--frequencies xs : The tuples of each character in xs, and their frequency
frequencies2 :: (Eq a) => [a] -> [(a,Int)] 
frequencies2 xs = [(x,n)|x<- remDup xs, n<- [length(filter (\a->x==a) xs)]]

--remDup xs: the list xs with all duplicates removed
remDup :: (Eq a) => [a] -> [a]
remDup xs =remDup' xs []

--remDup xs acc: the list of xs with all duplicates removed, prepended to acc
remDup' :: (Eq a) => [a] -> [a] -> [a]
remDup' [] acc = acc
remDup' (x:xs) acc = remDup'(filter (\a -> a/=x) xs) (x:acc)



--pytrips : the list of pythagorian triplets

pytrips :: [(Int,Int,Int)]

pytrips=[(a,b,c)| b <- [1..],
                  a <- [1..b-1],
                  c <- [b+1..a+b],
                  ((a*a)+(b*b)) == (c*c),
                  gcd a b == 1]


--as x : a list of approximate square roots of x

as :: Float -> [Float]

as x = as' x 1.0


--as' x y :: the list of approximations of root (x), starting from ys

as' :: Float -> Float -> [Float]

as' x y = y : as' x ((y+(x/y))/2.0)

--squareroot x : the approximate square root of x

squareroot :: Float -> Float

squareroot x = closeterms (as x)


--closeterms xs : the first element in a list that is 0.0001 in difference to
--                its previous neighbour

closeterms :: [Float]-> Float

closeterms (a:b:xs) = if abs (a-b) <0.0001
    then b
    else closeterms (b:xs)


--my first try of frequencies, works but isn't as simple as the other one
--frequencies xs : The tuples of each character in xs, and their frequency

--frequencies :: (Eq a) => [a] -> [(a,Int)]

--frequencies (x:xs) = freq' x xs [] []


--freq' cur xs acc1 acc2 : the number of times each character in xs, and cur, 
--                         appear, added to the corresponding tuple values in 
--                         acc1 and acc2

--freq' :: (Eq a) => a -> [a] -> [(a,Int)] -> [(a,Int)] -> [(a,Int)]

--freq' cur (x:xs) [] acc2 = freq' x xs ((cur,1):acc2) []

--freq' cur [] ((this,num):acc) acc2 = 
--    if cur==this
--        then (cur,num+1):acc++acc2
--        else freq' cur [] acc ((this,num):acc2)

--freq' cur [] [] acc2 = (cur,1):acc2

--freq' cur (x:xs) ((this,num):acc) acc2 = 
--    if cur == this 
--        then freq' x xs ((cur,num+1):acc ++ acc2) []
--        else freq' cur (x:xs) acc ((this,num):acc2)