在这里，我在 Haskell 中实现了 Nat（正整数）数据类型和 Sign（对于所有整数）数据类型，并分别实现了它们的 Plus 函数，它们工作正常。现在，我正在考虑将这些扩展到有理数，因此引入了 Ratio 数据类型来覆盖这些数据类型并分别实现了它的 plus 函数，但是我从这些得到的回报将是分数中的值，这些值可能不是它们各自最简单的形式，因此我只想从这些函数返回，分数很简单，即分子和分母的 gcd 为 1。我实现的代码如下->
data Nat = Zero | Succ Nat -- the user defined data type for Nat for non negative integers
data Sign = Neg Nat | Pos Nat -- another user defined data type which checks for the sign of the particular number and helps in converting numbers to integers with use of Sign
data Ratio = Ratio Sign Sign -- the new ratio defined datatype to use for defining the rationals
plus :: Nat -> Nat -> Nat -- the original plus function which adds two Nat datatypes
plus m Zero = m -- when one is zero, hence returning other
plus Zero m = m
plus m (Succ (n)) = Succ (plus m n) -- respectively adding using successor property
plusSign :: Sign -> Sign -> Sign -- defining the plus function which would add the two numbers
plusSign (Pos m) (Pos Zero) = Pos (plus m Zero) -- if one number is zero then give the other number only and cheking for all the respective sign cases
plusSign (Neg m) (Pos Zero) = Neg (plus m Zero)
plusSign (Pos m) (Neg Zero) = Pos (plus m Zero)
plusSign (Neg m) (Neg Zero) = Neg (plus m Zero)
plusSign (Pos m) (Pos n) = Pos (plus m n) -- if both are positive then add them respectively and give that number Pos only
plusSign (Neg m) (Neg n) = Neg (plus m n) -- if both are negative then add them, and give the part neg respectively
plusSign (Neg m) (Pos n) -- if any one particular is postive and other is negative, then adding on the basis of the conditoins given
| mvalue == nvalue = Pos Zero -- if both are equal in magnitude, then retrun Zero
| mvalue > nvalue = Neg (sub m n) -- if negative is larger, then subracting them and giving the respective sign
| mvalue < nvalue = Pos (sub m n) -- if postive is larger, then subracting them and giving the respective sign
where
mvalue = convtointSign (Neg m) -- checking on the basis of integer value
nvalue = convtointSign (Pos n)
plusSign (Pos m) (Neg n) -- hence respectively checking the other case ono the basis of opposite arguments
| mvalue == nvalue = Pos Zero -- similar defination
| mvalue < nvalue = Neg (sub m n)
| mvalue > nvalue = Pos (sub m n)
where
mvalue = convtointSign (Pos m) -- checking the magnitudes
nvalue = convtointSign (Neg n)
plusRatio :: Ratio -> Ratio -> Ratio -- defining the required rational adding funtion which would add the rationals as per required conditions
plusRatio (Ratio (Pos Zero) m) p = p -- if one of the number's numerators is zero, then give the second number back
plusRatio p (Ratio (Pos Zero) m) = p
plusRatio (Ratio (Neg Zero) m) p = p
plusRatio p (Ratio (Neg Zero) m) = p
plusRatio (Ratio m (Pos Zero)) p = error "One of the rationals defined has Denominator as Zero." -- if for one of the numbers the denominator is Zero, hence returnign error
plusRatio p (Ratio m (Pos Zero)) = error "One of the rationals defined has Denominator as Zero."
plusRatio (Ratio m (Neg Zero)) p = error "One of the rationals defined has Denominator as Zero."
plusRatio p (Ratio m (Neg Zero)) = error "One of the rationals defined has Denominator as Zero."
plusRatio (Ratio m n) (Ratio x y) = (Ratio (plusSign (multSign m y) (multSign n x)) (multSign n y)) -- else using the defined multSign function depending on the provided arguments