The following snippet from Wikipedia describes the correct usage of terminology for mathematical statements (which I often get confused about).

Theorems are often indicated by several other terms: the actual label "

" is reserved for the most important results, whereas results which are less important, or distinguished in other ways, are named by different terminology.theorem

- A
is a statement not associated with any particular theorem. This term sometimes connotes a statement with a simple proof, or a basic consequence of a definition that needs to be stated, but is obvious enough to require no proof. The wordpropositionpropositionis sometimes used for the statement part of atheorem.- A
is a "pre-theorem", a statement that forms part of the proof of a larger theorem. The distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. Gauss's lemma and Zorn's lemma, for example, are interesting enough that some authors present the nominal lemma without going on to use it in the proof of a theorem.lemma- A
is a proposition that follows with little or no proof from one other theorem or definition. That is, propositioncorollaryBis a corollary of a propositionAifBcan readily be deduced fromA.- A
is a necessary or independently interesting result that may be part of the proof of another statement. Despite the name, claims must be proved.claimThere are other terms, less commonly used, which are conventionally attached to proven statements, so that certain theorems are referred to by historical or customary names. For examples:

, used for theorems which state an equality between two mathematical expressions. Examples include Euler's identity and Vandermonde's identity.Identity, used for certain theorems such as Bayes' rule and Cramer's rule, that establish useful formulas.Rule. Examples include the law of large numbers, the law of cosines, and Kolmogorov's zero-one law.Law^{[3]}. Examples include Harnack's principle, the least upper bound principle, and the pigeonhole principle.Principle- A
is a reverse theorem. For example, If a theorem states that A is related to B, its converse would state that B is related to A. The converse of a theorem need not be always true.ConverseA few well-known theorems have even more idiosyncratic names. The

division algorithmis a theorem expressing the outcome of division in the natural numbers and more general rings. TheBanach–Tarski paradoxis a theorem in measure theory that is paradoxical in the sense that it contradicts common intuitions about volume in three-dimensional space.An unproven statement that is believed to be true is called a

(or sometimes aconjecturehypothesis, but with a different meaning from the one discussed above). To be considered a conjecture, a statement must usually be proposed publicly, at which point the name of the proponent may be attached to the conjecture, as with Goldbach's conjecture. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis.

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