Typing the Y combinator in Dart

y.dart

// Typing the Y combinator in Dart.
class Fix<A> {
  A Function(Fix<A>) f;
  Fix.inject(this.f);

  A Function(Fix<A>) get project => f;
}

// fix : (Fix<A> -> a) -> Fix<A>.
Fix<A> fix<A>(A Function(Fix<A>) f) => Fix.inject(f);

// Y : ((a -> b) -> a -> b) -> a -> b.
B Function(A) Y<A, B>(B Function(A) Function(B Function(A)) f) {
  B Function(A) Function(Fix<B Function(A)>) abs =
      (Fix<B Function(A)> x) => (A a) => f(x.project(x))(a);
  Fix<B Function(A)> arg =
      fix<B Function(A)>((Fix<B Function(A)> x) => (A a) => f(x.project(x))(a));
  return abs(arg);
}

// fact : (int -> int) -> int -> int.
int Function(int) fact(int Function(int) self) =>
    (int n) => n == 0 ? 1 : n * self(n - 1);

// A representation of natural numbers.
abstract class Nat {}
class Succ implements Nat {
  Nat n;
  Succ(this.n);

  String toString() => "Succ($n)";
}
class Zero implements Nat {
  Zero();
  String toString() => "Zero";
}

// int2nat : (int -> nat) -> int -> nat.
Nat Function(int) int2nat(Nat Function(int) self) =>
    (int n) => n == 0 ? Zero() : Succ(self(n-1));

void main() {
  int result = Y<int, int>(fact)(6);
  print("$result");
  Nat natural = Y<int, Nat>(int2nat)(6);
  print("$natural");
}

// Output:
// $ dart y.dart
// 720
// Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))