I'm an undergrad trying to publish a result concerning Conway's Base-13 function. Essentially, it's a direct proof that it's possible to formulate a closed-form expression for the function over specific subsets of $\mathbb{R}$.
Regardless, the proof requires defining a dozen or so functions, which, using finite arithmetic, serve as a variety of digit-manipulation operations.
I'm struggling to grasp how detailed the proof needs to be when it comes to proving that each of these functions do what I say they do, based on their definitions. Having dozens of them, which do relatively mundane things, I don't want the proof to be pages upon pages of lemma after lemma. That being said, I don't know how much can be left as an "exercise for the reader" either.
As a simple example, here's one of the first functions I define:
Definition: Let $D$, "the digit function", be defined as $D_b^n(x)=\big\lfloor\frac{x}{b^n}\big\rfloor-b\big\lfloor\frac{x}{b^{n+1}}\big\rfloor\ \forall x\in\mathbb{N}_0$, for any base $b\in\mathbb{Z}^{>1}$, and any digit-index $n\in\mathbb{N_0}$.
Aside from not knowing if this is good formatting/notation for definiting such a function, I'm uncertain how detailed of a proof I need to give to prove that this "digit function" does what I intend it do to. Which is, return the $n^{th}$ digit from the left of the radix point ($n=0$ being the units' place) for any base-$b$ expansion of a given integer $x$.
As I said, this is just a simple example. There are other functions further in the paper that do more complicated tasks such as replacing a specific digit with a decimal point, etc. I'm confident all these functions do what I've intended them to do, and any reader could prove such if they really wanted to.
Hence, for the sake of brevity, how much of a proof can be left for the reader? Is there a rule of thumb?
For more information on the situation, this is a follow-up to a previous question: Ambiguity of publishing as an undergrad.
It depends on the reader. If the reader is "Conway", then "all of it" can be left to the reader.
If the reader is your fellow students, then more likely the answer is "some of it, but less than you think." And if you can communicate the main idea of the function, and why it should do what you want, then you may reasonably be able to omit the proof.
When you say "any reader could prove such if they really wanted to," think back to yourself in 8th grade, say. If you were like me, you could understand the ideas (generalizing from an example or two), but couldn't write a proof to save your life. So making a claim like this is an excuse for not thinking carefully, and I advise you to avoid it.
It often helps to start an article with your assumptions about the reader ("I assume the reader is familiar with the usual definitions from elementary differential geometry of surfaces in 3-space, including the first and second fundamental form; I'll use the notation from [ref1]."); the helps set expectations, and helps remind you of what you promised.