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Normally for direct proof of equality we have the form:

Prove $$a = b$$

Proof (Style):

We start with $a$ (or $b$) and show through a sequence of logically connected steps that $a$ is $b$ (or the other way around). $_{_\square}$

But, since I'm not great with proofs I just wanted to have someone validate the following direct proof style, or comment on it's relative legitimacy compared with the first proof style:

Prove $$a = b$$

Proof (Style):
$$x = x$$ $$\vdots \tag{logical steps}$$ $$a = b$$

Furthermore, could someone recommend an elementary text concerning the validity of proof methods (or whatever it's actually called). Thank.

Matthew Towers
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Zduff
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    Why don't you try proving $a \leq b$ and $b \leq a$. This is automatically imply that $a = b$ – Aniruddha Deshmukh Oct 06 '17 at 17:49
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    @AniruddhaDeshmukh This is fine if $a$ and $b$ come from a set with a partial order, but this isn't always the case. – Xander Henderson Oct 06 '17 at 17:50
  • As long as the logical steps are valid, this is fine. – Zach Boyd Oct 06 '17 at 17:51
  • @XanderHenderson: I did not know about this. But, thanks for the information! – Aniruddha Deshmukh Oct 06 '17 at 18:15
  • A good proof is like a well-told story. Write your proofs in complete sentences. Especially do not omit the verb "implies" . Do not omit punctuation. State a justification for each assertion that may not be obvious. ("Obvious" is not really definable. It depends on you and on your probable audience.).....In principle every proof can of the form $S_1\to S_2\to .... \to S_n $. – DanielWainfleet Oct 07 '17 at 02:57

2 Answers2

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Instead of starting with $x=x$, start with $a=a$ or $b=b$

But yes, sometimes that's exactly what you need to do. For example, suppose I have that $b=a$ and I want to show that $a=b$. Then I can do:

  1. $b=a \quad $ Premise
  2. $b=b \quad $ = Intro
  3. $a=b \quad$ = Elim 1,2
Bram28
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    Is there anything wrong with starting at $x=x$? – Zach Boyd Oct 06 '17 at 18:25
  • @ZachBoyd Well, I'd need to know the specifics of your proof system .. but if you actually want to use the identity as a way to get to $a=b$, it needs to be about $a$ or $b$ ... but maybe you can substitute those once you have $x=x$? Again, it all depends on how your proof system works. – Bram28 Oct 06 '17 at 18:45
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    I guess he means something like this one.

    Theorem: $1/\sqrt{2}=\sqrt{2}/2$. Proof: $1/2=1/2$ $\Rightarrow$ $2/4=1/2$ $\Rightarrow$ $\sqrt{2}/\sqrt{4}=1/\sqrt{2}$ $\Rightarrow$ $\sqrt{2}/2=1/\sqrt{2}$.

     – A.Γ. Oct 06 '17 at 19:05
  • @A.Γ. I see, yes that could work! – Bram28 Oct 06 '17 at 19:45
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The second proof is logically correct too, but since you are talking about proof style I wouldn't call it good or preferable. In this aspect, I would like to cite a piece about writing proofs forward from the famous article How to write Mathematics by Paul Halmos:

A familiar trick of bad teaching is to begin a proof by saying: ``Given $\epsilon$, let $\delta$ be $\left(\frac{\epsilon}{3M^2+2}\right)^{1/2}$''. This is the traditional backward proof-writing of classical analysis. It has the advantage of being easily verifiable by a machine (as opposed to understandable by a human being), and it has the dubious advantage that something at the end comes out to be less than $\epsilon$, instead of less than, say, $\left(\frac{(3M^2+7)\epsilon}{24}\right)^{1/3}$. The way to make the human reader’s task less demanding is obvious: write the proof forward. Start, as the author always starts, by putting something less than $\epsilon$, and then do what needs to be done — multiply by $3M^2 + 7$ at the right time and divide by $24$ latter, etc., etc. — till you end up with what you end up with. Neither arrangement is elegant, but the forward one is graspable and rememberable.

A.Γ.
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