Survey of proof techniques.
I recently came across a survey of proof techniques that I remembered
reading a few years back. Many of you have probably already seen it but
I thought that it might be a helpful guide for anyone whose educational
background has been so lacking as to have missed it. The survey was
written by Dana Angluin. His professional association was not given.
Luckily, the survey is short enough to present here:
Proof by example:
The author gives only the case n=2 and suggests that it contains most
of the ideas of the general proof.
Proof by intimidation:
'Trivial.'
Proof by vigorous handwaving:
Works well in a classroom or seminar setting.
Proof by cumbersome notation:
Best done with access to at least four alphabets and special symbols.
Proof by exhaustion:
An issue or two of a journal devoted to your proof is useful.
Proof by omission:
'The reader may easily supply the details.'
'The other 253 cases are analogous.'
'...'
Proof by obfuscation:
A long plotless sequence of true and\or meaningless syntactically related
statements.
Proof by wishful citation:
The author cites the negation, converse, or generalization of a theorem
from the literature to support his claims.
Proof by funding:
How could three different government agencies be wrong?
Proof by eminent authority:
'I saw Karp in the elevator and he said it was probably NP-complete.'
Proof by personal communication:
'Eight-dimensional colored cycle stripping is NP-complete [Karp, personal
commmunication].
Proof by reduction to the wrong problem:
'To see that infinite-dimensional colored cycle stripping is decidable,
we reduce it to the halting problem.'
Proof by reference to inaccessible literature:
The author cites a simple corollary of a theorem to be found in a privately
circulated memoir of the Slovenian Philological Society, 1883.
Proof by importance:
A large body of useful consequences all follow from the proposition in
question.
Proof by accumulated evidence:
Long and diligent search has not revealed a counterexample.
Proof by cosmology:
The negation of the proposition is unimaginable or meaningless. Popular
for proofs of the existence of God.
Proof by mutual reference:
In reference A, Theorem 5 is said to follow from Theorem 3 in reference B,
which is shown to follow from Corollary 6.2 in reference C, which is an
easy consequence of Theorem 5 in reference A.
Proof by metaproof:
A method is given to construct the desired proof. The correctness of the
method is proved by any of these techniques.
Proof by picture:
A more convincing form of proof by example. Combines well with proof by
omission.
Proof by vehement assertion:
It is useful to have some kind of authority relation to the audience.
Proof by ghost reference:
Nothing even remotely resembling the cited theorem appears in the reference
given.
Proof by forward reference:
Reference is usually to a forthcoming paper of the author, which is often
not as forthcoming as at first.
Proof by semantic shift:
Some standard but inconvenient definitions are changed for the statement
of the result.
Proof by appeal to intuition:
Cloud-shaped drawings frequently help here.