Lemmas

A lemma is a proven proposition used as a stepping stone toward proving a larger result. The word comes from Greek lemma, meaning something taken for granted or assumed. In mathematics, lemmas are not the final goal themselves but are proved to facilitate the proof of a main theorem. Some lemmas are of independent interest; others are primarily technical tools.

Within mathematics, lemmas vary in importance. Euclid's lemma — if a prime divides a product ab, then it divides a or b — is a classic example; Zorn's lemma is another influential result used to establish existence statements. Lemmas may be stated and proved before the main theorem, or introduced within a proof as needed.

In linguistics and natural language processing, a lemma is the canonical dictionary form of a word—the base form used to index and analyze words. Inflected forms such as ran, runs, and running share the same lemma run. Lemmatization is the process of converting text to lemmas for analysis.

Relation to related terms: a corollary is a result that follows directly from a theorem; a proposition is a proved statement that may be less central than a theorem. The term lemma emphasizes its role as a useful, intermediary step in a logical argument or formal proof, rather than the primary statement of interest.