Writing lambda expressions in terms of pure functions

Question

I am trying to create some functions that I want to comfortably compose together using the pure functions.

The functions were originally written in $\lambda$-expressions so I would like to keep the format as close to it as possible. They are

• ${\rm id}:=(\lambda x . x)$

• ${\rm almost\$factorial}:=(\lambda f.(\lambda n. \text{If } (n=0) \text{ then } 1 \text{ else } n*f(n-1)))$

• ${\rm factorial0}:= {\rm almost\$factorial} ({\rm id})$

I realized that I can write

id:= (#)& 

for ${\rm id}$.

So id[x] gives me x, which is what I expected. However, I'm at lost on how to write $almost-factorial$. I tried

almost$factorial := If[#1==0, 1, #1*(#2[#1 - 1])]&

But it only works when it takes two arguments at once. I would to be able to define almost$factorial as a pure function that takes only one argument and outputs another pure function, which I can apply.

almost$factorial[id] [0] = 1. 

Is this possible to do in Mathematica?

Answer 1

Here are two ways. First with pure functions:

idP = Function[x, x];
almostFactorialP = Function[f, Function[n, If[n == 0, 1, n*f[n - 1]]]]

almostFactorialP[idP][4]
(* 12 *)

In general, this is the same method you tried with the Slot (#) notation. Unfortunately, in a construct like (#+(#+#&))& there is no way to refer to the outer # in the inner function or fill the inner slots. This is the reason I used Function explicitly.

Another way is to use SubValues:

id[x_] := x;
almostFactorial[f_][n_] := If[n == 0, 1, n*f[n - 1]];

almostFactorial[id][4]
(* 12 *)