    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.SBV.Examples.Puzzles.Coins
    -- Copyright   :  (c) Levent Erkok
    -- License     :  BSD3
    -- Maintainer  :  erkokl@gmail.com
    -- Stability   :  experimental
    --
    -- Solves the following puzzle:
    --
    -- @
    -- You and a friend pass by a standard coin operated vending machine and you decide to get a candy bar.
    -- The price is US $0.95, but after checking your pockets you only have a dollar (US $1) and the machine
    -- only takes coins. You turn to your friend and have this conversation:
    --   you: Hey, do you have change for a dollar?
    --   friend: Let's see. I have 6 US coins but, although they add up to a US $1.15, I can't break a dollar.
    --   you: Huh? Can you make change for half a dollar?
    --   friend: No.
    --   you: How about a quarter?
    --   friend: Nope, and before you ask I cant make change for a dime or nickel either.
    --   you: Really? and these six coins are all US government coins currently in production?
    --   friend: Yes.
    --   you: Well can you just put your coins into the vending machine and buy me a candy bar, and I'll pay you back?
    --   friend: Sorry, I would like to but I cant with the coins I have.
    -- What coins are your friend holding?
    -- @
    --
    -- To be fair, the problem has no solution /mathematically/. But there is a solution when one takes into account that
    -- vending machines typically do not take the 50 cent coins!
    --
    -----------------------------------------------------------------------------

    module Data.SBV.Examples.Puzzles.Coins where

    import Data.SBV

    -- | We will represent coins with 16-bit words (more than enough precision for coins).
    type Coin = SWord16

    -- | Create a coin. The argument Int argument just used for naming the coin. Note that
    -- we constrain the value to be one of the valid U.S. coin values as we create it.
    mkCoin :: Int -> Symbolic Coin
    mkCoin i = do c <- exists $ 'c' : show i
                  constrain $ bAny (.== c) [1, 5, 10, 25, 50, 100]
                  return c

    -- | Return all combinations of a sequence of values.
    combinations :: [a] -> [[a]]
    combinations coins = concat [combs i coins | i <- [1 .. length coins]]
      where combs 0 _      = [[]]
            combs _ []     = []
            combs k (x:xs) = map (x:) (combs (k-1) xs) ++ combs k xs

    -- | Constraint 1: Cannot make change for a dollar.
    c1 :: [Coin] -> SBool
    c1 xs = sum xs ./= 100

    -- | Constraint 2: Cannot make change for half a dollar.
    c2 :: [Coin] -> SBool
    c2 xs = sum xs ./= 50

    -- | Constraint 3: Cannot make change for a quarter.
    c3 :: [Coin] -> SBool
    c3 xs = sum xs ./= 25

    -- | Constraint 4: Cannot make change for a dime.
    c4 :: [Coin] -> SBool
    c4 xs = sum xs ./= 10

    -- | Constraint 5: Cannot make change for a nickel
    c5 :: [Coin] -> SBool
    c5 xs = sum xs ./= 5

    -- | Constraint 6: Cannot buy the candy either. Here's where we need to have the extra knowledge
    -- that the vending machines do not take 50 cent coins.
    c6 :: [Coin] -> SBool
    c6 xs = sum (map val xs) ./= 95
       where val x = ite (x .== 50) 0 x

    -- | Solve the puzzle. We have:
    --
    -- >>> puzzle
    -- Satisfiable. Model:
    --   c1 = 50 :: Word16
    --   c2 = 25 :: Word16
    --   c3 = 10 :: Word16
    --   c4 = 10 :: Word16
    --   c5 = 10 :: Word16
    --   c6 = 10 :: Word16
    --
    -- i.e., your friend has 4 dimes, a quarter, and a half dollar.
    puzzle :: IO SatResult
    puzzle = sat $ do
            cs <- mapM mkCoin [1..6]
            -- Assert each of the constraints for all combinations that has
            -- at least two coins (to make change)
            mapM_ constrain [c s | s <- combinations cs, length s >= 2, c <- [c1, c2, c3, c4, c5, c6]]
            -- the following constraint is not necessary for solving the puzzle
            -- however, it makes sure that the solution comes in decreasing value of coins,
            -- thus allowing the above test to succeed regardless of the solver used.
            constrain $ bAnd $ zipWith (.>=) cs (tail cs)
            -- assert that the sum must be 115 cents.
            return $ sum cs .== 115