# Toy theorem

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Simplified instance of a general theorem

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

In many cases, a toy theorem is used to illustrate the claim of a theorem, while in other cases, studying the [proofs](/source/Mathematical_proof) 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.

## Examples

A toy theorem of the [Brouwer fixed-point theorem](/source/Brouwer_fixed-point_theorem) is obtained by restricting the [dimension](/source/Dimension) to one. In this case, the Brouwer fixed-point theorem follows almost immediately from the [intermediate value theorem](/source/Intermediate_value_theorem).

Another example of toy theorem is [Rolle's theorem](/source/Rolle's_theorem), which is obtained from the [mean value theorem](/source/Mean_value_theorem) by equating the [function](/source/Function_(mathematics)) values at the endpoints.

## See also

- [Corollary](/source/Corollary)

- [Fundamental theorem](/source/Fundamental_theorem)

- [Lemma (mathematics)](/source/Lemma_(mathematics))

- [Toy model](/source/Toy_model)

## References

*This article incorporates material from toy theorem on [PlanetMath](/source/PlanetMath), which is licensed under the [Creative Commons Attribution/Share-Alike License](https://en.wikipedia.org/wiki/Wikipedia:CC-BY-SA).*

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