Lambda calculus and church numerals confusion

Question

I'm trying to understand the basics of lambda calculus and Church numerals. I have been doing a lot of reading and practising, but I seem to keep getting stuck with trying to see how some functions work.

The example I am stuck on is as follows. Perhaps someone can explain where I have gone wrong.

The Church numeral for 1 can be represented as:

λf. λx. f x

The exponentiation function on Church numerals (mⁿ) can be given as:

λm. λn. n m

All I want to do is show that by applying the exponentiation function to 1 and 1, I get back 1, since 1¹ = 1. I am doing this, so I understand better how these functions work. My working is as follows and I get stuck every time:

// Exp (1 1)
(λm. λn. n m) (λf1. λx1. f1 x1) (λf2. λx2. f2 x2)
// Substitute for m
(λn. n (λf1. λx1. f1 x1)) (λf2. λx2. f2 x2)
// Substitute for n
(λf2. λx2. f2 x2) (λf1. λx1. f1 x1)
// Substitute for f2
(λx2. (λf1. λx1. f1 x1) x2)
// Substitute for f1
λx2. (λx1. x2 x1)

And there I am stuck. I have lost both f's, left with x's only, and I haven't got 1 back. Where am I going wrong?

Answer 1

Where am I going wrong?

Nowhere! You're done. Remember, the variable names aren't important; it's the structure that is important. The names f or x2 aren't meaningful. It only matters how they are used. The Church numeral for 1 is

λf. λx. f x

and you have

λx2. (λx1. x2 x1)

Rename x2 to f and x1 to x and voilà! You have

λf. (λx. f x)
= λf. λx. f x