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I am relatively new to number theory, and wanted to ask how to appropriately write a “subproof” within a proof. I searched the internet for a solution and came across “lemmas” but am not sure if I can create my own lemma within a proof and refer to it later. Any help would be appreciated:

Let’s say I need to prove that a number x satisfies the inequality 8 < x < 12

Is there a way I can prove this by writing a proof that goes something like this:

Subproof 1(proves x < 12): blablabla random proof stuff… that results in: x < 12

Subproof 2(proves x > 8): blablabla random proof stuff… that results in x > 8

Then have a final statement: “due to subproof 1 and subproof 2: 8 < x < 12”

Shaurya Jeevagan
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  • check out any standard textbook in analysis or number theory. They'll have examples of how to write like this. – whoisit Jan 06 '24 at 22:26
  • This has nothing to do with number theory. Of course, $x<12$ and $x>8$ will give $8<x<12$. What means that "$x$ is a number"? Is it an integer? Is it just a real number? I suppose this is rather calculus than number theory. – Dietrich Burde Jan 06 '24 at 22:27
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    You can use paragraphs and natural language to structure the overall proof and separate the subproofs. If you're working on a larger project you may want to number the theorems as you go to refer back to them. – CyclotomicField Jan 06 '24 at 22:29

2 Answers2

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For smaller proofs, you could write something like

Proof. First, we prove that $x < 12$. [proof that $x < 12$].

Next, we prove that $x > 8. $ [proof that $x > 8$].

Therefore $8 < x < 12$. $\square$

For a larger proof, typically the "subcase" is its own separate lemma that is proven before the main theorem. For a lemma within a proof, the following format is an option (and certainly not the only option):

Theorem. [some statement]

Proof. [a partial proof]

We will now need the following lemma:

Lemma. [lemma statement]

Proof of lemma. [the proof of the lemma] $\square$

Now we return to the proof of the theorem. [completes proof of theorem] $\square$

Generally speaking, as long as other readers can read your proof without much difficulty, you're fine.

Ryan Tay
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  • The way you formatted your answer was extremely extremely helpful and clarified this for me. Thanks a lot. – Shaurya Jeevagan Jan 06 '24 at 22:59
  • One question- For the longer proof format you gave, what if I want to use the lemma again later in the proof? Would I have to reprove the lemma? – Shaurya Jeevagan Jan 06 '24 at 23:00
  • @ShauryaJeevagan You'd label the lemma, e.g. calling it "Lemma 1". You can then refer to it as Lemma 1 in other proofs. – Ryan Tay Jan 06 '24 at 23:05
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    Usually one will write something more along the lines of, "[...] but before we prove Theorem X, we need the following lemma Y". I haven't yet seen a textbook that interrupts a proof to first state a lemma. – soggycornflakes Jan 06 '24 at 23:58
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    @soggycornflakes Yeah it is much more common in textbooks to prove all necessary lemmas first before doing the proof of the big theorem. I have seen lemmas interrupt proofs in some papers though. – Ryan Tay Jan 07 '24 at 00:36
  • The way you write out a proof formally doesn't have to match the process you went through when finding the proof, either. It's likely that you start trying stuff, then half-way through realise "I need to prove this intermediate result" and take care of that. But when you write the proof out, you would turn that result into a lemma and put it at the start of the proof. – ConMan Jan 07 '24 at 22:59
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Lets assume you only operate in natural numbers, $\mathbb{N}$. IMHO, $8 < x < 12$ is just an abbreviation for there exists an $x$ such that $8 < x \land x < 12$. That is, $(\exists x)(p(8, x) \land p(x, 12))$, where the predicate $p$ is interpreted as the "$<$" relation.

IMHO, if you want to write in formal first order logic, the proof would look something like this:

  1. $\vdash (\exists x)p(8, x)$ (assume you proved it above)
  2. $\vdash (\exists x)p(x, 12)$ (assume you proved it above)
  3. $\vdash p(8, a)$ (C-Rule)
  4. $\vdash p(a, 12)$ (C-Rule)
  5. $\vdash p(8, a) \land p(a, 12)$ (AND introduction)
  6. $\vdash (\exists x)(p(8, x) \land p(x, 12)$ (Converse of C-Rule)

Then it is formally proved.

(I'm also not sure with the procedure; Any corrections are welcomed!)

WiggleLittleLi
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