Writing clear proofs involving multiple theorems and conditions

Question

Suppose the problem is that given $A$ and $C$ holds, prove $D$ holds.

Some theorems that we can use are:

$A \to B$

$(B,C) \to D$

I feel what I said may be unclear:

Because $A$ holds and $A$ implies $B$, $B$ holds. Because $B$ and $C$ holds and $B$ and $C$ imply $D$, $D$ holds.

Usually $A$, $B$, $C$ and $D$ are long sentences and may contain "and" again.

Also there sometimes can be more conditions and theorems for proving $D$.

In all of these cases, the above quote can be very long and I am afraid may be confusing.

How would you write it?

Do you mainly use words or formulas?

Thanks.

Answer 1

If you're concerned about long complex statements, you should give names for statements just like you've done in your post. For some real-life examples:

• Property A
• Property T
• Theorems A and B

Better yet, make an adjective describing when conditions hold; e.g. given an integer $n$,

• if (long statement $A$) is true of $n$, it is called a "wiggly number"
• if (long statement $B$) is true of $n$, it is called a "porcupine number"
• etc.

and then phrase your argument like

Because $n$ is wiggly, Lemma 3 implies it is also porcupine. Because $n$ is both porcupine and applesauce, Lemma 5 allows us to conclude that $n$ is fuschia.

Answer 2

My suggestion is to be liberal in the use of the word “then”. When you write “If A, B, C”, it’s not clear whether the conclusion is B and C, or whether the hypothesis is A and B. If you mean “If A, then B and C”, write that; if you mean “If A and B, then C”, write just that. I try (and I’ll bet I don’t always succeed) to preface the conclusion with the word “then” in every case.

Answer 3

I suppose that it depends on who your audience is and how formal you want your proof to be. I usually assume that my audience is aware of how Modus Ponens works. So I would organize it as follows.

---

Given that:

\begin{align} A &\implies B \tag 1 \\ B \land C &\implies D \tag 2 \end{align}

we want to prove that:

$$A \land C \implies D$$

To this end, suppose that $A \land C$ holds so that $A$ holds and $C$ holds. Then since $A$ holds, it follows by $(1)$ that $B$ holds. But since $C$ holds, we know that $B \land C$ holds. Hence, by $(2)$, we conclude that $D$ holds, as desired. $~~\blacksquare$

Answer 4

What you wrote is clear to me.

You can readily turn this into "proofical" form by changing

A holds and A implies B, B holds. Because B and C holds and B and C imply D, D holds

into

1. $ (A \land (A \implies B)) \implies B$

2. $((B \land C) \land ((B \land C) \implies D)) \implies D $

Therefore

3. $\left(A \land C \land (A \implies B) \land ((B \land C) \land ((B \land C) \implies D)\right) \implies D $

Answer 5

Since you say that you have certain theorems available, I assume that they have some name ("Theorem 42" or "Smith's foobar-lemma") that yuo can refer to. This should allow to avoid lengthy repetitions.

We have that (lengthy property $A$ holds). By thoerem1, this implies that (lengthy property B holds). Together with (lengthy property $C$), this allows us to conclude from theorem 2 that (lengthy property $D$).

To summarize, a proof (outside certain specifically formal requirements) should not only be formally correct, but also readable by the audience. Writing full sentences with interjected formulas is very helpful for the reader, even for the mathematical reader who is well acquainted with $\forall$, $\land$ and $\Rightarrow$. In fact, many submissions to exercsies that consist only of a dozen lines of equations are not readable in this sense, because the intention of the author is not conveyed: Only too often, people forget the connectives between the lines (is it $\Rightarrow$ or $\Leftrightarrow$? Often it shold be $\Leftarrow$!) that are necessary for following the argument, and when correctly inserted would make gaps in the argument apparent (also to the author).