TypeScript XOR with multiple operands

XOR.ts

export type XOR<T extends any[]> = XORInternal1<
  T,
  XORInternal2<T, never>,
  never
>;

type XORInternal2<T extends any[], R> = T["length"] extends 0
  ? R
  : XORInternal2<RemoveFirst<T>, keyof T[0] | R>;

type RemoveFirst<T extends any[]> = T extends [any, ...infer I] ? I : never;

type XORInternal1<
  T extends any[],
  K extends keyof any,
  R
> = T["length"] extends 0
  ? R
  : XORInternal1<
      RemoveFirst<T>,
      K,
      R | (Partial<Record<Exclude<K, keyof T[0]>, undefined>> & T[0])
    >;

XORTest.ts

import type { XOR } from './XOR.ts';

type ExpandRecursively<T> = T extends object
  ? T extends infer O
    ? { [K in keyof O]: ExpandRecursively<O[K]> }
    : never
  : T;

type Test = ExpandRecursively<
  XOR<[{ a: string }, { b: number }, { c: string; d: number }]>
>;