Safe Haskell	Trustworthy
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
- induction :: f 'Z -> (forall m. SNatI m => f m -> f ('S m)) -> f n
snat :: SNatI n => SNat n
withSNat :: SNat n -> (SNatI n => r) -> r
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 ...
discreteNat :: forall n m. (SNatI n, SNatI m) => Dec (n :~: m)
cmpNat :: forall n m. (SNatI n, SNatI m) => GOrdering n m
induction1 :: forall n f a. SNatI n => f 'Z a -> (forall m. SNatI m => f m a -> f ('S m) a) -> f n a
unfoldedFix :: forall n a proxy. SNatI 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 Mult2 (n :: Nat) :: Nat where ...
type family DivMod2 (n :: Nat) :: (Nat, Bool) 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 Source #

Nat natural numbers.

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

Constructors

Z
S Nat

Instances

Instances details

Arbitrary Nat Source #
Instance details Defined in Data.Nat Methods arbitrary :: Gen Nat # shrink :: Nat -> [Nat] #
CoArbitrary Nat Source #
Instance details Defined in Data.Nat Methods coarbitrary :: Nat -> Gen b -> Gen b #
Function Nat Source #
Instance details Defined in Data.Nat Methods function :: (Nat -> b) -> Nat :-> b #
Data Nat Source #
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 :: 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 Source #
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] #
Num Nat Source #
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 #
Integral Nat Source #
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 #
Real Nat Source #
Instance details Defined in Data.Nat Methods toRational :: Nat -> Rational #
Show Nat Source #	`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 Source #
Instance details Defined in Data.Nat Methods rnf :: Nat -> () #
Eq Nat Source #
Instance details Defined in Data.Nat Methods (==) :: Nat -> Nat -> Bool # (/=) :: Nat -> Nat -> Bool #
Ord Nat Source #
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 #
Hashable Nat Source #
Instance details Defined in Data.Nat Methods hashWithSalt :: Int -> Nat -> Int # hash :: Nat -> Int #
Universe Nat Source #	`>>>` `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
Instance details Defined in Data.Nat Methods universe :: [Nat] #
Category LEProof Source #	The other variant (`LEPRoof`) isn't `Category`, because `leRefl` requires `SNat` evidence.
Instance details 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 Source #
Instance details Defined in Data.Type.Nat Methods testEquality :: forall (a :: k) (b :: k). SNat a -> SNat b -> Maybe (a :~: b) #
EqP Fin Source #	`>>>` `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
Instance details Defined in Data.Fin Methods eqp :: forall (a :: k) (b :: k). Fin a -> Fin b -> Bool #
EqP SNat Source #	Since: 0.2.2
Instance details Defined in Data.Type.Nat Methods eqp :: forall (a :: k) (b :: k). SNat a -> SNat b -> Bool #
GNFData SNat Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods grnf :: forall (a :: k). SNat a -> () #
GCompare SNat Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods gcompare :: forall (a :: k) (b :: k). SNat a -> SNat b -> GOrdering a b #
GEq SNat Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods geq :: forall (a :: k) (b :: k). SNat a -> SNat b -> Maybe (a :~: b) #
GShow Fin Source #	Since: 0.2.2
Instance details Defined in Data.Fin Methods gshowsPrec :: forall (a :: k). Int -> Fin a -> ShowS #
GShow SNat Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods gshowsPrec :: forall (a :: k). Int -> SNat a -> ShowS #
OrdP Fin Source #	`>>>` `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
Instance details Defined in Data.Fin Methods comparep :: forall (a :: k) (b :: k). Fin a -> Fin b -> Ordering #
OrdP SNat Source #	Since: 0.2.2
Instance details Defined in Data.Type.Nat Methods comparep :: forall (a :: k) (b :: k). SNat a -> SNat b -> Ordering #

toNatural :: Nat -> Natural Source #

Convert Nat to Natural

>>> toNatural 0
0

>>> toNatural 2
2

>>> toNatural $ S $ S $ Z
2

fromNatural :: Natural -> Nat Source #

Convert Natural to Nat

>>> fromNatural 4
4

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

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

Fold Nat.

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

Showing

explicitShow :: Nat -> String Source #

show displaying a structure of Nat.

>>> explicitShow 0
"Z"

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

explicitShowsPrec :: Int -> Nat -> ShowS Source #

showsPrec displaying a structure of Nat.

Singleton

data SNat (n :: Nat) where Source #

Singleton of Nat.

Constructors

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

Instances

Instances details

TestEquality SNat Source #
Instance details Defined in Data.Type.Nat Methods testEquality :: forall (a :: k) (b :: k). SNat a -> SNat b -> Maybe (a :~: b) #
EqP SNat Source #	Since: 0.2.2
Instance details Defined in Data.Type.Nat Methods eqp :: forall (a :: k) (b :: k). SNat a -> SNat b -> Bool #
GNFData SNat Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods grnf :: forall (a :: k). SNat a -> () #
GCompare SNat Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods gcompare :: forall (a :: k) (b :: k). SNat a -> SNat b -> GOrdering a b #
GEq SNat Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods geq :: forall (a :: k) (b :: k). SNat a -> SNat b -> Maybe (a :~: b) #
GShow SNat Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods gshowsPrec :: forall (a :: k). Int -> SNat a -> ShowS #
OrdP SNat Source #	Since: 0.2.2
Instance details Defined in Data.Type.Nat Methods comparep :: forall (a :: k) (b :: k). SNat a -> SNat b -> Ordering #
Show (SNat p) Source #
Instance details Defined in Data.Type.Nat Methods showsPrec :: Int -> SNat p -> ShowS # show :: SNat p -> String # showList :: [SNat p] -> ShowS #
SNatI n => Boring (SNat n) Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods boring :: SNat n #
NFData (SNat n) Source #	Since: 0.2.1
Instance details Defined in Data.Type.Nat Methods rnf :: SNat n -> () #
Eq (SNat a) Source #	Since: 0.2.2
Instance details Defined in Data.Type.Nat Methods (==) :: SNat a -> SNat a -> Bool # (/=) :: SNat a -> SNat a -> Bool #
Ord (SNat a) Source #	Since: 0.2.2
Instance details 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 Source #

Convert SNat to Nat.

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

snatToNatural :: forall n. SNat n -> Natural Source #

Convert SNat to Natural

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

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

Implicit

class SNatI (n :: Nat) where Source #

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 Source #

Arguments

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

Instances

Instances details

SNatI 'Z Source #
Instance details Defined in Data.Type.Nat Methods induction :: f 'Z -> (forall (m :: Nat). SNatI m => f m -> f ('S m)) -> f 'Z Source #
SNatI n => SNatI ('S n) Source #
Instance details Defined in Data.Type.Nat Methods induction :: f 'Z -> (forall (m :: Nat). SNatI m => f m -> f ('S m)) -> f ('S n) Source #

snat :: SNatI n => SNat n Source #

Construct explicit SNat value.

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

Constructor SNatI dictionary from SNat.

Since: 0.0.3

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

Reify Nat.

>>> reify nat3 reflect
3

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

Reflect type-level Nat to the term level.

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

As reflect but with any Num.

Equality

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

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 ... Source #

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) Source #

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 Source #

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 Source #

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 Source #

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 ... Source #

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 ... Source #

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 ... Source #

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 ... Source #

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 ... Source #

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 ... Source #

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 Source #

nat1 :: Nat Source #

nat2 :: Nat Source #

nat3 :: Nat Source #

nat4 :: Nat Source #

nat5 :: Nat Source #

nat6 :: Nat Source #

nat7 :: Nat Source #

nat8 :: Nat Source #

nat9 :: Nat Source #

promoted Nat

type Nat0 = 'Z Source #

type Nat1 = 'S Nat0 Source #

type Nat2 = 'S Nat1 Source #

type Nat3 = 'S Nat2 Source #

type Nat4 = 'S Nat3 Source #

type Nat5 = 'S Nat4 Source #

type Nat6 = 'S Nat5 Source #

type Nat7 = 'S Nat6 Source #

type Nat8 = 'S Nat7 Source #

type Nat9 = 'S Nat8 Source #

Proofs

proofPlusZeroN :: Plus Nat0 n :~: n Source #

0 + n = n

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

n + 0 = n

proofMultZeroN :: Mult Nat0 n :~: Nat0 Source #

0 * n = 0

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

n * 0 = 0

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

1 * n = n

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

n * 1 = n