## Lambda calculation and church figures

I have a question about the church figures and the successor function.

I think I understand the basics of beta reduction and that I can implement church figures (somehow)

So, it defines a general encoding of a value n as

``````n = 𝜆f.𝜆x.f^n x
``````

^ n is f raised to n, I do not know how to format by exponent / index here …

then

``````0 = 𝜆f.𝜆x.x
1 = 𝜆f.𝜆x.fx
2 = 𝜆f.𝜆x.f(fx)
``````

What I get as a number is just an account of the encapsulation of a function `f` more than `x`.

He then defines `SUC` as a way to add 1 to `n` as:

``````SUC = 𝜆n.𝜆f.𝜆x.nf(fx)

``````

So, apply `n` at `SUC`:

``````SUC_n = (𝜆n.𝜆f.𝜆x.nf(fx))(𝜆f.𝜆x.f^n x)
(n := (𝜆f.𝜆x.f^n x))
= 𝜆f.𝜆x.(𝜆f.𝜆x.f^n x)f(fx)
(f := f)
= 𝜆f.𝜆x.(𝜆x.f^n x)(fx)
(x := (fx))
= 𝜆f.𝜆x.f^n(fx)
= 𝜆f.𝜆x.f^(n+1)x        This step I get, we are increasing n as an encapsulation of f over x
= n + 1                 There is step here I don't get. How are we losing f^(n+1)x to get
just n+1
``````

So, my question is. This last step, how are you `𝜆f.𝜆x.f^(n+1)x` for just `n+1`.

Is it a beta reduction that I do not understand / miss here?

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