Dead simple Haskell

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Haskell molecules

Atoms

atoms

atoms

data H = H -- hydrogen
data O = O -- oxygen
data C = C -- carbon

Molecules

Atoms

Atoms

type H₂O = (H, O, H) -- water
type O₂  = (O, O)    -- oxygen (gas)
type CO₂ = (O, C, O) -- carbon dioxide

Reactions

Magic

More magic

More magic

makeWater :: H -> H -> O -> H₂O

makeOxygen :: O -> O -> O₂

burnOxygen :: C -> O₂ -> CO₂

makeWater :: H -> H -> O -> H₂O

makeWater h1 h2 o = (h1, o, h2)

~

makeOxygen :: O -> O -> O₂

makeOxygen o1 o2 = (o1, o2)

~

burnOxygen :: C -> O₂ -> CO₂

burnOxygen c (o1, o2) = (o1, c, o2)
.$

Plumbing

Partial application

λ> :type makeOxygen
makeOxygen :: O -> O -> O₂

λ> :type makeOxygen O
makeOxygen O :: O -> O₂

λ> :type makeOxygen O O
makeOxygen O O :: O₂

Combustion

λ> :type burnOxygen
burnOxygen :: C -> O₂ -> CO₂

λ> :type burnOxygen C
burnOxygen C :: O₂ -> CO₂

λ> :type burnOxygen C (O, O)
burnOxygen C O₂ :: CO₂

Two functions

f1 :: O -> O₂
f1 = makeOxygen O

f2 :: O₂ -> CO₂
f2 = burnOxygen C

Composition

f3 = f2 . f1

Composition2

f3 :: O -> CO₂

f1 :: { O -> O₂ }
f2 :: [ O₂ -> CO₂ ]
f3 :: {O -> [ O₂ } -> CO₂ ]
f3 :: O -> CO₂

f3' o = f2 . f1 o

Diagnostics:
1. • Couldn't match type: (O, O)
                    with: a -> O₂
     Expected: a -> O₂
       Actual: O₂
   • Possible cause: ‘f1’ is applied to too many arguments
     In the second argument of ‘(.)’, namely ‘f1 o’
     In the expression: f2 . f1 o
     In an equation for ‘f3'’: f3' o = f2 . f1 o
   • Relevant bindings include
       f3' :: O -> a -> CO₂
         (bound at Script.hs:112:1) [-Wdeferred-type-errors]

Composition3

f3' o = f2 . f1 o

Funnel

f3' o = f2 . f1 $ o

-- OR

f3' o = f2 $ f1 o

infixr 9 .
(.) :: (b -> c) -> (a -> b) -> (a -> c)
(f . g) x = f (g x)

infixr 0 $
($) :: (a -> b) -> a -> b
f $ x = f x

Isotopes

Heavy water

Isotopes

data H'
  = H¹-- protium
  | H²-- deuterium
  | H³-- tritium

type H2O' = (H', O, H') -- water

makeWater' :: H' -> H' -> O -> H2O'
makeWater' h1 h2 o = (h1, o, h2)

Algebra

How many values inhibit each of these?

TypePossibilities
()?
O?
H'?
(H', H')?

()

()

1

()

O

O

1

O

H'

H'

3

H¹ H² H³

(H', H')

(H', H')

9

(H¹, H¹) (H¹, H²) (H¹, H³)

(H², H¹) (H², H²) (H², H³)

(H³, H¹) (H³, H²) (H³, H³)

(H', Bool)

(H', Bool)

6

(H¹, True) (H¹, False)

(H², True) (H², False)

(H³, True) (H³, False)

TypeInhibitants
()1
Bool2
H'3
Aa
Bb
(A, B)a × b
A | Ba + b

Product types Π

TypeInhibitants
()1
Bool2
((), Bool)1 × 2 = 2
(H’, Bool)3 × 2 = 6
data Pair a b = Pair a b

type Pair a b = (a, b)

data Pair a b = Pair a b

data Pair a b = Pair
  { fieldA :: a
  , fieldB :: b
  }

Sum types Σ

TypeInhibitants
()1
Bool2
() | Bool1 + 2 = 3
H’ | Bool3 + 2 = 5
data Either a b = Left a | Right b

Why not both?

Why not both?

data Crazy a b
  = CrazyA a
  | CrazyB b
  | Both a b
  | Neither

data Crazy a b
  = CrazyA a
  | CrazyB b
  | Both a b
  | Neither
MemberInhibitants
CrazyA aa
CrazyB bb
Both a ba × b
Neither1
Σa + b + a × b + 1

Can there be a “0”?

Can there be a “0”?

Yes

Crazy Void ()
  = CrazyA Void
  | CrazyB ()
  | Both Void ()
  | Neither

Σ = 0 + 1 + 0 × 1 + 1 = 2

How do lists fit into this?

How do lists fit into this?

data List a
  = Nil
  | Cons a (List a)

-- [1, 2, 3] or 1 : 2 : 3 []
list1 = Cons 1 (Cons 2 (Cons 3 Nil))
-- []
list2 = Nil

data List a
  = Nil
  | Cons a (List a)

x = 1 + a * x

data List a
  = Nil
  | Cons a (List a)

x = 1 + a * x
x = 1 + a * (1 + a * x)
x = 1 + a * (1 + a * (1 + a * (...)))

data List a
  = Nil
  | Cons a (List a)

x = 1 + a * x
x = 1 + a * (1 + a * x)
x = 1 + a * (1 + a * (1 + a * (...)))

x = 1 + a + a² + a³ + a⁴ + a⁵ + ...

data List a
  = Nil
  | Cons a (List a)

x = 1 + a * x
x = 1 + a * (1 + a * x)
x = 1 + a * (1 + a * (1 + a * (...)))

x = 1 + a + a² + a³ + a⁴ + a⁵ + ...

data List a = () | (a) | (a, a) | (a, a, a) | ...

Traits

Neon colors

Element nameColor
Heliumorange
Neonred
Argonlavender
Kryptonwhite
Xenonblue
Radonred
data He = He -- Helium
data Ne = Ne -- Neon
data Ar = Ar -- Argon
data Kr = Kr -- Krypton
data Xe = Xe -- Xenon
data Rn = Rn -- Radon

How to convert noble gases to color?

toColor :: ? -> String

What about allowing everything in?

toColor :: a -> String
toColor a
  | a == He   = "orange"
  | a == Ne   = "red"
  | otherwise = undefined

-- ???
_ = toColor "anything in"
_ = toColor 1234

What about treating noble gases as a sum type?

data Noble
  = He
  | Ne
  | Ar
  | Kr
  | Xe
  | Rn

a -> String

^

???

^

Noble -> String

class Noble a where
  toColor :: a -> String

~

instance Noble He where
  toColor _ = "orange"

instance Noble Ne where
  toColor _ = "red"

instance Noble Ar where
  toColor _ = "lavender"

Works:

λ> toColor He
"orange"
λ> toColor Ne
"red"

Doesn’t work:

λ> toColor C

<interactive>:6:1: error:
    • No instance for (Noble C) arising from a use of ‘toColor’
    • In the expression: toColor C
      In an equation for ‘it’: it = toColor C

mixNoble :: (Noble n1, Noble n2) => n1 -> n2 -> String
mixNoble n1 n2 = toColor n1 <> "-" <> toColor n2

λ> mixNoble He Ne
"orange-red"

Shrodinger’s cat

Cat in a box

Cat in a box

data Box a
  = Has a
  | Empty
  deriving Show

Map

data Cat = Cat String deriving Show
data Dog = Dog String deriving Show

Map

class Mappable box where
  map' :: (a -> b) -> box a -> box b

class Mappable box where
  map' :: (a -> b) -> box a -> box b

instance Mappable Box where
  map' _ Empty   = Empty
  map' f (Has a) = Has (f a)

instance Mappable [] where
  map' _ [] = []
  map' f xs = [f x | x <- xs]

instance Mappable [] where
  map' _ [] = []
  map' f xs = [f x | x <- xs]

Cat list

Cat list

λ> a = Has $ Cat "cat"
λ> :type a
a :: Box Cat
λ> :type map' toDog a
map' toDog a :: Box Dog
λ> :type map toDog [Cat "a", Cat "b"]
map toDog [Cat "a", Cat "b"] :: [Dog]

toDog :: Cat -> Dog
toDog (Cat name) = Dog name

convertAllCats :: Mappable box => box Cat -> box Dog
convertAllCats cs = map' toDog cs

Any “box” can go in 👍

convertAllCats :: Mappable box => box Cat -> box Dog
convertAllCats cs = map' toDog cs

λ> convertAllCats [Cat "a", Cat "b"]
[Dog "a",Dog "b"]
λ> convertAllCats $ Has $ Cat "a"
Has (Dog "a")

Cat merge

Cat merge

merge :: Cat -> Cat -> Dog
merge (Cat c1)  (Cat c2) = Dog $ c1 <> " & " <> c2

catA = Cat "Meow"
catB = Cat "Nyaa"

λ> merge catA catB
Dog "Meow & Nyaa"

Cat merge

boxA  = Has $ Cat "Meow" :: Box Cat
boxB  = Has $ Cat "Nyaa" :: Box Cat
empty = Empty            :: Box Cat

Cat merge

boxA  = Has $ Cat "Meow" :: Box Cat
boxB  = Has $ Cat "Nyaa" :: Box Cat
empty = Empty            :: Box Cat

Cat merge

λ> merge boxA catB

<interactive>:7:7: error:
    • Couldn't match expected type ‘Cat’ with actual type ‘Box Cat’
    • In the first argument of ‘merge’, namely ‘boxA’
      In the expression: merge boxA catB
      In an equation for ‘it’: it = merge boxA catB

boxA  = Has $ Cat "Meow" :: Box Cat
boxB  = Has $ Cat "Nyaa" :: Box Cat
empty = Empty            :: Box Cat

Cat merge

merge' :: Box Cat -> Box Cat -> Box Dog
merge' Empty _ = Empty
merge' _ Empty = Empty
merge' (Has c1) (Has c2) = Has $ merge c1 c2

merge' :: Box Cat -> Box Cat -> Box Dog
merge' Empty _ = Empty
merge' _ Empty = Empty
merge' (Has c1) (Has c2) = merge c1 c2

Cat merge

λ> merge' boxA boxB
Has (Dog "Meow & Nyaa")
λ> merge' boxA (Has catB)
Has (Dog "Meow & Nyaa")
λ> merge' boxA empty
Empty

class Appliable box where
  wrap   :: a -> box a
  apply' :: box (a -> b) -> box a -> box b

Wrap apply

class Appliable box where
  wrap   :: a -> box a
  apply' :: box (a -> b) -> box a -> box b

instance Appliable Box where
  wrap = Has
  apply' Empty _ = Empty
  apply' _ Empty = Empty
  apply' (Has f) (Has a) = Has $ f a

instance Appliable [] where
  wrap x = [x]
  apply' [] _ = []
  apply' _ [] = []
  apply' fs xs = [f x | f <- fs, x <- xs]

instance Appliable Box where
  wrap = Has
  apply' Empty _ = Empty
  apply' _ Empty = Empty
  apply' (Has f) (Has a) = Has $ f a

instance Appliable [] where
  wrap x = [x]
  apply' [] _ = []
  apply' _ [] = []
  apply' fs xs = [f x | f <- fs, x <- xs]

λ> apply' (apply' (wrap merge) boxA) boxB
Has (Dog "Meow & Nyaa")
λ> apply' (apply' (wrap merge) boxA) empty
Empty

instance Appliable Box where
  wrap = Has
  apply' Empty _ = Empty
  apply' _ Empty = Empty
  apply' (Has f) (Has a) = Has $ f a

instance Appliable [] where
  wrap x = [x]
  apply' [] _ = []
  apply' _ [] = []
  apply' fs xs = [f x | f <- fs, x <- xs]

λ> apply' (apply' (wrap merge) [Cat "A"]) [Cat "B1", Cat "B2"]
[Dog "A & B1",Dog "A & B2"]
λ> apply' (apply' (wrap merge) [Cat "A"]) []
[]

λ> apply' (apply' (wrap merge) [Cat "A"]) [Cat "B1", Cat "B2"]
[Dog "A & B1",Dog "A & B2"]
λ> apply' (apply' (wrap merge) [Cat "A"]) []
[]

λ> wrap merge `apply'` [Cat "A"] `apply'` [Cat "B1", Cat "B2"]
[Dog "A & B1",Dog "A & B2"]
λ> wrap merge `apply'` [Cat "A"] `apply'` []
[]

λ> wrap merge `apply'` [Cat "A"] `apply'` [Cat "B1", Cat "B2"]
[Dog "A & B1",Dog "A & B2"]
λ> wrap merge `apply'` [Cat "A"] `apply'` []
[]

λ> wrap merge `apply'` boxA `apply'` boxB
Has (Dog "Meow & Nyaa")
λ> wrap merge `apply'` boxA `apply'` empty
Empty

Wrap apply

Wrap apply

merge4 :: Cat -> Cat -> Cat -> Cat -> Dog
merge4 (Cat a) (Cat b) (Cat c) (Cat d) = Dog $ a <> b <> c <> d

λ> wrap merge4 `apply'` boxA `apply'` boxB `apply'` boxA `apply'` boxB
Has (Dog "MeowNyaaMeowNyaa")
λ> wrap merge4 `apply'` boxA `apply'` boxB `apply'` empty `apply'` boxB
Empty

killOrSave :: Cat -> Box Cat
killOrSave cat@(Cat name) = case name of
  "Meow" -> Empty
  _      -> Has cat

(no animals were harmed in the making of this slideshow)

killOrSave :: Cat -> Box Cat
killOrSave cat@(Cat name) = case name of
  "Meow" -> Empty
  _      -> Has cat

Kill

Kill

λ> wrap killOrSave `apply'` Has (Cat "Meow")
Has Empty
λ> wrap killOrSave `apply'` Has (Cat "Nyaa")
Has (Has (Cat "Nyaa"))
λ> wrap killOrSave `apply'` Empty
Empty

class Chainable box where
  chain :: box a -> (a -> box b) -> box b

class Chainable box where
  chain :: box a -> (a -> box b) -> box b

instance Chainable Box where
  chain Empty _ = Empty
  chain (Has a) f = f a

instance Chainable [] where
  chain [] _ = []
  chain xs f = [x' | x <- xs, x' <- f x]

instance Chainable Box where
  chain Empty _ = Empty
  chain (Has a) f = f a

instance Chainable [] where
  chain [] _ = []
  chain xs f = [x' | x <- xs, x' <- f x]

Save-kill

Save-kill

kill :: Cat -> Box Cat
kill _ = Empty

save :: Cat -> Box Cat
save = Has

Save-kill

λ> boxA `chain` save `chain` save `chain` save
Has (Cat "Meow")
λ> boxA `chain` save `chain` kill `chain` save
Empty

kill' :: Cat -> [Cat]
kill' _ = []

save' :: Cat -> [Cat]
save' c = [c]

clone :: Cat -> [Cat]
clone (Cat c) = [Cat $ c <> "L" , Cat $ c <> "R"]

kill' :: Cat -> [Cat]
kill' _ = []

save' :: Cat -> [Cat]
save' c = [c]

clone :: Cat -> [Cat]
clone (Cat c) = [Cat $ c <> "L" , Cat $ c <> "R"]

λ> [catA] `chain` save' `chain` save' `chain` save'
[Cat "Meow"]
λ> [catA] `chain` save' `chain` clone `chain` save'
[Cat "MeowL",Cat "MeowR"]
λ> [catA] `chain` more' `chain` clone `chain` save'
[Cat "MeowLL",Cat "MeowLR",Cat "MeowRL",Cat "MeowRR"]
λ> [catA] `chain` clone `chain` clone `chain` kill'
[]

λ> [catA] `chain` save' `chain` save' `chain` save'
[Cat "Meow"]
λ> [catA] `chain` save' `chain` clone `chain` save'
[Cat "MeowL",Cat "MeowR"]
λ> [catA] `chain` more' `chain` clone `chain` save'
[Cat "MeowLL",Cat "MeowLR",Cat "MeowRL",Cat "MeowRR"]
λ> [catA] `chain` clone `chain` clone `chain` kill'
[]

Chain list

Callbacks

This was all dream

What’s a Mappable?

It’s a Functor

It’s a Functor

It’s a Functor

class Mappable box where
  map' :: (a -> b) -> box a -> box b

class Functor f where
  fmap :: (a -> b) -> f a -> f b
  (<$) :: a -> f b -> f a

It’s a Functor

λ> map' (+1) []
[]
λ> map' (+1) [1, 2, 3]
[2,3,4]
λ> map' (*3) [1, 2, 3]
[3,6,9]

λ> fmap (+1) []
[]
λ> fmap (+1) [1, 2, 3]
[2,3,4]
λ> fmap (*3) [1, 2, 3]
[3,6,9]
λ> 1 <$ [1, 2, 3]
[1,1,1]

It’s a Functor

Tardis

It’s a Functor

Tardis inside

main :: IO ()
main = print "Hello World!"

Tardis inside

What’s a Appliable?

It’s an Applicative

It’s an applicative

It’s an Applicative

class Appliable box where
  wrap   :: a -> box a
  apply' :: box (a -> b) -> box a -> box b

class (Functor f) => Applicative f where
  pure  :: a -> f a
  (<*>) :: f (a -> b) -> f a -> f b

It’s an Applicative

λ> wrap merge `apply'` [catA] `apply'` []
[]
λ> wrap merge `apply'` [catA] `apply'` [catB]
[Dog "Meow & Nyaa"]
λ> wrap merge `apply'` [catA] `apply'` [catB, Cat "C"]
[Dog "Meow & Nyaa",Dog "Meow & C"]

λ> pure merge <*> [catA] <*> []
[]
λ> pure merge <*> [catA] <*> [catB]
[Dog "Meow & Nyaa"]
λ> pure merge <*> [catA] <*> [catB, Cat "C"]
[Dog "Meow & Nyaa",Dog "Meow & C"]

λ> pure merge <*> [catA] <*> []
[]
λ> pure merge <*> [catA] <*> [catB]
[Dog "Meow & Nyaa"]
λ> pure merge <*> [catA] <*> [catB, Cat "C"]
[Dog "Meow & Nyaa",Dog "Meow & C"]

λ> merge <$> [catA] <*> []
[]
λ> merge <$> [catA] <*> [catB]
[Dog "Meow & Nyaa"]
λ> merge <$> [catA] <*> [catB, Cat "C"]
[Dog "Meow & Nyaa",Dog "Meow & C"]

merge :: Cat -> Cat -> Dog

_ = merge catA catB

What’s a Chainable?

It’s a monad

It’s a monad

It’s a monad

class Chainable box where
  chain :: box a -> (a -> box b) -> box b

class Applicative m => Monad m where
  (>>=) :: m a -> (a -> m b) -> m b

  (>>) :: m a -> m b -> m b
  m >> k = m >>= \_ -> k

  return :: a -> m a
  return = pure

It’s a monad

λ> [catA] `chain` save' `chain` save' `chain` save'
[Cat "Meow"]
λ> [catA] `chain` save' `chain` clone `chain` save'
[Cat "MeowL",Cat "MeowR"]
λ> [catA] `chain` save' `chain` clone `chain` kill'
[]

λ> [catA] >>= save' >>= save' >>= save'
[Cat "Meow"]
λ> [catA] >>= save' >>= clone >>= save'
[Cat "MeowL",Cat "MeowR"]
λ> [catA] >>= save' >>= clone >>= kill'
[]

Just one more typeclass bro

More here:

  • Learn You a Haskell for Great Good! by Miran Lipovaca
  • Programming in Haskell - 2nd Edition by Graham Hutton
  • Effective Haskell by Rebecca Skinner