Michael Saelee

Feb 6, 2019

Step 1: determine the type Step 2: list all the patterns Step 3: define the trivial cases Step 4: define the hard cases Step 5: generalize and simplify

```
fib :: Integer -> Integer
fib 0 = 0
fib 1 = 1
fib n = fib (n-1) + fib (n-2)
factorial :: Integer -> Integer
factorial 0 = 1
factorial n = n * factorial (n-1)
-- define integral `pow` (exponentiation) in terms of multiplication
pow :: Integral a => a -> a -> a
pow _ 0 = 1
pow n e = n * pow n (e-1)
-- define integral `add` only in terms of succ and pred
add :: Integral a => a -> a -> a
add m 0 = m
add m n | n > 0 = add (succ m) (pred n)
| otherwise = add (pred m) (succ n)
-- define integral `mod` using subtraction
mod' :: Integral a => a -> a -> a
mod' m n | m < n = m
| otherwise = mod' (m-n) n
```

```
last' :: [a] -> a
last' [] = error "empty list"
last' (x:[]) = x
last' (x:xs) = last' xs
(!!!) :: [a] -> Integer -> a
[] !!! _ = error "index too large"
(x:_) !!! 0 = x
(_:xs) !!! n = xs !!! pred n
length' :: [a] -> Int
length' [] = 0
length' (_:xs) = 1 + length' xs
-- elem' :: return True if a given element is found in a list, False otherwise
elem' :: Eq a => a -> [a] -> Bool
_ `elem'` [] = False
x `elem'` (y:ys) = x == y || x `elem'` ys
-- and' :: determine if all values in a list are True
and' :: [Bool] -> Bool
and' [] = True
and' (x:xs) = x && and' xs
-- sum' :: compute sum of a list of numbers
sum' :: Num a => [a] -> a
sum' [] = 0
sum' (x:xs) = x + sum' xs
```

```
(+++) :: [a] -> [a] -> [a]
[] +++ ys = ys
xs +++ [] = xs
(x:xs) +++ ys = x : (xs +++ ys)
take' :: Int -> [a] -> [a]
take' _ [] = []
take' 0 _ = []
take' n (x:xs) = x : (take' (n-1) xs)
drop' :: Int -> [a] -> [a]
drop' _ [] = []
drop' 0 xs = xs
drop' n (_:xs) = drop (n-1) xs
-- replicate' :: create a list of N copies of some value
replicate' :: Int -> a -> [a]
replicate' 0 _ = []
replicate' n x = x : replicate' (n-1) x
-- repeat' :: create an infinite list of some value
repeat' :: a -> [a]
repeat' x = x : repeat' x
-- concat' :: concatenate all lists in a list of lists
concat' :: [[a]] -> [a]
concat' [] = []
concat' (x:xs) = x ++ concat' xs
-- merge :: merge together two sorted lists to give a single sorted list
merge :: Ord a => [a] -> [a] -> [a]
merge xs [] = xs
merge [] ys = ys
merge l1@(x:xs) l2@(y:ys) | x < y = x : merge xs l2
| otherwise = y : merge l1 ys
-- mergeSort :: sort a list by recursively merging sorted halves of a list
mergeSort :: Ord a => [a] -> [a]
mergeSort [] = []
mergeSort [x] = [x]
mergeSort l = merge (mergeSort left) (mergeSort right)
where left = take half l
right = drop half l
half = length l `div` 2
-- zip' :: create a list of tuples drawn from elements of two lists
zip' :: [a] -> [b] -> [(a,b)]
zip' _ [] = []
zip' [] _ = []
zip' (x:xs) (y:ys) = (x,y) : zip' xs ys
fibonacci :: [Integer]
fibonacci = 0 : 1 : next fibonacci
where next (x0:x1:xs) = x0+x1 : next (x1:xs)
```