add x y = strSum '0' (fst (sameLength x y)) (snd (sameLength x y))

sameLength x y
 | length x < length y = ((resize (length y) x), y)
 | length x > length y = (x, (resize (length x) y))
 | otherwise = (x, y)

strSum _ "" "" = ""
strSum c x y = strSum (snd (bitSum c (last x) (last y))) (init x) (init y) ++ [fst (bitSum c (last x) (last y))]

bitSum '0' '0' '0' = ('0', '0')
bitSum '0' '1' '0' = ('1', '0')
bitSum '0' '0' '1' = ('1', '0')
bitSum '0' '1' '1' = ('0', '1')
bitSum '1' '0' '0' = ('1', '0')
bitSum '1' '1' '0' = ('0', '1')
bitSum '1' '0' '1' = ('0', '1')
bitSum '1' '1' '1' = ('1', '1')