Ringschluss

In mathematics, a Ringschluss is a mathematical proof technique where the equivalence of several statements can be proven without having to prove all pairwise equivalences directly. In English it is also sometimes called a cycle of implications, closed chain inference, or circular implication; however, it should be distinguished from circular reasoning, a logical fallacy.
In order to prove that the statements are each pairwise equivalent, proofs are given for the implications,,, and.
The pairwise equivalence of the statements then results from the transitivity of the material conditional.

Example

For the proofs are given for,, and. The equivalence of and results from the chain of conclusions that are no longer explicitly given:
That is.

Motivation

The technique saves writing effort above all. In proving the equivalence of statements, it requires the direct proof of only out of the implications between these statements. In contrast, for instance, choosing one of the statements as being central and proving that the remaining statements are each equivalent to the central one would require implications, a larger number. The difficulty for the mathematician is to find a sequence of statements that allows for the most elegant direct proofs possible.