From cate3.osbunorth@xerox.com Fri Aug 31 19:33:53 1990 From: cate3.osbunorth@xerox.com (Henry Cate III) Subject: How to prove something
Survey of proof techniques
This survey was written by Dana Angluin. Not really sure where it came from.
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.
Henry Cate III
(ucbvax!xerox.com!cate3.osbunorth) OR (cate3.osbunorth@Xerox.Com)
Everyone complains of his memory, no one of his judgment.