exercise-1-43.hs

{-
Exercise 1.43.  If f is a numerical function and n is a positive integer, then we can form the nth
repeated application of f, which is defined to be the function whose value at x is
f(f(...(f(x))...)). For example, if f is the function x -> x + 1, then the nth repeated application
of f is the function x -> x + n. If f is the operation of squaring a number, then the nth repeated
application of f is the function that raises its argument to the 2^n th power. Write a procedure
that takes as inputs a procedure that computes f and a positive integer n and returns the procedure
that computes the nth repeated application of f. Your procedure should be able to be used as
follows:

((repeated square 2) 5)
625

Hint: You may find it convenient to use compose from exercise 1.42.

*Main> (repeated square 2) 5
625
*Main> (repeatedI square 2) 5
625
-}

module Main where

repeated :: (a -> a) -> Integer -> (a -> a)
repeated f n = \x -> if n == 0 then x
                               else (f . repeated f (n-1)) x

repeatedI :: (a -> a) -> Int -> (a -> a)
repeatedI f n = \x -> iterate f x !! n

-----

square x = x * x