b :: Natural) :: Natural where ...
- type family (a :: Natural) * (b :: Natural) :: Natural where ...
- type family (a :: Natural) ^ (b :: Natural) :: Natural where ...
- type family (a :: Natural) - (b :: Natural) :: Natural where ...
- type family CmpNat (a :: Natural) (b :: Natural) :: Ordering where ...
- cmpNat :: forall (a :: Nat) (b :: Nat) proxy1 proxy2. (KnownNat a, KnownNat b) => proxy1 a -> proxy2 b -> OrderingI a b
- type family Div (a :: Natural) (b :: Natural) :: Natural where ...
- type family Mod (a :: Natural) (b :: Natural) :: Natural where ...
- type family Log2 (a :: Natural) :: Natural where ...

# Nat Kind

data Natural #

#### Instances

Instances details

PrintfArg Natural Source # | Since: base-4.8.0.0
Instance details Defined in Text.Printf Methods formatArg :: Natural -> FieldFormatter Source # parseFormat :: Natural -> ModifierParser Source #
Bits Natural | Since: base-4.8.0

Instance details

Defined in GHC.Internal.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 #

Data Natural | Since: base-4.8.0.0

Instance details

Defined in GHC.Internal.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 #

Enum Natural | Since: base-4.8.0.0

Instance details

Defined in GHC.Internal.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 | Since: base-4.8.0.0
Instance details Defined in GHC.Internal.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 | 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.Internal.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 | Since: base-4.8.0.0
Instance details Defined in GHC.Internal.Read Methods readsPrec :: Int -> ReadS Natural Source # readList :: ReadS [Natural] Source # readPrec :: ReadPrec Natural Source # readListPrec :: ReadPrec [Natural] Source #
Integral Natural | Since: base-4.8.0.0
Instance details Defined in GHC.Internal.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 | Since: base-4.8.0.0
Instance details Defined in GHC.Internal.Real Methods toRational :: Natural -> Rational Source #
Show Natural | Since: base-4.8.0.0
Instance details Defined in GHC.Internal.Show Methods showsPrec :: Int -> Natural -> ShowS Source # show :: Natural -> String Source # showList :: [Natural] -> ShowS 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 | Since: base-4.18.0.0
Instance details Defined in GHC.Internal.TypeNats Methods testCoercion :: forall (a :: Nat) (b :: Nat). SNat a -> SNat b -> Maybe (Coercion a b) Source #
TestEquality SNat | Since: base-4.18.0.0
Instance details Defined in GHC.Internal.TypeNats Methods testEquality :: forall (a :: Nat) (b :: Nat). SNat a -> SNat b -> Maybe (a :~: b) Source #
type Compare (a :: Natural) (b :: Natural) |
Instance details Defined in GHC.Internal.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 | Since: base-4.7.0.0
Instance details Defined in GHC.Internal.TypeNats Methods readsPrec :: Int -> ReadS SomeNat Source # readList :: ReadS [SomeNat] Source # readPrec :: ReadPrec SomeNat Source # readListPrec :: ReadPrec [SomeNat] Source #
Show SomeNat | Since: base-4.7.0.0
Instance details Defined in GHC.Internal.TypeNats Methods showsPrec :: Int -> SomeNat -> ShowS Source # show :: SomeNat -> String Source # showList :: [SomeNat] -> ShowS Source #
Eq SomeNat | Since: base-4.7.0.0
Instance details Defined in GHC.Internal.TypeNats Methods (==) :: SomeNat -> SomeNat -> Bool Source # (/=) :: SomeNat -> SomeNat -> Bool Source #
Ord SomeNat | Since: base-4.7.0.0
Instance details Defined in GHC.Internal.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 | Since: base-4.18.0.0
Instance details Defined in GHC.Internal.TypeNats Methods testCoercion :: forall (a :: Nat) (b :: Nat). SNat a -> SNat b -> Maybe (Coercion a b) Source #
TestEquality SNat | Since: base-4.18.0.0
Instance details Defined in GHC.Internal.TypeNats Methods testEquality :: forall (a :: Nat) (b :: Nat). SNat a -> SNat b -> Maybe (a :~: b) Source #
Show (SNat n) | Since: base-4.18.0.0
Instance details Defined in GHC.Internal.TypeNats Methods showsPrec :: Int -> SNat n -> ShowS Source # show :: SNat n -> String Source # showList :: [SNat n] -> ShowS Source #
Eq (SNat n) | Since: base-4.19.0.0
Instance details Defined in GHC.Internal.TypeNats Methods (==) :: SNat n -> SNat n -> Bool Source # (/=) :: SNat n -> SNat n -> Bool Source #
Ord (SNat n) | Since: base-4.19.0.0
Instance details Defined in GHC.Internal.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 = {- KnownNat 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