Data.Type.Nat

Nat numbers. DataKinds stuff.

This module re-exports Data.Nat, and adds type-level things.

Natural, Nat numbers

data Nat

Nat natural numbers.

Better than GHC's built-in Nat for some use cases.

Constructors

Z

S Nat

Instances

Arbitrary Nat

Defined in Data.Nat

Methods

arbitrary :: Gen Nat

shrink :: Nat -> [Nat]

CoArbitrary Nat

Defined in Data.Nat

Methods

coarbitrary :: Nat -> Gen b -> Gen b

Function Nat

Defined in Data.Nat

Methods

function :: (Nat -> b) -> Nat :-> b

Data Nat

Defined in Data.Nat

Methods

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

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

toConstr :: Nat -> Constr

dataTypeOf :: Nat -> DataType

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

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

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

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

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

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

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

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

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

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

Enum Nat

Defined in Data.Nat

Methods

succ :: Nat -> Nat

pred :: Nat -> Nat

toEnum :: Int -> Nat

fromEnum :: Nat -> Int

enumFrom :: Nat -> [Nat]

enumFromThen :: Nat -> Nat -> [Nat]

enumFromTo :: Nat -> Nat -> [Nat]

enumFromThenTo :: Nat -> Nat -> Nat -> [Nat]

Num Nat

Defined in Data.Nat

Methods

(+) :: Nat -> Nat -> Nat

(-) :: Nat -> Nat -> Nat

(*) :: Nat -> Nat -> Nat

negate :: Nat -> Nat

abs :: Nat -> Nat

signum :: Nat -> Nat

fromInteger :: Integer -> Nat

Integral Nat

Defined in Data.Nat

Methods

quot :: Nat -> Nat -> Nat

rem :: Nat -> Nat -> Nat

div :: Nat -> Nat -> Nat

mod :: Nat -> Nat -> Nat

quotRem :: Nat -> Nat -> (Nat, Nat)

divMod :: Nat -> Nat -> (Nat, Nat)

toInteger :: Nat -> Integer

Real Nat

Defined in Data.Nat

Methods

toRational :: Nat -> Rational

Show Nat

Nat is printed as Natural.

To see explicit structure, use explicitShow or explicitShowsPrec

Defined in Data.Nat

Methods

showsPrec :: Int -> Nat -> ShowS

show :: Nat -> String

showList :: [Nat] -> ShowS

NFData Nat

Defined in Data.Nat

Methods

rnf :: Nat -> ()

Eq Nat

Defined in Data.Nat

Methods

(==) :: Nat -> Nat -> Bool

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

Ord Nat

Defined in Data.Nat

Methods

compare :: Nat -> Nat -> Ordering

(<) :: Nat -> Nat -> Bool

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

(>) :: Nat -> Nat -> Bool

(>=) :: Nat -> Nat -> Bool

max :: Nat -> Nat -> Nat

min :: Nat -> Nat -> Nat

Hashable Nat

Defined in Data.Nat

Methods

hashWithSalt :: Int -> Nat -> Int

hash :: Nat -> Int

Universe Nat

>>> import qualified Data.Universe.Class as U
>>> take 10 (U.universe :: [Nat])
[0,1,2,3,4,5,6,7,8,9]

Since: 0.1.2

Defined in Data.Nat

Methods

universe :: [Nat]

Category LEProof

The other variant (LEPRoof) isn't Category, because leRefl requires SNat evidence.

Defined in Data.Type.Nat.LE.ReflStep

Methods

id :: forall (a :: k). LEProof a a

(.) :: forall (b :: k) (c :: k) (a :: k). LEProof b c -> LEProof a b -> LEProof a c

TestEquality SNat

Defined in Data.Type.Nat

Methods

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

EqP Fin

>>> eqp FZ FZ
True

>>> eqp FZ (FS FZ)
False

>>> let xs = universe @N.Nat4; ys = universe @N.Nat6 in traverse_ print [ [ eqp x y | y <- ys ] | x <- xs ]
[True,False,False,False,False,False]
[False,True,False,False,False,False]
[False,False,True,False,False,False]
[False,False,False,True,False,False]

Since: 0.2.2

Defined in Data.Fin

Methods

eqp :: forall (a :: k) (b :: k). Fin a -> Fin b -> Bool

EqP SNat

Since: 0.2.2

Defined in Data.Type.Nat

Methods

eqp :: forall (a :: k) (b :: k). SNat a -> SNat b -> Bool

GNFData SNat

Since: 0.2.1

Defined in Data.Type.Nat

Methods

grnf :: forall (a :: k). SNat a -> ()

GCompare SNat

Since: 0.2.1

Defined in Data.Type.Nat

Methods

gcompare :: forall (a :: k) (b :: k). SNat a -> SNat b -> GOrdering a b

GEq SNat

Since: 0.2.1

Defined in Data.Type.Nat

Methods

geq :: forall (a :: k) (b :: k). SNat a -> SNat b -> Maybe (a :~: b)

GShow Fin

Since: 0.2.2

Defined in Data.Fin

Methods

gshowsPrec :: forall (a :: k). Int -> Fin a -> ShowS

GShow SNat

Since: 0.2.1

Defined in Data.Type.Nat

Methods

gshowsPrec :: forall (a :: k). Int -> SNat a -> ShowS

OrdP Fin

>>> let xs = universe @N.Nat4; ys = universe @N.Nat6 in traverse_ print [ [ comparep x y | y <- ys ] | x <- xs ]
[EQ,LT,LT,LT,LT,LT]
[GT,EQ,LT,LT,LT,LT]
[GT,GT,EQ,LT,LT,LT]
[GT,GT,GT,EQ,LT,LT]

Since: 0.2.2

Defined in Data.Fin

Methods

comparep :: forall (a :: k) (b :: k). Fin a -> Fin b -> Ordering

OrdP SNat

Since: 0.2.2

Defined in Data.Type.Nat

Methods

comparep :: forall (a :: k) (b :: k). SNat a -> SNat b -> Ordering

toNatural :: Nat -> Natural

Convert Nat to Natural

>>> toNatural 0
0

>>> toNatural 2
2

>>> toNatural $ S $ S $ Z
2

fromNatural :: Natural -> Nat

Convert Natural to Nat

>>> fromNatural 4
4

>>> explicitShow (fromNatural 4)
"S (S (S (S Z)))"

cata :: a -> (a -> a) -> Nat -> a

Fold Nat.

>>> cata [] ('x' :) 2
"xx"

Showing

explicitShow :: Nat -> String

show displaying a structure of Nat.

>>> explicitShow 0
"Z"

>>> explicitShow 2
"S (S Z)"

explicitShowsPrec :: Int -> Nat -> ShowS

showsPrec displaying a structure of Nat.

Singleton

data SNat (n :: Nat) where

Singleton of Nat.

Constructors

SZ :: SNat 'Z

SS :: SNatI n => SNat ('S n)

Instances

TestEquality SNat

Defined in Data.Type.Nat

Methods

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

EqP SNat

Since: 0.2.2

Defined in Data.Type.Nat

Methods

eqp :: forall (a :: k) (b :: k). SNat a -> SNat b -> Bool

GNFData SNat

Since: 0.2.1

Defined in Data.Type.Nat

Methods

grnf :: forall (a :: k). SNat a -> ()

GCompare SNat

Since: 0.2.1

Defined in Data.Type.Nat

Methods

gcompare :: forall (a :: k) (b :: k). SNat a -> SNat b -> GOrdering a b

GEq SNat

Since: 0.2.1

Defined in Data.Type.Nat

Methods

geq :: forall (a :: k) (b :: k). SNat a -> SNat b -> Maybe (a :~: b)

GShow SNat

Since: 0.2.1

Defined in Data.Type.Nat

Methods

gshowsPrec :: forall (a :: k). Int -> SNat a -> ShowS

OrdP SNat

Since: 0.2.2

Defined in Data.Type.Nat

Methods

comparep :: forall (a :: k) (b :: k). SNat a -> SNat b -> Ordering

Show (SNat p)

Defined in Data.Type.Nat

Methods

showsPrec :: Int -> SNat p -> ShowS

show :: SNat p -> String

showList :: [SNat p] -> ShowS

SNatI n => Boring (SNat n)

Since: 0.2.1

Defined in Data.Type.Nat

Methods

boring :: SNat n

NFData (SNat n)

Since: 0.2.1

Defined in Data.Type.Nat

Methods

rnf :: SNat n -> ()

Eq (SNat a)

Since: 0.2.2

Defined in Data.Type.Nat

Methods

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

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

Ord (SNat a)

Since: 0.2.2

Defined in Data.Type.Nat

Methods

compare :: SNat a -> SNat a -> Ordering

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

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

(>) :: SNat a -> SNat a -> Bool

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

max :: SNat a -> SNat a -> SNat a

min :: SNat a -> SNat a -> SNat a

snatToNat :: forall n. SNat n -> Nat

Convert SNat to Nat.

>>> snatToNat (snat :: SNat Nat1)
1

snatToNatural :: forall n. SNat n -> Natural

Convert SNat to Natural

>>> snatToNatural (snat :: SNat Nat0)
0

>>> snatToNatural (snat :: SNat Nat2)
2

Implicit

class SNatI (n :: Nat) where

Implicit SNat.

In an unorthodox singleton way, it actually provides an induction function.

The induction should often be fully inlined. See test/Inspection.hs.

>>> :set -XPolyKinds
>>> newtype Const a b = Const a deriving (Show)
>>> induction (Const 0) (coerce ((+2) :: Int -> Int)) :: Const Int Nat3
Const 6

Methods

induction

Arguments

:: f 'Z

zero case

-> (forall m. SNatI m => f m -> f ('S m))

induction step

-> f n

Instances

SNatI 'Z

Defined in Data.Type.Nat

Methods

induction :: f 'Z -> (forall (m :: Nat). SNatI m => f m -> f ('S m)) -> f 'Z

SNatI n => SNatI ('S n)

Defined in Data.Type.Nat

Methods

induction :: f 'Z -> (forall (m :: Nat). SNatI m => f m -> f ('S m)) -> f ('S n)

snat :: SNatI n => SNat n

Construct explicit SNat value.

withSNat :: SNat n -> (SNatI n => r) -> r

Constructor SNatI dictionary from SNat.

Since: 0.0.3

reify :: forall r. Nat -> (forall n. SNatI n => Proxy n -> r) -> r

Reify Nat.

>>> reify nat3 reflect
3

reflect :: forall n proxy. SNatI n => proxy n -> Nat

Reflect type-level Nat to the term level.

reflectToNum :: forall n m proxy. (SNatI n, Num m) => proxy n -> m

As reflect but with any Num.

Equality

eqNat :: forall n m. (SNatI n, SNatI m) => Maybe (n :~: m)

Decide equality of type-level numbers.

>>> eqNat :: Maybe (Nat3 :~: Plus Nat1 Nat2)
Just Refl

>>> eqNat :: Maybe (Nat3 :~: Mult Nat2 Nat2)
Nothing

type family EqNat (n :: Nat) (m :: Nat) where ...

Type family used to implement == from Data.Type.Equality module.

Equations

EqNat 'Z 'Z = 'True

EqNat ('S n) ('S m) = EqNat n m

EqNat n m = 'False

discreteNat :: forall n m. (SNatI n, SNatI m) => Dec (n :~: m)

Decide equality of type-level numbers.

>>> decShow (discreteNat :: Dec (Nat3 :~: Plus Nat1 Nat2))
"Yes Refl"

Since: 0.0.3

cmpNat :: forall n m. (SNatI n, SNatI m) => GOrdering n m

Decide equality of type-level numbers.

>>> cmpNat :: GOrdering Nat3 (Plus Nat1 Nat2)
GEQ

>>> cmpNat :: GOrdering Nat3 (Mult Nat2 Nat2)
GLT

>>> cmpNat :: GOrdering Nat5 (Mult Nat2 Nat2)
GGT

Induction

induction1

Arguments

:: forall n f a. SNatI n

=> f 'Z a

zero case

-> (forall m. SNatI m => f m a -> f ('S m) a)

induction step

-> f n a

Induction on Nat, functor form. Useful for computation.

Example: unfoldedFix

unfoldedFix :: forall n a proxy. SNatI n => proxy n -> (a -> a) -> a

Unfold n steps of a general recursion.

Note: Always benchmark. This function may give you both bad properties: a lot of code (increased binary size), and worse performance.

For known n unfoldedFix will unfold recursion, for example

unfoldedFix (Proxy :: Proxy Nat3) f = f (f (f (fix f)))

Arithmetic

type family Plus (n :: Nat) (m :: Nat) :: Nat where ...

Addition.

>>> reflect (snat :: SNat (Plus Nat1 Nat2))
3

Equations

Plus 'Z m = m

Plus ('S n) m = 'S (Plus n m)

type family Mult (n :: Nat) (m :: Nat) :: Nat where ...

Multiplication.

>>> reflect (snat :: SNat (Mult Nat2 Nat3))
6

Equations

Mult 'Z m = 'Z

Mult ('S n) m = Plus m (Mult n m)

type family Mult2 (n :: Nat) :: Nat where ...

Multiplication by two. Doubling.

>>> reflect (snat :: SNat (Mult2 Nat4))
8

Equations

Mult2 'Z = 'Z

Mult2 ('S n) = 'S ('S (Mult2 n))

type family DivMod2 (n :: Nat) :: (Nat, Bool) where ...

Division by two. False is 0 and True is 1 as a remainder.

>>> :kind! DivMod2 Nat7 == '(Nat3, True)
DivMod2 Nat7 == '(Nat3, True) :: Bool
= 'True

>>> :kind! DivMod2 Nat4 == '(Nat2, False)
DivMod2 Nat4 == '(Nat2, False) :: Bool
= 'True

Equations

DivMod2 'Z = '('Z, 'False)

DivMod2 ('S 'Z) = '('Z, 'True)

DivMod2 ('S ('S n)) = DivMod2' (DivMod2 n)

Conversion to GHC Nat

type family ToGHC (n :: Nat) :: Nat where ...

Convert to GHC Nat.

>>> :kind! ToGHC Nat5
ToGHC Nat5 :: GHC.Nat...
= 5

Equations

ToGHC 'Z = 0

ToGHC ('S n) = 1 + ToGHC n

type family FromGHC (n :: Nat) :: Nat where ...

Convert from GHC Nat.

>>> :kind! FromGHC 7
FromGHC 7 :: Nat
= 'S ('S ('S ('S ('S ('S ('S 'Z))))))

Equations

FromGHC 0 = 'Z

FromGHC n = 'S (FromGHC (n - 1))

Aliases

Nat

nat0 :: Nat

nat1 :: Nat

nat2 :: Nat

nat3 :: Nat

nat4 :: Nat

nat5 :: Nat

nat6 :: Nat

nat7 :: Nat

nat8 :: Nat

nat9 :: Nat

promoted Nat

type Nat0 = 'Z

type Nat1 = 'S Nat0

type Nat2 = 'S Nat1

type Nat3 = 'S Nat2

type Nat4 = 'S Nat3

type Nat5 = 'S Nat4

type Nat6 = 'S Nat5

type Nat7 = 'S Nat6

type Nat8 = 'S Nat7

type Nat9 = 'S Nat8

Proofs

proofPlusZeroN :: Plus Nat0 n :~: n

0 + n = n

proofPlusNZero :: SNatI n => Plus n Nat0 :~: n

n + 0 = n

proofMultZeroN :: Mult Nat0 n :~: Nat0

0 * n = 0

proofMultNZero :: forall n proxy. SNatI n => proxy n -> Mult n Nat0 :~: Nat0

n * 0 = 0

proofMultOneN :: SNatI n => Mult Nat1 n :~: n

1 * n = n

proofMultNOne :: SNatI n => Mult n Nat1 :~: n

n * 1 = n