typing for "brother" field in TypeScript (just a scribble); union-type for each permutation for subsets of object properties in tuple `SubsAlpha` & `type TupleAtMost3<T> = [] | SubsAlpha<T>;`

brother.ts

import { AssertTrue, Has, IsExact } from "conditional-type-checks";

// interface A<T1 extends { [key: string]: any }, T2 extends T1> {
//   a: T1;
//   // b: (keyof T2)[]; // 現実解
//   b: Workaround<keyof T2>[]; // 現実解
//   // b: TupleAtMost3<T2>; // 重複を防ぐようにするにはすべての組み合わせを列挙する？
// }

// interface A<T1 extends { [key: string]: any }, T2 extends T1> {
interface A<T extends { [key: string]: any }, Keys extends (keyof T)[]> {
  a: T;
  // b: (keyof T2)[]; // 現実解
  // b: Workaround<keyof T2>[]; // 現実解
  b: Keys; // 現実解
}

// type Workaround<K> = K[] & { length: 1 };

// declare function f<T1 extends { [key: string]: any }, T2 extends T1>(
declare function f<T extends { [key: string]: any }, Keys extends (keyof T)[]>(
  obj: A<T, Keys>
): A<T, Keys>;

const a = f({ a: { c: 1, d: 2 }, b: ["c", "d"] });
const b = f({ a: { c: 1, d: 2 }, b: ["x"] });
const c = f({ a: { x: 1 }, b: ["x"] });
const d = f({ a: { x: 1 }, b: ["x", "x"] });
const e = f({ a: { x: 1, y: 2 }, b: ["x", "y"] });
const g = f({ a: { x: 1, y: 2 }, b: ["x", "y", "z"] });
const h = f({ a: { x: 1, y: 2 }, b: ["x"] });
const i = f({ a: { x: 1, y: 2 }, b: ["x", "x"] });
const j = f({ a: { x: 1, y: 2, z: 3 }, b: ["x"] });

interface V<MInput extends any, MOutput extends MInput> {
  data(): MInput extends any ? MInput : never;

  methods: { [key: string]: (this: { readonly $state: MOutput }) => void };
}

declare function v<M1 extends any, M2 extends M1>(obj: V<M1, M2>): V<M1, M2>;

const component = v({
  data() {
    return { a: 1 };
  },
  methods: {
    a() {
      this.$state;
    }
  }
});

type C<First, Rest> = [First, ...Rest[]];
const cc: C<string, number> = ["1", 1, 2];
declare const ddd: keyof typeof cc;
declare const eee: keyof [];

// [] | ["0"] | ["0", "1"] | ["1", "0"]

// type CCC = [] | [...("0" | "1")[]];
type DDD<T extends [...string[]]> = T extends { length: 0 }
  ? []
  : T extends { length: 1 }
  ? [] | ["0"]
  : T extends { length: 2 }
  ? [] | ["0"] | ["1"] | ["0", "1"] | ["1", "0"]
  : never;

namespace TypeChecks {
  type doTest0 = AssertTrue<IsExact<DDD<[]>, []>>;
  type doTest1 = AssertTrue<Has<DDD<["a"]>, ["0"]>>;
}

// type Hoge<Union> = Union extends infer T1 | infer T2 ? T1 : never;

// type TupleAtMost2<
//   Union
// > = /*Union[] extends (infer T1 | infer T2 | infer T3)[]
//   ? [T1, T2, T3]
//   :*/ Union[] extends (
//   | infer T1
//   | infer T2)[]
//   ? [T1, T2]
//   : Union extends infer T
//   ? [T]
//   : never;

type TupleAtMost2<
  Union
> = /*Union[] extends (infer T1 | infer T2 | infer T3)[]
  ? [T1, T2, T3]
  :*/ Union[] extends (
  | infer T1
  | infer T2)[]
  ? [T1, T2]
  : Union extends infer T
  ? [T]
  : never;

type AAAAA<T, K extends keyof T> = K extends keyof T ? [Pick<T, K>] : never;

type BBBBB<T, K extends keyof T> = K extends keyof T ? never : never;

type KnownKeys<T> = {
  [K in keyof T]: string extends K ? never : number extends K ? never : K
} extends { [_ in keyof T]: infer U }
  ? ({} extends U ? never : U)
  : never;

// union distribution + not distributed

// type TupleAtMost2<U> = Sub<U>;

type Sub1<T, K extends keyof T = keyof T> = K extends keyof T
  ? Pick<T, Exclude<keyof T, K>>
  : never;

type Subs<
  T,
  K1 extends keyof T = keyof T,
  K2 extends keyof T = keyof T,
  K3 extends keyof T = keyof T,
  K4 extends keyof T = keyof T,
  K5 extends keyof T = keyof T,
  K6 extends keyof T = keyof T,
  K7 extends keyof T = keyof T,
  K8 extends keyof T = keyof T
> = K1 extends keyof T
  ? K2 extends keyof T
    ? K3 extends keyof T
      ? K4 extends keyof T
        ? K5 extends keyof T
          ? K6 extends keyof T
            ? K7 extends keyof T
              ? K8 extends keyof T
                ? Pick<
                    T,
                    Exclude<keyof T, K1 | K2 | K3 | K4 | K5 | K6 | K7 | K8>
                  >
                : never
              : never
            : never
          : never
        : never
      : never
    : never
  : never;

namespace TypeChecks {
  type doTest0 = AssertTrue<IsExact<DDD<[]>, []>>;
  type doTest1 = AssertTrue<Has<DDD<["a"]>, ["0"]>>;
}

// type AtLeastOne<T> = Spread<T, keyof T>;
// type Spread<T, K extends keyof T> = K extends keyof T
//   ? PartiallyPartial<T, Exclude<keyof T, K>>
//   : never;
// type PartiallyPartial<T, K extends keyof T> = Partial<Pick<T, K>> &
//   Pick<T, Exclude<keyof T, K>>;

// negate type?
// type A<Union> = { 0: Union[0], 1: Union[1] }

// type Target = { a: number; b: number; c: number; d: number };
type Target = { a: number; b: number; c: number };
type Middle = Subs<Target> | Target;

const m: Middle = {};

type SubsAlpha<
  T,
  K1 extends keyof T = keyof T,
  K2 extends keyof T = keyof T,
  K3 extends keyof T = keyof T
> = K1 extends keyof T
  ? K2 extends keyof T
    ? K3 extends keyof T
      ? (K1 extends K2
          ? (/* K1 == K2, then */ K2 extends K3
              ? /* K1 == K2 == K3, so */ [K1]
              : /* K1 == K2 != K3, so */ [K1, K3])
          : /* K1 != K2, then */ K2 extends K3
          ? never
          : /* K1 != K2 && K2 != K3, then */ K3 extends K1
          ? never
          : /* K1,K2,K3 are distinguish, so */ [K1, K2, K3])
      : never
    : never
  : never;

type Omit<T, K extends keyof T> = Pick<T, Exclude<keyof T, K>>;

type SubsBeta<
  T,
  Keys extends any[] = [],
  NewKey extends keyof T = keyof T
> = Exclude<T, Keys[number]> extends never
  ? never
  : NewKey extends keyof T
  ? { 1: NewKey; 0: SubsBeta<Omit<T, NewKey>, [NewKey]> }
  : never;

// type SubBeta_<T, Keys extends any[] = [], NewKey extends keyof T = keyof T> = SubsAlpha<T, NewKey>;

type MiddleAlpha = [] | SubsAlpha<Target>;
type MiddleBeta = [] | SubsBeta<Target>;
type TupleAtMost3<T> = [] | SubsAlpha<T>;

const wewrq: { "0": 1 } = [1];

type UnionToTuple<U> = UnionToTupleRecursively<[], U>;

type UnionToTupleRecursively<T extends any[], U> = {
  1: T;
  0: Math;
}[[U] extends [never] ? 1 : 0];

type ASSS = UnionToTuple<"a" | "b">;

type Values<T> = T extends { [index: number]: infer E } ? E : never;

const aaa = ["a", "b"] as const;

type As = Values<typeof aaa>;

// type A2<T1 extends { [key: T2[number]]: any }, T2 extends string[]> = T2 extends (infer T3)[] ? {
//   a: T1;
//   b: T3;
// } : never;
//
// declare function f2<T1 extends { [key: string]: any }, T2>(
//   obj: A2<T1, T2>
// ): A2<T1, T2>;
//
// const a2 = f2({ a: { c: 1, d: 2 }, b: ["c", "d"] as const });
// const b2 = f2({ a: { c: 1, d: 2 }, b: ["x"] as const });
// const c2 = f2({ a: { x: 1 }, b: ["x"] });
// const d2 = f2({ a: { x: 1 }, b: ["x", "x"] });
// const e2 = f2({ a: { x: 1, y: 2 }, b: ["x", "y"] });
// const g2 = f2({ a: { x: 1, y: 2 }, b: ["x", "y", "z"] });
// const h2 = f2({ a: { x: 1, y: 2 }, b: ["x"] });
// const i2 = f2({ a: { x: 1, y: 2 }, b: ["x", "x"] });
// const j2 = f2({ a: { x: 1, y: 2, z: 3 }, b: ["x"] });