module TypesPractice where
import Prelude hiding ((.),Maybe(..),even,flip,map)
— * Notes
— 1. On the quiz, I’ll only give you the types of relevant functions, not
— their implementations. I assume that you know Haskell’s Bool and list
— data types, and so won’t repeat their definitions here or on the quiz.
— All other data types and relevant function types will be provided.
—
— 2. There are three kinds of problems, broken into three sets.
—
— 3. All problems are commented out since some are not type correct. You can
— check your answers using the instructions at the beginning of each set.
—
— 4. Many of these problems (especially near the end of the first set) are
— more difficult than the problems you’ll see on the quiz. So if you feel
— comfortable with all of the problems in this practice, you’re in good shape.
—
— 5. You can pretty easily generate your own typing problems by just combining
— these functions and data constructors in various ways.
— * Definitions
data Maybe a
= Nothing
| Just a
deriving (Eq,Show)
data Tree a b
= Leaf a
| Node b (Tree a b) (Tree a b)
deriving (Eq,Show)
one :: Int
one = 1
two :: Int
two = 2
even :: Int -> Bool
even i = mod i 2 == 0
bit :: Bool -> Int
bit b = if b then 1 else 0
gt :: Int -> Int -> Bool
gt x y = x > y
flip :: (a -> b -> c) -> b -> a -> c
flip f b a = f a b
(.) :: (b -> c) -> (a -> b) -> a -> c
f . g = \x -> f (g x)
infixr 9 . — this says . is right-associative
map :: (a -> b) -> [a] -> [b]
map _ [] = []
map f (x:xs) = f x : map f xs
treeMap :: (a -> c) -> (b -> d) -> Tree a b -> Tree c d
treeMap f _ (Leaf b) = Leaf (f b)
treeMap f g (Node a l r) = Node (g a) (treeMap f g l) (treeMap f g r)
— * Problems
— ** Determine the type
— In this problem set, you’re given an expression and have to determine its
— type. If the expression is type correct, write down its type. If not,
— write “type error”.
—
— You can check your answer for a problem by uncommenting the expression and
— reloading the file in GHCi. If you do not get a type error, you can check
— the type of the expression with the :type command.
ex1 :: Tree Int b
ex1 = Leaf one
ex2 :: [Maybe Bool]
ex2 = [Just True, Nothing]
— type error!
— ex3 = [Leaf one, Leaf True]
ex4 :: [Tree Int (Maybe a)]
ex4 = [Leaf one, Node Nothing (Leaf one) (Leaf two)]
— type error!
— ex5 = Just bit True
ex6 :: Bool -> Maybe Int
ex6 = Just . bit
ex7 :: Tree (Tree Int b1) b2
ex7 = Leaf (Leaf one)
ex8 :: Int -> Int
ex8 = bit . even
— type error!
— ex9 = even . gt one
ex10 :: Bool -> Bool
ex10 = gt one . bit
ex11 :: Int -> Tree Int Bool
ex11 = Node True (Leaf one) . Leaf
— type error!
— ex12 = flip Just
ex13 :: (Int -> b) -> [b]
ex13 = flip map [one,two]
ex14 :: Tree Bool Int -> Tree Int Bool
ex14 = treeMap bit even
ex15 :: Tree Int Bool -> Tree Bool Int
ex15 = flip treeMap bit even
— type error!
— ex16 = flip (treeMap bit) (Leaf one)
ex17 :: (b -> d) -> Tree Bool d
ex17 = flip (treeMap even) (Leaf one)
ex18 :: Int -> Int
ex18 = flip (.) even bit
— ** Create an expression
— In this problem set, you’re given a type. Your goal is to construct an
— expression of the given type using *only* the definitions in this file.
— If this is possible, write the expression. If it’s not, write “impossible”.
—
— For types that are not impossible, you can check your answers by adding
— the definitions to the file without changing the type. If the file
— loads successfully in GHCi, your answer is correct.
ex19 :: Maybe (Bool -> Int)
ex19 = Just bit
ex20 :: Maybe (Maybe a)
ex20 = Just Nothing
ex21 :: Maybe (Maybe Int)
ex21 = Just (Just one)
— ex22 :: Tree Int
— impossible!
ex23 :: Tree (Maybe a) b
ex23 = Leaf Nothing
ex24 :: [Int] -> [Bool]
ex24 = map even
ex25 :: Tree a Bool -> Tree (Maybe a) Int
ex25 = treeMap Just bit
ex26 :: Tree Int b -> (b -> d) -> Tree Bool d
ex26 = flip (treeMap even)
ex27 :: (a -> c) -> Tree a Int -> Tree c Bool
ex27 = flip treeMap even
— ** Figure out the type of “secret”
— In this problem set, you’re given a type and an expression with an
— undefined ‘secret’ value. Determine whether the expression can possibly
— have the indicated type. If so, write the type of ‘secret’ that makes the
— expression type correct. Note that you do *not* need to define the value
— of ‘secret’, only its type. If the expression cannot possibly be type
— correct, write “impossible”.
—
— For expressions that are not impossible, you can check your answers by
— adding an expicit type annotation to the undefined secret value as
— illustrated below. If the file loads successfully in GHCi, your answer is
— correct.
— | I’ve given you the answer for this one already so you can see how to check
— your own solutions in GHCi.
ex28 :: Maybe Int
ex28 = Just secret
where secret = undefined :: Int
ex29 :: Tree Char Bool
ex29 = Leaf secret
where secret = undefined :: Char
ex30 :: Bool
ex30 = even secret
where secret = undefined :: Int
— ex31 :: Bool -> Int
— ex31 = bit secret
— where secret = undefined — impossible!
ex32 :: Char -> Bool -> Int
ex32 = flip secret
where secret = undefined :: Bool -> Char -> Int
ex33 :: [Char]
ex33 = map secret [one,two]
where secret = undefined :: Int -> Char
ex34 :: Tree Bool Bool -> Tree Int Char
ex34 = treeMap bit secret
where secret = undefined :: Bool -> Char
ex35 :: [Char]
ex35 = map secret (map even [one,two])
where secret = undefined :: Bool -> Char
ex36 :: [Bool]
ex36 = map even (map secret [one,two])
where secret = undefined :: Int -> Int