ts-nat-comm/typeLevelNats.ts

export type Zero = { tag: "Zero" };
export type Suc<Prev> = { tag: "Suc"; prev: Prev };

const zero: Zero = { tag: "Zero" };
function suc<Prev>(p: Prev): Suc<Prev> {
  return { tag: "Suc", prev: p };
}

type BuildTuple<L extends keyof RealNats, T extends any[] = []> = T extends {
  length: L;
}
  ? T
  : BuildTuple<L, [...T, "x"]>;
const _12: BuildTuple<1> = ["x"];
const _13: BuildTuple<2> = ["x"];
const _14: BuildTuple<2> = ["x", "x"];

type Tail<L extends any[]> = L extends [unknown, ...infer rest] ? rest : never;
const _6: Tail<[1, 2, 3]> = [2, 3];
const _7: Tail<[1, 2, 3]> = [2, 4];

type Length<L extends any[]> = L extends { length: infer len } ? len : never;
const _8: Length<[3, 4, 5]> = 3;
const _9: Length<[3, 4, 5]> = 4;

type LengthOfTail<L extends any[]> = Length<Tail<L>>;
const _10: LengthOfTail<[3, 4, 5]> = 2;
const _11: LengthOfTail<[3, 4, 5]> = 3;

type Nat<Num extends number> = BuildTuple<Num> extends ["x", ...infer rest]
  ? Suc<Nat<Length<rest>>>
  : Zero;
const _15: ["x"] extends ["x", ...infer _] ? "true" : "false" = "true";
const _16: ["x"] extends ["x", ...infer _] ? "true" : "false" = "false";

const _17: [] extends ["x", ...infer _] ? "true" : "false" = "true";
const _18: [] extends ["x", ...infer _] ? "true" : "false" = "false";

const _1: Nat<0> = zero;

const _3: Nat<1> = suc(zero); // should pass
const _2: Nat<1> = zero; // should fail

const _5: Nat<4> = suc(suc(suc(suc(zero)))); // should pass
const _4: Nat<4> = suc(suc(suc(zero))); // should fail

type RealNats = [
  0,
  1,
  2,
  3,
  4,
  5,
  6,
  7,
  8,
  9,
  10,
  11,
  12,
  13,
  14,
  15,
  16,
  17,
  18,
  19,
  20,
  21,
  22,
  23,
  24,
  25,
  26,
  27,
  28,
  29,
  30,
  31,
  32,
  33,
  34,
  35,
  36,
  37,
  38,
  39,
  40,
  41,
  42,
  43,
  44,
  45,
  46,
  47,
  48,
  49,
  50,
  51,
  52,
  53,
  54,
  55,
  56,
  57,
  58,
  59,
  60,
  61,
  62,
  63,
  64,
  65,
  66,
  67,
  68,
  69,
  70,
  71,
  72,
  73,
  74,
  75,
  76,
  77,
  78,
  79,
  80,
  81,
  82,
  83,
  84,
  85,
  86,
  87,
  88,
  89,
  90,
  91,
  92,
  93,
  94,
  95,
  96,
  97,
  98,
  99,
  100,
  101,
  102,
  103,
  104,
  105,
  106,
  107,
  108,
  109,
  110,
  111,
  112,
  113,
  114,
  115,
  116,
  117,
  118,
  119,
  120,
  121,
  122,
  123,
  124,
  125,
  126,
  127,
  128,
  129,
  130,
  131,
  132,
  133,
  134,
  135,
  136,
  137,
  138,
  139,
  140,
  141,
  142,
  143,
  144,
  145,
  146,
  147,
  148,
  149,
  150,
  151,
  152,
  153,
  154,
  155,
  156,
  157,
  158,
  159,
  160,
  161,
  162,
  163,
  164,
  165,
  166,
  167,
  168,
  169,
  170,
  171,
  172,
  173,
  174,
  175,
  176,
  177,
  178,
  179,
  180,
  181,
  182,
  183,
  184,
  185,
  186,
  187,
  188,
  189,
  190,
  191,
  192,
  193,
  194,
  195,
  196,
  197,
  198,
  199,
  200,
  201,
  202,
  203,
  204,
  205,
  206,
  207,
  208,
  209,
  210,
  211,
  212,
  213,
  214,
  215,
  216,
  217,
  218,
  219,
  220,
  221,
  222,
  223,
  224,
  225,
  226,
  227,
  228,
  229,
  230,
  231,
  232,
  233,
  234,
  235,
  236,
  237,
  238,
  239,
  240,
  241,
  242,
  243,
  244,
  245,
  246,
  247,
  248,
  249,
  250,
  251,
  252,
  253,
  254,
  255,
  256
];