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> I'm not sure "try not to use [proof by contradiction] unless you really need it" is the optimal conclusion.

Oh, that's only a guideline for presenting your proves. Not for coming up with them in the first place. Mathematicians do exactly what you are suggesting already.

The first prove one finds is usually quite ugly, and then comes the simplification. Writers call it editing, programmers refactoring.



Many proofs by contradiction are simply results of wrapping a constructive proof:

Suppose premises/axioms (B, C, ...) which don't depend on the value of A. Then, B=true AND C=true ... implies A=true by some logical argument, such as

A Or Not (B And C And ...) = True

In this case, the proof by contradiction is

Assume A=False But B=True, C=True, ... (Since they dont depend on A) Apply above constructive argument So A is true, A CONTRADICTION

Proofs by contradiction can often be of this type so the first and last lines can be removed.

The question is, what other types can there be?




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