Toy theorem

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In mathematics, a toy theorem is a simplified instance (special case) of a more general theorem, which can be useful in providing a handy representation of the general theorem, or a framework for proving the general theorem.[1] One way of obtaining a toy theorem is by introducing some simplifying assumptions in a theorem.[2]

In many cases, a toy theorem is used to illustrate the claim of a theorem, while in other cases, studying the proofs of a toy theorem (derived from a non-trivial theorem) can provide insight that would be hard to obtain otherwise.

Toy theorems can also have educational value as well. For example, after presenting a theorem (with, say, a highly non-trivial proof), one can sometimes give some assurance that the theorem really holds, by proving a toy version of the theorem.[2]


A toy theorem of the Brouwer fixed-point theorem is obtained by restricting the dimension to one. In this case, the Brouwer fixed-point theorem follows almost immediately from the intermediate value theorem.[2]

Another example of toy theorem is Rolle's theorem, which is obtained from the mean value theorem by equating the function value at the endpoints.[1]

See also[edit]


  1. ^ a b "The Definitive Glossary of Higher Mathematical Jargon — Theorem". Math Vault. 2019-08-01. Retrieved 2019-11-26.
  2. ^ a b c "toy theorem". Retrieved 2019-11-26.

This article incorporates material from toy theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.