How To Prove a Theorem

This list has been compiled by Dana Angluin and was originally published in SIGACT News, Winter-Spring, 1983, Volume 15 #1.

  1. Proof by example
    The author gives only the case n=2 and suggests that it contains most of the ideas of the general proof.
  2. Proof by intimidation
  3. Proof by vigorous handwaving
    Works well in a classroom or seminar setting.
  4. Proof by cumbersome notation
    Best done with access to at least four alphabets and special symbols.
  5. Proof by exhaustion
    An issue or two of a journal devoted to your proof is useful.
  6. Proof by omission
    “The reader may easily supply the details.”
    “The other 253 cases are analogous.”
  7. Proof by obfuscation
    A long plotless sequence of true and/or meaningless syntactically related statements.
  8. Proof by wishful citation
    The author cites the negation, converse, or generalization of a theorem from the literature to support his claim.
  9. Proof by funding
    How could three different government agencies be wrong?
  10. Proof by eminent authority
    “I saw Karp in the elevator and he said it was probably NP-complete.”
  11. Proof by personal communication
    “Eight-dimensional coloured cycle stripping is NP-complete (Karp, personal communication).”
  12. Proof by reduction to the wrong problem
    “To see that infinite-dimensional coloured cycle stripping is decidable, we reduce it to the halting problem.”
  13. 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.
  14. Proof by importance
    A large body of useful consequences all follow from the proposition in question.
  15. Proof by accumulation of evidence
    Long and diligent search has not revealed a counterexample.
  16. Proof by cosmology
    The negation of the proposition is unimaginable or meaningless. Popular for proofs of the existence of God.
  17. 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.
  18. Proof by metaproof
    A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.
  19. Proof by picture
    A more convincing form of proof by example. Combines well with proof by omission.
  20. Proof by vehement assertion
    It is useful to have some kind of authority relation to the audience.
  21. Proof by ghost reference
    Nothing even remotely resembling the cited theorem appears in the reference given.
  22. Proof by forward reference
    Reference is usually to a forthcoming paper by the author.
  23. Proof by semantic shift
    Some of the standard but inconvenient definitions are changed for the statement of the result.
  24. Proof by appeal to intuition
    Cloud-shaped drawings frequently help here.

The following methods have been used successfully and have been consequently submitted by fellow readers and members.

  1. Proof by lack of imagination
    How could a life form not be carbon based?