Help:Equivalence Proofs

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Proofs for the equivalence between two or more statements require special attention.

Page structure

Two statements

In the case of only two statements, one can opt for either the same page structure used for multiple statements as described below, or for a simple sentence using Template:Iff. If there is a direct proof of the equivalence, the second option is simplest and cleanest. If the two implications require a different proof, or if there is a second proof which does the two separately, the first option is the way to go, because it is very convenient to refer to the statements with a number.

Multiple statements

The structure of an equivalence proof with multiple statements is as follows:

== Theorem ==

Theorem intro


:$(1):\quad$ First statement.

:$(2):\quad$ Second statement.

:$(3):\quad$ ...


== Proof ==

Followed by the rest of the page. See Help:Page Structure.

Use of the TFAE Template is encouraged. It produces:

The following statements are equivalent:

Long lists

In the case of a large amount of equivalent statements, the implications are best proved on individual pages, and the proof of the overall equivalence then consists purely of links to those pages, without any additional elements of proof. This is because:

  • A lot of subpages is unwieldy.
  • Some of the equivalences may be considerably harder to prove, so that they may rely on other equivalences being established already, which makes referencing inside the proof difficult.
  • Referring to the statements with numbers is less informative and makes them harder to search for than equivalence proofs whose title describes the two statements.

Definition Equivalences

If there are Multiple Definitions for one thing, their equivalence has to be proved on a page with the title:

Equivalence of Definitions of Concept that is Defined

See also Help:Page Naming

The structure of such a page is roughly as follows:

== Theorem ==

Theorem intro

{{TFAE|def = Name of Definition Page}}

=== First transcluded definition ===
=== Second transcluded definition ===
=== ... ===

== Proof ==

Note that:

  • the TFAE template adds the page to the Definition Equivalences category. For instructions and additional syntax, see the TFAE template.
  • there are no blank lines between the transcluded definitions.

For the proof, the general guidelines for equivalence proofs apply. There are multiple ways to name the proofs of the implications:

  • 1 implies 2
  • Definition 1 implies Definition 2
  • Definition 1 implies Definition 2, with links

It has not been discussed which one of these is preferred and why.

Also see