A functional solution to https://gist.github.com/Mroik/a536468705191a1321c0f63ec5d114a8

q.ml

type natural = Z | S of natural;;
exception NoSolutionInNatural;;

let rec sum a b =
    match b with
    | Z -> a
    | S x -> sum (S a) x
;;

let rec sub a b =
    match b with
    | Z -> a
    | S x ->
        match a with
        | Z -> raise NoSolutionInNatural
        | S y -> sub y x
;;

let rec mult a b =
    match b with
    | Z -> Z
    | S Z -> a
    | S x -> mult a x |> sum a
;;

let rec enumerate_n =
    let rec loop x = fun () -> Seq.Cons (x, loop (S x)) in
    loop Z
;;

type rational = Q of natural * natural;;
exception NoSolutionInRational;;

let rec qsum a b =
    match a, b with
    | Q (_, Z), _ -> raise NoSolutionInRational
    | _, Q (_, Z) -> raise NoSolutionInRational
    | Q (x, d1), Q (y, d2) when d1 = d2 -> Q (sum x y, d1)
    | Q (x, d1), Q (y, d2) ->
        let num1 = mult x d2 in
        let num2 = mult y d1 in
        let num = sum num1 num2 in
        let den = mult d1 d2 in
        Q (num, den)
;;

let rec qmult a b =
    match a, b with
    | Q (_, Z), _ -> raise NoSolutionInRational
    | _, Q (_, Z) -> raise NoSolutionInRational
    | Q (n1, d1), Q (n2, d2) -> Q (mult n1 n2, mult d1 d2)
;;