Function to auto-curry another function, with strong typing

currying.ts

interface Curry {
  <T1, R>(func: (t1: T1) => R): CurriedFunction1<T1, R>;
  <T1, T2, R>(func: (t1: T1, t2: T2) => R): CurriedFunction2<T1, T2, R>;
  <T1, T2, T3, R>(func: (t1: T1, t2: T2, t3: T3) => R): CurriedFunction3<
    T1,
    T2,
    T3,
    R
  >;
  <T1, T2, T3, T4, R>(
    func: (t1: T1, t2: T2, t3: T3, t4: T4) => R
  ): CurriedFunction4<T1, T2, T3, T4, R>;
  <T1, T2, T3, T4, T5, R>(
    func: (t1: T1, t2: T2, t3: T3, t4: T4, t5: T5) => R
  ): CurriedFunction5<T1, T2, T3, T4, T5, R>;
  (func: (...args: any[]) => any): (...args: any[]) => any;
}

interface CurriedFunction1<T1, R> {
  (): CurriedFunction1<T1, R>;
  (t1: T1): R;
}
interface CurriedFunction2<T1, T2, R> {
  (): CurriedFunction2<T1, T2, R>;
  (t1: T1): CurriedFunction1<T2, R>;
  (t1: T1, t2: T2): R;
}
interface CurriedFunction3<T1, T2, T3, R> {
  (): CurriedFunction3<T1, T2, T3, R>;
  (t1: T1): CurriedFunction2<T2, T3, R>;
  (t1: T1, t2: T2): CurriedFunction1<T3, R>;
  (t1: T1, t2: T2, t3: T3): R;
}
interface CurriedFunction4<T1, T2, T3, T4, R> {
  (): CurriedFunction4<T1, T2, T3, T4, R>;
  (t1: T1): CurriedFunction3<T2, T3, T4, R>;
  (t1: T1, t2: T2): CurriedFunction2<T3, T4, R>;
  (t1: T1, t2: T2, t3: T3): CurriedFunction1<T4, R>;
  (t1: T1, t2: T2, t3: T3, t4: T4): R;
}
interface CurriedFunction5<T1, T2, T3, T4, T5, R> {
  (): CurriedFunction5<T1, T2, T3, T4, T5, R>;
  (t1: T1): CurriedFunction4<T2, T3, T4, T5, R>;
  (t1: T1, t2: T2): CurriedFunction3<T3, T4, T5, R>;
  (t1: T1, t2: T2, t3: T3): CurriedFunction2<T4, T5, R>;
  (t1: T1, t2: T2, t3: T3, t4: T4): CurriedFunction1<T5, R>;
  (t1: T1, t2: T2, t3: T3, t4: T4, t5: T5): R;
}

const curry: Curry = (fn) => {
  const arity = fn.length;
  return function $curry(...args) {
    if (args.length >= arity) {
      return fn(...args);
    }
    return $curry.bind(null, ...args);
  };
};

// Example
const fn = curry(
  (a: number, b: number, c: number, d: number, e: number) => a + b + c + d + e
)
console.log(fn(1)(2, 3, 4, 5));
console.log(fn(1, 2)(3, 4, 5));
console.log(fn(1, 2, 3)(4, 5));
console.log(fn(1, 2, 3, 4, 5));