Toy theorem

{{Short description|Simplified instance of a general theorem}} {{Unreferenced|date=March 2023}} 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. 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 [[mathematical proof|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.

== Examples == 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]].

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

==See also== *[[Corollary]] *[[Fundamental theorem]] *[[Lemma (mathematics)]] *[[Toy model]]

== References == <references />{{PlanetMath attribution|id=4684|title=toy theorem}}

[[Category:Mathematical theorems]] [[Category:Mathematical terminology]]