-   Scrivere una funzione “prodM” che realizza il
--  prodotto di due matrici di numeri rappresentate come liste di
--  liste. La funzione deve verificare che le due matrici (cioè le
--  loro rappresentazioni) siano ben formate e le loro dimensioni
--  compatibili.

--Es matrici
--A = [[1,0,2],[0,3,-2]]  B = [[4,1],[-2,2],[0,3]]
-- AB =[[4,7],[-6,3]]

prodM :: Num a => [[a]] -> [[a]] -> [[a]]
prodM xss yss
    | not (areGoodFormatted xss yss) = error "matrices are bad formatted"
    | otherwise = prodMat xss yss

prodMat :: Num a => [[a]] -> [[a]] -> [[a]]
prodMat _ [] = []
prodMat [] _ = []
prodMat m1@(xs:xss) m2@(ys:yss)
    | null xs = []
    | otherwise = prodRowMatrix xs m2 : prodMat xss m2

prodRowMatrix :: Num a => [a] -> [[a]] -> [a]
prodRowMatrix xs mat@(ys:yss)
      | length ys == 0 = []
      | otherwise = prodRowCol xs mat : (prodRowMatrix xs (map tail mat))


prodRowCol :: Num a => [a] -> [[a]]  -> a
prodRowCol [] _ = 0
prodRowCol (x:xs) (ys:yss)
        | not (null ys) = (x * head ys + prodRowCol xs yss)
        | otherwise = 0

areGoodFormatted :: Num a => [[a]] -> [[a]] -> Bool
areGoodFormatted [[]] [[]] = True
areGoodFormatted xss yss = hasSameColSize xss && hasSameColSize yss

hasSameColSize :: Num a => [[a]] -> Bool
hasSameColSize [[]] = True
hasSameColSize [[_]] = True
hasSameColSize (xs:xss)
    |not (null xss) = length xs == length(head xss) && hasSameColSize xss
    |otherwise = True

areCompatible :: Num a => [[a]] -> [[a]] -> Bool
areCompatible [[]] [[]] = True;
areCompatible (xs:xss) (ys:yss) = length xs == length (ys:yss)
                              &&  length (xs:xss) == length ys