Lazy state monads.

This module is inspired by the paper Functional Programming with Overloading and Higher-Order Polymorphism, Mark P Jones (http://web.cecs.pdx.edu/~mpj/) Advanced School of Functional Programming, 1995.

A function to increment a counter. Taken from the paper Generalising Monads to Arrows, John Hughes (http://www.math.chalmers.se/~rjmh/), November 1998:

tick :: State Int Int
tick = do n <- get
          put (n+1)
          return n

Add one to the given number using the state monad:

plusOne :: Int -> Int
plusOne n = execState tick n

A contrived addition example. Works only with positive numbers:

plus :: Int -> Int -> Int
plus n x = execState (sequence $ replicate n tick) x

An example from The Craft of Functional Programming, Simon Thompson (http://www.cs.kent.ac.uk/people/staff/sjt/), Addison-Wesley 1999: "Given an arbitrary tree, transform it to a tree of integers in which the original elements are replaced by natural numbers, starting from 0. The same element has to be replaced by the same number at every occurrence, and when we meet an as-yet-unvisited element we have to find a 'new' number to match it with:"

data Tree a = Nil | Node a (Tree a) (Tree a) deriving (Show, Eq)
type Table a = [a]

numberTree :: Eq a => Tree a -> State (Table a) (Tree Int)
numberTree Nil = return Nil
numberTree (Node x t1 t2)
       =  do num <- numberNode x
             nt1 <- numberTree t1
             nt2 <- numberTree t2
             return (Node num nt1 nt2)
    where
    numberNode :: Eq a => a -> State (Table a) Int
    numberNode x
       = do table <- get
            (newTable, newPos) <- return (nNode x table)
            put newTable
            return newPos
    nNode::  (Eq a) => a -> Table a -> (Table a, Int)
    nNode x table
       = case (findIndexInList (== x) table) of
         Nothing -> (table ++ [x], length table)
         Just i  -> (table, i)
    findIndexInList :: (a -> Bool) -> [a] -> Maybe Int
    findIndexInList = findIndexInListHelp 0
    findIndexInListHelp _ _ [] = Nothing
    findIndexInListHelp count f (h:t)
       = if (f h)
         then Just count
         else findIndexInListHelp (count+1) f t

numTree applies numberTree with an initial state:

numTree :: (Eq a) => Tree a -> Tree Int
numTree t = evalState (numberTree t) []

testTree = Node "Zero" (Node "One" (Node "Two" Nil Nil) (Node "One" (Node "Zero" Nil Nil) Nil)) Nil
numTree testTree => Node 0 (Node 1 (Node 2 Nil Nil) (Node 1 (Node 0 Nil Nil) Nil)) Nil

sumTree is a little helper function that does not use the State monad:

sumTree :: (Num a) => Tree a -> a
sumTree Nil = 0
sumTree (Node e t1 t2) = e + (sumTree t1) + (sumTree t2)