Safe Haskell	None
Language	Haskell2010

Data.Type.Nat

Contents

Natural, Nat numbers
Showing
Singleton
Implicit
Equality
Induction
- Example: unfoldedFix
Arithmetic
Conversion to GHC Nat
Aliases
- Nat
- promoted Nat
Proofs

Description

Nat numbers. DataKinds stuff.

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

Synopsis

data Nat
- = Z
- | S Nat
toNatural :: Nat -> Natural
fromNatural :: Natural -> Nat
cata :: a -> (a -> a) -> Nat -> a
explicitShow :: Nat -> String
explicitShowsPrec :: Int -> Nat -> ShowS
data SNat (n :: Nat) where
- SZ :: SNat Z
- SS :: SNatI n => SNat (S n)
snatToNat :: forall n. SNat n -> Nat
snatToNatural :: forall n. SNat n -> Natural
class SNatI (n :: Nat) where
- snat :: SNat n
reify :: forall r. Nat -> (forall n. SNatI n => Proxy n -> r) -> r
reflect :: forall n proxy. SNatI n => proxy n -> Nat
reflectToNum :: forall n m proxy. (SNatI n, Num m) => proxy n -> m
eqNat :: forall n m. (SNatI n, SNatI m) => Maybe (n :~: m)
type family EqNat (n :: Nat) (m :: Nat) where ...
induction :: forall n f. SNatI n => f Z -> (forall m. SNatI m => f m -> f (S m)) -> f n
induction1 :: forall n f a. SNatI n => f Z a -> (forall m. SNatI m => f m a -> f (S m) a) -> f n a
class SNatI n => InlineInduction (n :: Nat) where
- inlineInduction1 :: f Z a -> (forall m. InlineInduction m => f m a -> f (S m) a) -> f n a
inlineInduction :: forall n f. InlineInduction n => f Z -> (forall m. InlineInduction m => f m -> f (S m)) -> f n
unfoldedFix :: forall n a proxy. InlineInduction n => proxy n -> (a -> a) -> a
type family Plus (n :: Nat) (m :: Nat) :: Nat where ...
type family Mult (n :: Nat) (m :: Nat) :: Nat where ...
type family ToGHC (n :: Nat) :: Nat where ...
type family FromGHC (n :: Nat) :: Nat where ...
nat0 :: Nat
nat1 :: Nat
nat2 :: Nat
nat3 :: Nat
nat4 :: Nat
nat5 :: Nat
nat6 :: Nat
nat7 :: Nat
nat8 :: Nat
nat9 :: 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
proofPlusZeroN :: Plus Nat0 n :~: n
proofPlusNZero :: SNatI n => Plus n Nat0 :~: n
proofMultZeroN :: Mult Nat0 n :~: Nat0
proofMultNZero :: forall n proxy. SNatI n => proxy n -> Mult n Nat0 :~: Nat0
proofMultOneN :: SNatI n => Mult Nat1 n :~: n
proofMultNOne :: SNatI n => Mult n Nat1 :~: n

Natural, Nat numbers

data Nat #

Nat natural numbers.

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

Constructors

Z
S Nat

Instances

Enum Nat #
Instance details 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] #
Eq Nat #
Instance details Defined in Data.Nat Methods (==) :: Nat -> Nat -> Bool # (/=) :: Nat -> Nat -> Bool #
Integral Nat #
Instance details 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 #
Data Nat #
Instance details 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 :: (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 #
Num Nat #
Instance details 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 #
Ord Nat #
Instance details 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 #
Real Nat #
Instance details Defined in Data.Nat Methods toRational :: Nat -> Rational #
Show Nat #	`Nat` is printed as `Natural`. To see explicit structure, use `explicitShow` or `explicitShowsPrec`
Instance details Defined in Data.Nat Methods showsPrec :: Int -> Nat -> ShowS # show :: Nat -> String # showList :: [Nat] -> ShowS #
NFData Nat #
Instance details Defined in Data.Nat Methods rnf :: Nat -> () #
Hashable Nat #
Instance details Defined in Data.Nat Methods hashWithSalt :: Int -> Nat -> Int # hash :: Nat -> Int #
TestEquality SNat #
Instance details Defined in Data.Type.Nat Methods testEquality :: SNat a -> SNat b -> Maybe (a :~: b) #

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 #
Instance details Defined in Data.Type.Nat Methods testEquality :: SNat a -> SNat b -> Maybe (a :~: b) #
Show (SNat p) #
Instance details Defined in Data.Type.Nat Methods showsPrec :: Int -> SNat p -> ShowS # show :: SNat p -> String # showList :: [SNat p] -> ShowS #

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 #

Convenience class to get SNat.

Methods

snat :: SNat n #

Instances

SNatI Z #
Instance details Defined in Data.Type.Nat Methods snat :: SNat Z #
SNatI n => SNatI (S n) #
Instance details Defined in Data.Type.Nat Methods snat :: SNat (S n) #

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

Induction

induction #

Arguments

:: SNatI n
=> f Z	zero case
-> (forall m. SNatI m => f m -> f (S m))	induction step
-> f n

Induction on Nat.

Useful in proofs or with GADTs, see source of proofPlusNZero.

induction1 #

Arguments

:: 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.

>>> induction1 (Tagged 0) $ retagMap (+2) :: Tagged Nat3 Int
Tagged 6

class SNatI n => InlineInduction (n :: Nat) where #

The induction will be fully inlined.

See test/Inspection.hs.

Methods

inlineInduction1 :: f Z a -> (forall m. InlineInduction m => f m a -> f (S m) a) -> f n a #

Instances

InlineInduction Z #
Instance details Defined in Data.Type.Nat Methods inlineInduction1 :: f Z a -> (forall (m :: Nat). InlineInduction m => f m a -> f (S m) a) -> f Z a #
InlineInduction n => InlineInduction (S n) #
Instance details Defined in Data.Type.Nat Methods inlineInduction1 :: f Z a -> (forall (m :: Nat). InlineInduction m => f m a -> f (S m) a) -> f (S n) a #

inlineInduction #

Arguments

:: InlineInduction n
=> f Z	zero case
-> (forall m. InlineInduction m => f m -> f (S m))	induction step
-> f n

See InlineInduction.

Example: unfoldedFix

unfoldedFix :: forall n a proxy. InlineInduction 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)

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 = n

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

1 * n = n

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

n * 1 = n