Lambda calculus in Typescript

lambda.ts

type _bool<X, Y> = (X, Y) => X | Y;

let _true = <X, Y>(x: X, y: Y) => x;
let _false = <X, Y>(x: X, y: Y) => y;
let _not = <X, Y>(b: _bool<Y, X>) => (x: X, y: Y) => b(y, x);

let _and = <X, Y>(b1: _bool<X | Y, Y>, b2: _bool<X, Y>) => (x: X, y: Y) => b1(b2(x, y), _false(x, y));
let _or = <X, Y>(b1: _bool<X, X | Y>, b2: _bool<X, Y>) => (x: X, y: Y) => b1(_true(x, y), b2(x, y));
// Using
//     let __compose = <X, Y, U, V, W>(f: (U, V) => W, g: (X, Y) => U, h: (X, Y) => V) => (x: X, y: Y) => f(g(x, y), h(x, y)); 
// we can write
//     let _and = <X, Y>(b1: _bool<X | Y, Y>, b2: _bool<X, Y>) => __compose(b1, b2, _false);
//     let _or  = <X, Y>(b1: _bool<X | Y, Y>, b2: _bool<X, Y>) => __compose(b1, _true, b2);


type _nat<X> = (f: (X) => X, x: X) => X;

let _zero = <X>(f: (X) => X, x: X) => x; 
let _succ = <X>(n: _nat<X>) => (f: (X) => X, x: X) => n(f, f(x));

let _one = <X>(f: (X) => X, x: X) => _succ(_zero);

let __curry = <X, Y, Z>(f: (X, Y) => Z) => (x: X) => (y: Y) => f(x, y);

let _plus = <X>(n1: _nat<X>, n2: _nat<X>) => (f: (X) => X, x: X) => n1(f, n2(f, x));
let _times = <X>(n1: _nat<X>, n2: _nat<X>) => (f: (X) => X, x: X) => n1(__curry(n2)(f), x);