Typing the Y combinator in C#.

y.cs

// Typing the Y combinator in C#.
// $ mcs y.cs
// $ mono y.exe
// 720
// Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))

using System;

class YCombinator {

    class Fix<A> {
        Func<Fix<A>, A> f;
        public Fix(Func<Fix<A>, A> f) {
            this.f = f;
        }

        public Func<Fix<A>,A> Unfix {
            get => f;
        }
    }

    // Y : ((A -> B) -> A -> B) -> A -> B
    public static Func<A, B> Y<A,B>(Func<Func<A,B>, Func<A,B>> f) {
        Func<Fix<Func<A,B>>,Func<A,B>> g =
            (Fix<Func<A,B>> x) =>
            (A a) => f(x.Unfix(x))(a);

        return g(new Fix<Func<A,B>>(g));
    }

    // The factorial function.
    static Func<Func<int,int>,Func<int,int>> Fact
        = (Func<int, int> self) =>
          (int n) => n == 0 ? 1 : n * self(n - 1);


    // A representation of natural numbers.
    abstract class Nat {}
    class Zero : Nat {
        public Zero() {}

        override public String ToString() => "Zero";
    }
    class Succ : Nat {
        Nat n;
        public Succ(Nat n) {
            this.n = n;
        }

        override public String ToString() {
            return "Succ(" + n.ToString() + ")";
        }
    }

    static Func<Func<int,Nat>,Func<int,Nat>> Int2Nat
        = (Func<int,Nat> self) =>
        (int n) => n == 0 ? (Nat)new Zero() : (Nat)new Succ(self(n-1));

    public static void Main(String[] args) {
        int result0 = Y(Fact)(6);
        Console.WriteLine(result0);
        Nat result1 = Y(Int2Nat)(6);
        Console.WriteLine(result1.ToString());
    }
}