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)