I'm restricting it to ℕ so it will work, though you do have dividing by 2 in the proc.
U is essentially the Successor Function.
I know. I guess I was wrong, it's just NUMBER that has U built in. So, yeah, that's basically Church numerals only with named functions (which matters once you want to do recursion).
For a given number n, to add number q, call the successor of n q times.
That can't quite be right. The successor of n will be n+1 no matter how many times you call it. It's a good thought, though, but remember that a number is a function of functions.
Think about this:
Yeah but HOW?
I mean, try plugging things into there and see what results you get. Like, what's 3(0)? What's 3(3)? What's 3(successor)? Etc.
Right. Thanks for the tip. I actually posted a reply to a post right as you moved it to a new topic. I was talking about the "Exploring how computers work" YT series. Sorry 'bout that.
I found that:
I'm not quite sure what I'm doing, but I came up with
Though, it acts more like messed-up exponents.
Given F(x) = x+1, n plus q will be F(n) wrapped in F() q times, right?
2 + 3 = F(F(F(2))) = F(F(3)) = F(4) = 5
I think I may have misworded my claim.
Yes. It won't be F(n) repeatedly; it'll be, as you say, F(F(F(...(N)))). We don't have CASCADE in this game, so you (that is to say, @bluebaritone21) have to find a way to do that given the available tools.
Indeed. You seem to have solved the exercise after next. :~) You want a bigger hint?
Yes, please. I can't seem to figure this out, no matter how much random junk I try. I've tried adding the successor around the second item, but that didn't work. I tried hard-coding it, but that's cheating. I have no idea what I'm doing.
Okay. What does this do?
Don't just TRY it -- you won't learn anything from that. Actually plug SUCCESSOR into the input to THREE and see what function it reports.
I got this:
Does 
go wherever 
is? Like:
Or expanded out:
(I see what you mean about ugly programs.)
This. So, tell me in words, what does that function do when you call it with a number. Let's say f(7)?
It plugs 7 into successor, then plugs THAT into successor, and finally, plugs that into Successor, aka S(S(S(7))) = 10
OH!
would this work?
(Just tested it. YES! IT DOES!)
Thank You!!!!
I've almost got times:
(EDIT: No I didn't)
Cool! So, remember what 3(f) means, and we want to do 3*5=15, so how do you make 15(f) out of 3(f)?
3+3+3+3+3 (3+3 5 times)? So:
3+(3+(3+(3+(3))))
Hmm....
Remember, 3 is a function, so it's more like 3(3(3(3(3(something))))).