Kata

There are a few Katas about Church Numerals so let's talk about booleans.

In lambda calculus, the only primitive are lambdas. No numbers, no strings, and of course no booleans. Everything is reduced to anonymous functions.

Booleans are defined thusly (this definition is preloaded for you) :

const True = T => F => T;
const False = T => F => F;

type Boolean = forall a. a -> a -> a -- this requires RankNTypes

false,true :: Boolean
false = \ t f -> f
true  = \ t f -> t

true  = lambda t: lambda f: t
false = lambda t: lambda f: f

True  = \ t f . t
False = \ t f . f

Your task will be to implement basic operators on booleans (using only lambdas and function application) : Not, And, Or and Xor.

To help, the function unchurch comes preloaded, and returns the native boolean given a church boolean :

unchurch(True); //true;

Note: You should not use the following:

• numbers
• strings
• booleans
• boolean operators
• objects (curly brackets) or arrays (square brackets)
• regexp
• "new"