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chapter-03.hs
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chapter-03.hs
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import Test.QuickCheck
-- Exercise 3.1
original_xor True x = not x
original_xor False x = x
new_xor x y = (x && not y) || (not x && y)
-- Exercise 3.2
{- Box Diagram
+-----+
x ------------------------>| && |-----------------+
| +-----+ |
| ^ |
| | |
| +-----+ | |
+--------->| not |-----------+ | |
+-----+ | | |
| | +-----+
| | | || |
| | +-----+
+-----+ | | |
+ -------->| not |-----------+--+ |
| +-----+ | |
| | |
| v |
| +-----+ |
y ------------------------>| && |-----------------+
+-----+
-}
-- Exercise 3.3
literal_xor True True = False
literal_xor True False = True
literal_xor False True = True
literal_xor False False = False
-- Exercise 3.4
my_and True True = True
my_and True False = False
my_and False True = False
my_and False False = False
-- Exercise 3.5
nAnd_1 True True = False
nAnd_1 True False = True
nAnd_1 False True = True
nAnd_1 False False = True
nAnd_2 x y = not (x && y)
{-
Diagram
x ---------+
|
v
+-----+ +-----+
| && |------>| not |------>
+-----+ +-----+
^
|
y ---------+
-}
-- Exercise 3.6
{- Line-by-line calculations of
nAnd_2 True True
~ not (True && True)
~ not (True)
~ False
nAnd_2 True False
~ not (True && False)
~ not (False)
~ True
-}
-- Exercise 3.7
-- Quickcheck of the functions above
prop_xors x y = (original_xor x y == new_xor x y) &&
(original_xor x y == literal_xor x y)
-- Exercise 3.8
mystery :: Integer -> Integer -> Integer -> Bool
mystery m n p = not ((m == n) && (n == p))
{-
ghci> mystery 3 4 5
True
ghci> mystery 3 3 3
False
ghci> mystery 3 3 1
True
mystery tells us that the 3 numbers are different.
-}
-- Exercise 3.9
threeDifferent :: Integer -> Integer -> Integer -> Bool
threeDifferent m n p = (m /= n) && (m /= p) && (n /= p)
{-
ghci> threeDifferent 3 4 5
True
ghci> threeDifferent 3 4 3
False
ghci> threeDifferent 5 4 5
False
ghci>
-}
-- Exercise 3.10
threeEqual :: Integer -> Integer -> Integer -> Bool
threeEqual m n p = (m == n) && (n == p)
fourDifferent :: Integer -> Integer -> Integer -> Integer -> Bool
fourDifferent m n p q = threeDifferent m n p && threeDifferent n p q && threeDifferent m p q
-- fourDifferent' m n p q = not threeEqual
-- Exercise 3.11
-- Exercise 3.12
-- Started out with this:
prop_threeDifferent m n p = (threeDifferent m n p == threeDifferent n p m)
&& (threeDifferent m n p == threeDifferent p m n)
&& (threeDifferent m n p == threeDifferent m p n)
&& (threeDifferent m n p == threeDifferent n m p)
&& (threeDifferent m n p == threeDifferent p n m)
&& (threeDifferent m n n == False)
&& (threeDifferent m m n == False)
&& (threeDifferent m m m == False)
&& (threeDifferent m (m + 1) (m + 2) == True)
prop_fourDifferent m n p q = (fourDifferent m n p q == fourDifferent m n q p)
&& (fourDifferent m n p q == fourDifferent m p n q)
-- ... there are 24 of these checks based on combinations
-- of m n p q
&& (fourDifferent m n n n == False)
&& (fourDifferent m m n n == False)
&& (fourDifferent m m m n == False)
&& (fourDifferent m m m m == False)
&& (fourDifferent m (m + 1) (m + 2) (m + 3) == True)
-- Exercise 3.13
{-
ghci> max (3-2) (3*8)
24
ghci> maxTree (4+5) (2*6) (100 `div` 7)
14
-}
-- Exercise 3.14
min' :: Int -> Int -> Int
min' x y
| x < y = x
| otherwise = y
minThree :: Int -> Int -> Int -> Int
minThree x y z
| x <= y && x <= z = x
| y <= z = y
| otherwise = z
-- Exercise 3.15
prop_min' x y = (min' x y == min' y x)
&& (min' x (x + 1) == x)
&& (min' (x + 1) x == x)
prop_minThree x y z = (minThree x y z == minThree x z y)
&& (minThree x y z == minThree y x z)
&& (minThree x y z == minThree z x y)
&& (minThree x y z == minThree z y x)
&& (minThree x (x + 1) (x + 2) == x)
-- Exercise 3.16
offset :: Int
offset = fromEnum 'A' - fromEnum 'a'
toUpper :: Char -> Char
toUpper ch = toEnum(fromEnum ch + offset)
isLower :: Char -> Bool
isLower ch = 'a' <= ch && ch <= 'z'
smallToCapitals :: Char -> Char
smallToCapitals ch
| isLower ch = toUpper ch
| otherwise = ch
-- Exercise 3.17
isDigit :: Char -> Bool
isDigit ch = ('0' <= ch) && (ch <= '9')
charToNum :: Char -> Int
charToNum ch
| isDigit ch = fromEnum ch - fromEnum '0'
| otherwise = 0
-- Exercise 3.18
onThreeLines :: String -> String -> String -> String
onThreeLines a b c = a ++ "\n" ++ b ++ "\n" ++ c ++ "\n"
-- Exercise 3.19
romanDigit :: Char -> String
romanDigit ch
| ch == '1' = "I"
| ch == '2' = "II"
| ch == '3' = "III"
| ch == '4' = "IV"
| ch == '5' = "V"
| ch == '6' = "VI"
| ch == '7' = "VII"
| ch == '8' = "VIII"
| ch == '9' = "IX"
| otherwise = "???"
-- Exercise 3.20
averageThree :: Integer -> Integer -> Integer -> Float
averageThree x y z = ( fromInteger x + fromInteger y + fromInteger z) / 3.0
howManyAboveAverage :: Integer -> Integer -> Integer -> Integer
howManyAboveAverage x y z = (if (fromInteger x > averageThree x y z) then 1 else 0)
+ (if (fromInteger y > averageThree x y z) then 1 else 0)
+ (if (fromInteger z > averageThree x y z) then 1 else 0)
-- Exercise 3.21
-- QuickChecks
prop_averageThree x y z = (averageThree x x x == fromInteger(x))
&& (averageThree x (x + 1) (x + 2) == fromInteger(x + 1))
&& (averageThree x y z == averageThree x z y)
&& (averageThree x y z == averageThree y x z)
&& (averageThree x y z == averageThree y z x)
&& (averageThree x y z == averageThree z y x)
prop_hasManyAboveAverage x y z = (howManyAboveAverage x x x == 0)
&& (howManyAboveAverage x (x + 1) (x + 2) == 1)
&& (howManyAboveAverage x (x + 100) (x + 101) == 2)
&& (howManyAboveAverage x y z == howManyAboveAverage x z y)
&& (howManyAboveAverage x y z == howManyAboveAverage y x z)
&& (howManyAboveAverage x y z == howManyAboveAverage y z x)
&& (howManyAboveAverage x y z == howManyAboveAverage z y x)
-- Exercise 3.22
numberNDroots :: Float -> Float -> Float -> Integer
numberNDroots a b c
| b * b > 4.0 * a * c = 2
| b * b == 4.0 * a * c = 1
| b * b < 4.0 * a * c = 0
-- Exercise 3.23
numberRoots :: Float -> Float -> Float -> Integer
numberRoots a b c
| a /= 0 = numberNDroots a b c
| otherwise = if b /= 0.0 then 1
else if b == 0.0 && c /= 0.0 then 0
else 3
-- Exercise 3.24
smallerRoot:: Float -> Float -> Float -> Float
smallerRoot a b c = (-b - sqrt(b*b - 4.0 * a * c))/(2*a)
largerRoot:: Float -> Float -> Float -> Float
largerRoot a b c = (-b + sqrt(b*b - 4.0 * a * c))/(2*a)
-- Exercise 3.25
-- Quickchecks
prop_roots a b c = (smallerRoot a b c <= largerRoot a b c)
-- TODO: doesn't work for imaginary roots
-- Exercise 3.26
-- Rewrite solutions using recommended layout
-- Exercise 3.27
funny x = x * x
peculiar y = y
-- removing the space fixes the "parse error on input"