-- | Homework 2 template. See the homework description page for more details,
--   hints, and things to think about for each part.
module HW2 where

import HW1

-- Copied from my HW1 for your convenience!
fold' :: (Int -> a) -> (a -> a -> a) -> (a -> a -> a) -> Expr -> a
fold' f1 f2 f3 = rec
    where 
        rec (Lit i) = f1 i
        rec (Add l r) = f2 (rec l) (rec r)
        rec (Mul l r) = f3 (rec l) (rec r)

--
-- * Part 1: Reverse Polish Notation
-- 

-- | Takes an expression and returns a string encoding of that expression in
--   Reverse Polish Notation (RPN).
--
--   >>> toRPN (Lit 3)
--   "3"
--
--   >>> toRPN e1
--   "2 3 4 * +"
--
--   >>> toRPN e2
--   "7 6 + 5 *"
--
--   >>> toRPN e3
--   "3 2 * 5 4 * +"
--
--   >>> elem (toRPN e4) ["8 7 9 * + 6 +", "8 7 9 * 6 + +"]
--   True
--   
toRPN :: Expr -> String
toRPN = fold' show (rpn "+") (rpn "*")
    where rpn op l r = concat [l, " ", r, " ", op]


-- | Takes a string that is an RPN-encoded expression and produces the same
--   expression represented as an abstract syntax tree.
--
--   You can assume that your function will only be given valid RPN-encodings
--   of expressions. That is, it need not fail gracefully if it encounters an
--   error. However, if you would like to improve the error handling, you are
--   welcome to change the type of your function and the doctests.
--
--   >>> fromRPN "3"
--   Lit 3
--
--   >>> fromRPN "2 3 +"
--   Add (Lit 2) (Lit 3)
--
--   >>> fromRPN "2 3 4 + +"
--   Add (Lit 2) (Add (Lit 3) (Lit 4))
--
--   >>> all (\e -> e == fromRPN (toRPN e)) [e1,e2,e3,e4]
--   True
--
fromRPN :: String -> Expr
fromRPN = head . foldl step [] . words
    where
        step (l:r:es) "+" = (Add r l):es
        step (l:r:es) "*" = (Mul r l):es
        step es s = (Lit (read s)):es


--
-- * Part 2: Syntactic Sugar
--

-- | Takes an expression and returns an expresion that evaluates to its
--   negation. Notice that this function does *not* evaluate the expression!
--   It returns a new expression that, when evaluated, will evaluate to the
--   negation of the original expression.
--
--   >>> eval e2
--   65
--
--   >>> eval (neg e2)
--   -65
--
neg = fold' (Lit . negate) Add (Mul . neg) -- Not efficient, but short :^)


-- | Takes two expressions and returns an expression that evalautes to the
--   second expression subtracted from the first. Once again, note that the
--   return type is an expression.
--
--   >>> eval e1
--   14
--
--   >>> eval (sub e2 e1)
--   51
--
sub = (. neg) . Add -- (a -> b) is a Functor