We continue our introduction to Haskell. Today we talk about parametric types and list comprehensions.

1. Haskell
A Whirlwind Tour
(Part II)
William Taysom ~ 2011

3. Haskell is a non-strict,
purely functional
programming language
with strong,
static type inference.

4. Review

5. Recursive Data
data Color = Red | Green | Blue
| Mix Color Color

6. Recursive Functions
hue :: Color -> Double
hue Red =0
hue Green = 120
hue Blue = 240

7. h' = 60
hue (Mix c c') = let
h = hue c
h' = hue c'
m = average h h' m = 180
m' = norm (m + 180) m' = 0
d = distance h m
in case compare d 90 of
LT -> m
EQ -> nan h = 300
GT -> m'
norm h | h < 360 = h
| otherwise = norm (h - 360)

8. Parametric
Types

9. Tuple (Product)
data Point = Point Double Double

10. Tuple (Product)
data Pair a b = Pair a b

11. Tuple (Product)
data (a, b) = (a, b)
-- Built-in syntax:
-- definition only for illustrative purposes.

12. Tuple (Product)
ghci> (Blue, False)
data (a, b) = (a, b)
-- Built-in syntax:
-- definition only for illustrative purposes.

13. ghci> (Blue, False)
(Blue,False)
ghci>

14. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False

15. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False
(Blue,False)
ghci>

16. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False
(Blue,False)
ghci> :t it

17. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False
(Blue,False)
ghci> :t it
it :: (Color, Bool)
ghci>

18. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False
(Blue,False)
ghci> :t it
it :: (Color, Bool)
ghci> :t (,)

19. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False
(Blue,False)
ghci> :t it
it :: (Color, Bool)
ghci> :t (,)
(,) :: a -> b -> (a, b)
ghci>

20. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False
(Blue,False)
ghci> :t it
it :: (Color, Bool)
ghci> :t (,)
(,) :: a -> b -> (a, b)
ghci> :kind (,)

21. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False
(Blue,False)
ghci> :t it
it :: (Color, Bool)
ghci> :t (,)
(,) :: a -> b -> (a, b)
ghci> :kind (,)
(,) :: * -> * -> *
ghci>

22. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False
(Blue,False)
ghci> :t it
it :: (Color, Bool)
ghci> :t (,)
(,) :: a -> b -> (a, b)
ghci> :kind (,)
(,) :: * -> * -> *
ghci> :k Color

23. ghci> (Blue, False)
(Blue,False)
ghci> (,) Blue False
(Blue,False)
ghci> :t it
it :: (Color, Bool)
ghci> :t (,)
(,) :: a -> b -> (a, b)
ghci> :kind (,)
(,) :: * -> * -> *
ghci> :k Color
Color :: *
ghci>

24. Color :: *
ghci>

25. Color :: *
ghci> :t (Red, Blue, Green)

26. Color :: *
ghci> :t (Red, Blue, Green)
(Red, Blue, Green) :: (Color, Color, Color)
ghci>

27. Color :: *
ghci> :t (Red, Blue, Green)
(Red, Blue, Green) :: (Color, Color, Color)
ghci> :t (,,)

28. Color :: *
ghci> :t (Red, Blue, Green)
(Red, Blue, Green) :: (Color, Color, Color)
ghci> :t (,,)
(,,) :: a -> b -> c -> (a, b, c)
ghci>

29. Color :: *
ghci> :t (Red, Blue, Green)
(Red, Blue, Green) :: (Color, Color, Color)
ghci> :t (,,)
(,,) :: a -> b -> c -> (a, b, c)
ghci> :t (,,,)

30. Color :: *
ghci> :t (Red, Blue, Green)
(Red, Blue, Green) :: (Color, Color, Color)
ghci> :t (,,)
(,,) :: a -> b -> c -> (a, b, c)
ghci> :t (,,,)
(,,,) :: a -> b -> c -> d -> (a, b, c, d)
ghci>

31. Color :: *
ghci> :t (Red, Blue, Green)
(Red, Blue, Green) :: (Color, Color, Color)
ghci> :t (,,)
(,,) :: a -> b -> c -> (a, b, c)
ghci> :t (,,,)
(,,,) :: a -> b -> c -> d -> (a, b, c, d)
ghci> :t ()

32. Color :: *
ghci> :t (Red, Blue, Green)
(Red, Blue, Green) :: (Color, Color, Color)
ghci> :t (,,)
(,,) :: a -> b -> c -> (a, b, c)
ghci> :t (,,,)
(,,,) :: a -> b -> c -> d -> (a, b, c, d)
ghci> :t ()
() :: ()
ghci>

33. Color :: *
ghci> :t (Red, Blue, Green)
(Red, Blue, Green) :: (Color, Color, Color)
ghci> :t (,,)
(,,) :: a -> b -> c -> (a, b, c)
ghci> :t (,,,)
(,,,) :: a -> b -> c -> d -> (a, b, c, d)
ghci> :t ()
() :: ()
ghci> -- Trivial is similar to void.

34. Either (Sum)
Color :: *
ghci> :t (Red, Blue, Green)
(Red, Blue, Green) :: (Color, Color, Color)
ghci>Either a b = Left a | Right b
data :t (,,)
(,,) :: a -> b -> c -> (a, b, c)
ghci> :t (,,,)
(,,,) :: a -> b -> c -> d -> (a, b, c, d)
ghci> :t ()
() :: ()
ghci> -- Trivial is similar to void.
ghci>

35. Either (Sum)
data Either a b = Left a | Right b

36. Maybe (Optional)
data Maybe a = Nothing | Just a

37. hue :: Color -> Double
hue Red =0
hue Green = 120
hue Blue = 240
hue (Mix c c') = ... sometimes NaN

38. hue :: Color -> Double
hue Red =0
hue Green = 120
hue Blue = 240
hue (Mix c c') = ... sometimes nothing

39. hue :: Color -> Maybe Double
hue Red =0
hue Green = 120
hue Blue = 240
hue (Mix c c') = ... sometimes nothing

40. hue :: Color -> Maybe Double
hue Red = Just 0
hue Green = Just 120
hue Blue = Just 240
hue (Mix c c') = ... sometimes nothing

41. hue :: Color -> Maybe Double
hue Red = Just 0
hue Green = Just 120
hue Blue = Just 240
hue (Mix c c') = let
... stuff ...
in case compare d 90 of
LT -> m
EQ -> nan
GT -> m'

42. hue :: Color -> Maybe Double
hue Red = Just 0
hue Green = Just 120
hue Blue = Just 240
hue (Mix c c') = let
... stuff ...
in case compare d 90 of
LT -> Just m
EQ -> nan
GT -> Just m'

43. hue :: Color -> Maybe Double
hue Red = Just 0
hue Green = Just 120
hue Blue = Just 240
hue (Mix c c') = let
... stuff ...
in case compare d 90 of
LT -> Just m
EQ -> Nothing
GT -> Just m'

44. hue :: Color -> Maybe Double
hue Red = Just 0
hue Green = Just 120
hue Blue = Just 240
hue (Mix c c') = let
h = hue c
h' = hue c'
... stuff ...
in case compare d 90 of
LT -> Just m
EQ -> Nothing
GT -> Just m'

45. hue (Mix c c') = let
h = hue c
h' = hue c'
...

46. hue (Mix c c') = let
h = hue c
h' = hue c'
...

47. hue (Mix c c') = let
(h, h') = (hue c, hue c')
...

48. hue (Mix c c') = case (hue c, hue c') of
(h, h') -> let
...

49. hue (Mix c c') = case (hue c, hue c') of
(Just h, Just h') -> let
...

50. hue (Mix c c') = case (hue c, hue c') of
(Just h, Just h') -> let
...
(Just h, Nothing) -> ...
(Nothing, Just h') -> ...
(Nothing, Nothing) -> ...

51. hue (Mix c c') = case (hue c, hue c') of
(Just h, Just h') -> let
...
(Just h, Nothing) -> Nothing
(Nothing, Just h') -> Nothing
(Nothing, Nothing) -> Nothing

52. hue (Mix c c') = case (hue c, hue c') of
(Just h, Just h') -> let
...
_ -> Nothing

53. hue (Mix c c') = case (hue c, hue c') of
(Just h, Just h') -> let
m = average h h'
m' = norm (m + 180)
d = distance h m
in case compare d 90 of
LT -> Just m
EQ -> Nothing
GT -> Just m'
_ -> Nothing

54. Parametric
Polymorphism
id x = x
const x _ = x
flip f x y = f y x

55. Parametric
Polymorphism
id :: a -> a
id x = x
const x _ = x
flip f x y = f y x

56. Parametric
Polymorphism
id :: a -> a
id x = x
const :: a -> b -> a
const x _ = x
flip f x y = f y x

57. Parametric
Polymorphism
id :: a -> a
id x = x
const :: a -> b -> a
const x _ = x
flip :: (a -> b -> c) -> b -> a -> c
flip f x y = f y x

58. Parametric
Polymorphism
infixr . -- defaults to 9
(f . g) x = f (g x)

59. Parametric
Polymorphism
(.) :: (b -> c) -> (a -> b) -> a -> c
infixr . -- defaults to 9
(f . g) x = f (g x)

60. Parametric
Polymorphism
(.) :: (b -> c) -> (a -> b) -> a -> c
infixr . -- defaults to 9
(f . g) x = f (g x)
isN :: Double -> Bool
isN = not . isNaN

61. Parametric
Polymorphism
(.) :: (b -> c) -> (a -> b) -> a -> c
infixr . -- defaults to 9
(f . g) x = f (g x)
isN :: Double -> Bool
isN = not . isNaN
celsiusToFahrenheit :: Double -> Double
celsiusToFahrenheit t = 9/5 * t + 32

62. Parametric
Polymorphism
(.) :: (b -> c) -> (a -> b) -> a -> c
infixr . -- defaults to 9
(f . g) x = f (g x)
isN :: Double -> Bool
isN = not . isNaN
celsiusToFahrenheit :: Double -> Double
celsiusToFahrenheit t = 9/5 * t + 32
celsiusToFahrenheit' = (32 +) . (9/5 *)

63. Parametric
Polymorphism
infixr 0 $ -- very low precedence.
f$x=fx

64. Parametric
Polymorphism
($) :: (a -> b) -> a -> b
infixr 0 $ -- very low precedence.
f$x=fx

65. Parametric
Polymorphism
($) :: (a -> b) -> a -> b
infixr 0 $ -- very low precedence.
f$x=fx
-- Allows you to drop parenthesis:
n = sqrt ( abs ( cos 1))
n' = sqrt $ abs $ cos 1

66. List (Stream)
data List a = Nil | Cons a (List a)

67. List (Stream)
data [a] = [] | a:[a]
-- Built-in syntax:
-- definition only for illustrative purposes.

68. List (Stream)
ghci> [1,2,3]
data [a] = [] | a:[a]
-- Built-in syntax:
-- definition only for illustrative purposes.

69. ghci> [1,2,3]
[1,2,3]
ghci>

70. ghci> [1,2,3]
[1,2,3]
ghci> 1:[2,3]

71. ghci> [1,2,3]
[1,2,3]
ghci> 1:[2,3]
[1,2,3]
ghci>

72. ghci> [1,2,3]
[1,2,3]
ghci> 1:[2,3]
[1,2,3]
ghci> 1:2:3:[]

73. ghci> [1,2,3]
[1,2,3]
ghci> 1:[2,3]
[1,2,3]
ghci> 1:2:3:[]
[1,2,3]
ghci>

74. ghci> [1,2,3]
[1,2,3]
ghci> 1:[2,3]
[1,2,3]
ghci> 1:2:3:[]
[1,2,3]
ghci> :t (:)

75. ghci> [1,2,3]
[1,2,3]
ghci> 1:[2,3]
[1,2,3]
ghci> 1:2:3:[]
[1,2,3]
ghci> :t (:)
(:) :: a -> [a] -> [a]
ghci>

76. ghci> [1,2,3]
[1,2,3]
ghci> 1:[2,3]
[1,2,3]
ghci> 1:2:3:[]
[1,2,3]
ghci> :t (:)
(:) :: a -> [a] -> [a]
ghci> :t []

77. Characters
ghci> [1,2,3]
[1,2,3]
ghci> 1:[2,3]
[1,2,3]
c = 'c'
ghci> 1:2:3:[]
six = '6'
[1,2,3]
tab = 't'
ghci> :t (:)
(:) :: aisAlpha, isDigit :: Char -> Bool
isSpace, -> [a] -> [a]
ghci> Char -> Int
ord :: :t []
[] :: Int -> Char
chr :: [a]
ghci>

78. Characters
c = 'c'
six = '6'
tab = 't'
isSpace, isAlpha, isDigit :: Char -> Bool
ord :: Char -> Int
chr :: Int -> Char

79. Strings (Type Synonym)
type String = [Char]

80. Strings (Type Synonym)
ghci> :info String
type String = [Char]

81. ghci> :info String
type String = [Char] ! Defined in
--
GHC.Base
ghci>

82. ghci> :info String
type String = [Char] ! Defined in
--
GHC.Base
ghci> ['h', 'e', 'l', 'l', 'o']

83. ghci> :info String
type String = [Char] ! Defined in
--
GHC.Base
ghci> ['h', 'e', 'l', 'l', 'o']
"hello"
ghci>

84. ghci> :info String
type String = [Char] ! Defined in
--
GHC.Base
ghci> ['h', 'e', 'l', 'l', 'o']
"hello"
ghci> :t error

85. ghci> :info String
type String = [Char] ! Defined in
--
GHC.Base
ghci> ['h', 'e', 'l', 'l', 'o']
"hello"
ghci> :t error
error :: [Char] -> a
ghci>

86. ghci> :info String
type String = [Char] ! Defined in
--
GHC.Base
ghci> ['h', 'e', 'l', 'l', 'o']
"hello"
ghci> :t error
error :: [Char] -> a
ghci> error "oops"

87. Arithmetic Sequences
ghci> :info String
type String = [Char] ! Defined in
--
GHC.Base
ghci> ['h', 'e', 'l', 'l', 'o']
one_ten = [1..10]
"hello"
a_z = ['a'..'z']
ghci> :t error
error :: [Char] -> a
ghci> error "oops"
*** Exception: oops
ghci>

88. Arithmetic Sequences
one_ten = [1..10]
a_z = ['a'..'z']

89. Arithmetic Sequences
nats = [1..]

90. Length
length :: [a] -> Int
length [] =0
length (_:xs)= 1 + length xs
length one_ten --> 10
length a_z --> 26

91. Map
map :: (a -> b) -> [a] -> [b]
map f [] = []
map f (x:xs) = f x : map f xs
map ord "abc" --> [97, 98, 99]

92. Append
(++) :: [a] -> [a] -> [a]
infixr 5 ++
[] ++ ys = ys
(x:xs) ++ ys = x : (xs ++ ys)
"hello" ++ " " ++ "world" --> "hello world"

93. Filter
filter :: (a -> Bool) -> [a] -> [a]
filter p [] = []
filter p (x:xs)
|px = x : filter p xs
| otherwise = filter p xs
filter (> 7) one_ten --> [8, 9, 10]

94. Fold
foldr :: (a -> b -> b) -> b -> [a] -> b
foldr f z [] =z
foldr f z (x:xs) = f x (foldr f x xs)
foldr (*) 1 [1..5] --> 120

95. Concat (Flattening)
concat :: [[a]] -> [a]
concat = foldr (++) []
concat ["hello", ", ", "world"] --> "hello, world"

96. List
Comprehensions

97. List Comprehensions
divides x y = rem y x == 0
divisors x = [d | d <- [1..x], d `divides` x]

98. Generators
[ (a, a, a) | a <- nats]
[(1,1,1),(2,2,2),(3,3,3),⋯]

99. Generators
[ (a, b, c) | a <- nats, b <- nats, c <- nats]
[(1,1,1),(1,1,2),(1,1,3),⋯]

100. Generators
[ (a, b, c) | a <- nats, b <- nats, c <- nats]
[(1,1,1),(1,1,2),(1,1,3),⋯]
(1,2,3) (1,3,2)
(2,1,3) (3,1,2)
(2,3,1) (3,2,1)

101. Generators
[ (a, b, c) | a <- nats, b <- [1..a], c <- [1..b]]
[(1,1,1),(2,1,1),(2,2,1),⋯]

102. Generators
[ (a, b, c) | c <- nats, b <- [1..c], a <- [1..b]]
[(1,1,1),(1,1,2),(1,2,2),⋯]

103. Guards
[ (a, b, c) | c <- nats, b <- [1..c], a <- [1..b],
a^2 + b^2 == c^2]
[(3,4,5),(6,8,10),(5,12,13),⋯]

104. Local Declaration
[ (a, b, c) | c <- nats, b <- [1..c], a <- [1..b],
a^2 + b^2 == c^2,
...
commonDivisors == [1]]
[(3,4,5),(5,12,13),(8,15,17),⋯]

105. Local Declaration
[ (a, b, c) | c <- nats, b <- [1..c], a <- [1..b],
a^2 + b^2 == c^2,
let commonDivisors = [d | d <- divisors a,
d `divides` b, d `divides` c],
commonDivisors == [1]]
[(3,4,5),(5,12,13),(8,15,17),⋯]

106. Local Declaration
[ (a, b, c) | c <- nats, b <- [1..c], a <- [1..b],
a^2 + b^2 == c^2,
let d = gcd a b,
d == 1]
[(3,4,5),(5,12,13),(8,15,17),⋯]

107. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]

108. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
primes

109. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
sieve [2..]

110. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
sieve (2:[3..])

111. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : sieve [x | x <- [3..], rem x 2 /= 0]

112. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : sieve [x | x <- 3:[4..], rem x 2 /= 0]

113. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : sieve (3:[x | x <- [4..], rem x 2 /= 0])

114. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : sieve [x |
x <- [x | x <- [4..], rem x 2 /= 0],
rem x 3 /= 0]

115. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : sieve [x |
x <- [x | x <- 4:[5..], rem x 2 /= 0],
rem x 3 /= 0]

116. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : sieve [x |
x <- [x | x <- [5..], rem x 2 /= 0],
rem x 3 /= 0]

117. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : sieve [x |
x <- [x | x <- 5:[6..], rem x 2 /= 0],
rem x 3 /= 0]

118. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : sieve [x |
x <- 5:[x | x <- [6..], rem x 2 /= 0],
rem x 3 /= 0]

119. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : sieve (5:[x |
x <- [x | x <- [6..], rem x 2 /= 0],
rem x 3 /= 0])

120. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : 5 : sieve [x |
x <- [x |
x <- [x | x <- [6..], rem x 2 /= 0],
rem x 3 /= 0]),
rem x 5 /= 0]

121. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : 5 : sieve [x |
x <- [x |
x <- [x | x <- 6:[7..], rem x 2 /= 0],
rem x 3 /= 0]),
rem x 5 /= 0]

122. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : 5 : sieve [x |
x <- [x |
x <- [x | x <- [7..], rem x 2 /= 0],
rem x 3 /= 0]),
rem x 5 /= 0]

123. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : 5 : sieve [x |
x <- [x |
x <- [x | x <- 7:[8..], rem x 2 /= 0],
rem x 3 /= 0]),
rem x 5 /= 0]

124. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : 5 : sieve [x |
x <- [x |
x <- 7:[x | x <- [8..], rem x 2 /= 0],
rem x 3 /= 0]),
rem x 5 /= 0]

125. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : 5 : sieve [x |
x <- 7:[x |
x <- [x | x <- [8..], rem x 2 /= 0],
rem x 3 /= 0]),
rem x 5 /= 0]

126. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : 5 : sieve (7:[x |
x <- [x |
x <- [x | x <- [8..], rem x 2 /= 0],
rem x 3 /= 0]),
rem x 5 /= 0])

127. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
2 : 3 : 5 : 7 : sieve ⋯

128. Recursion
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]
[2, 3, 5, 7, sieve ⋯]

129. To be continued...

130. Summary
Haskell has parametric types.
List Comprehensions are cool.

131. Preview: Infinite Lists
primes = sieve [2..] where
sieve (p:xs) =
p : sieve [x | x <- xs, rem x p /= 0]

132. Preview: IO (echo.hs)
import System.Environment (getArgs)
import Data.List (intercalate)
main = do
args <- getArgs
putStrLn (intercalate " " args)

133. Preview: Parsers

134. Preview: Parse JSON
data Value = String String
| Number Double
| Object [(String, Value)]
| Array [Value]
| Bool Bool
| Null

135. Preview: Parse JSON
value = String <$>jsstring
<|> Number <$>number
<|> Object <$>commaGroup '{' pair '}'
<|> Array <$>commaGroup '[' value ']'
<|> Bool True <$ string "true"
<|> Bool False <$ string "false"
<|> Null <$ string "null"

136. Preview: Parse JSON
pair :: Parser (String, Value)
pair = do
s <- jsstring
sp_char_sp ':'
v <- value
spaces
return (s, v)

137. To be continued...