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Academic Guidance Academic Guidance - Academic guidance for those pursuing a college degree... what college? Grad school? PhD help?


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February 8th, 2019, 10:03 AM   #1
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A lemma which is a special case of a theorem

It is often the case that a theorem is called using a lemma which in turn is an easy consequence of the theorem (in other words, is a special case of the theorem).

Which term could you suggest specifically for such a lemma?

I think about using my own coined word (like "specialia") to denote such lemmas in my book. Is it worth to coin a new word?
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February 8th, 2019, 10:23 AM   #2
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It is often the case that a theorem is called using a lemma which in turn is an easy consequence of the theorem (in other words, is a special case of the theorem).
You're thinking of a corollary, an easy consequence of a theorem.

A lemma is a minor theorem that helps you prove a major theorem.

In written presentations, corollaries come after their corresponding theorem; and lemmas come before the theorem they're helping to prove
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February 8th, 2019, 10:25 AM   #3
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Originally Posted by Maschke View Post
You're thinking of a corollary, an easy consequence of a theorem.

A lemma is a minor theorem that helps you prove a major theorem.

In written presentations, corollaries come after their corresponding theorem; and lemmas come before the theorem they're helping to prove
But I want such a corollary to be named "theorem" not "corollary". It improves emphasis of importance of such a theorem, rather than calling it just a corollary.
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February 8th, 2019, 12:39 PM   #4
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Originally Posted by Maschke View Post
You're thinking of a corollary, an easy consequence of a theorem.

A lemma is a minor theorem that helps you prove a major theorem.

In written presentations, corollaries come after their corresponding theorem; and lemmas come before the theorem they're helping to prove
Also "corollary" is an easy consequence of the premise. In my case the theorem may be not an easy consequence of the lemma. What I say is that the reverse implication (from the theorem to the lemma, in the order reverse to the order it is proved in the text) is easy.
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February 8th, 2019, 02:32 PM   #5
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Call them “tip” or “hint” .
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February 8th, 2019, 02:33 PM   #6
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Also "corollary" is an easy consequence of the premise. In my case the theorem may be not an easy consequence of the lemma. What I say is that the reverse implication (from the theorem to the lemma, in the order reverse to the order it is proved in the text) is easy.
That's right, a corollary is a relatively easy consequence of a theorem.

But a theorem is generally not an easy consequence of a lemma. Rather, a lemma is kind of a building block. It's something you need to prove along the way to the main theorem; but it's self-contained enough that you can pull it out. Sometimes because you want to use it for something else; or because it simplifies the narrative flow of the main theorem. It's sort of like the programming practice of pulling out a chunk of code into its own subroutine.

You're right, a lemma need not make the main theorem easy.

But really it's just terminology, these aren't meaningful distinctions, as the axiom of choice, Zorn's lemma, and the well-ordering theorem show. A lot of the naming is historical accident.
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