Problem with Theorem, Lemma and Corollary

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In summary: Yes, I agree.In summary, the conversation discusses the use of terminology in mathematical proofs, specifically the usage of "corollary" and "lemma" in relation to theorems. It is mentioned that a corollary should depend only on the content of the theorem, but in some cases, a lemma may also be necessary to prove a proposition. It is noted that there is no strict rule for how to label or name these results, and it ultimately depends on the style and preference of the author and potential referees.
  • #1
DaTario
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Hi All,

I would like to know if is there any problem to present and prove a theorem and a Lemma (in this order) and after that use this theorem and this lemma to prove a corollary (which is simpler to prove and not so important as the theorem).

I have looked up in some papers but with no success.

Thank you

Best wishes

DaTario
 
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  • #2
The word 'corollary' is usually used to refer to a result that follows easily from a theorem that has just been proved. If you need both the Lemma and the Theorem to prove the Corollary, but don't need the Lemma to prove the original Theorem, it would be unusual to call the Corollary a Corollary, rather than just another Theorem.

Simple rule of thumb: If you need to quote any other results, other than other corollaries, of theorem A before you can prove theorem B, then it would be nonstandard and confusing to call B a corollary of A.
 
  • #3
So, a corollary of A should depend only on the content of Theorem A. Is it?

Even if it is very simple to prove, it would be better to call it a Theorem B (that needs Theorem A and the Lemma ), is it?

Thank you
 
  • #4
DaTario said:
Even if it is very simple to prove, it would be better to call it a Theorem B (that needs Theorem A and the Lemma ), is it?
Yes
 
  • #5
Thank you very much, andrewkirk.
DaTario
 
  • #6
DaTario said:
So, a corollary of A should depend only on the content of Theorem A. Is it?

Even if it is very simple to prove, it would be better to call it a Theorem B (that needs Theorem A and the Lemma ), is it?

Thank you

I don't think anybody really cares what you call what.
 
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  • #7
You could also use Proposition instead of Theorem. You can reserve Theorem for the important stuff, Proposition for the less important stuff which are not lemmas. You can also usually get away with a larger theorem, listing several results into one. Instead of having related statements Theorem A, Corollary B, Lemma C and Theorem D, just stuff everything into Theorem A indexing each notable statement. You can have Lemma (or Claim) C inside the proof if you want. It is easier to read and grasp a big theorem rather than several smaller ones.
 
  • #8
What happens in my case is that the theorem 1 is the great one, more difficult to prove - no lemma required. The theorem 2 is easier to prove. One proves it with the theorem 1 and the lemma. But after both theorems are on the table, they look very similar with respect to their function in the whole context. They look like brothers. The lemma is really a different result, smaller in importance.
 
  • #9
micromass said:
I don't think anybody really cares what you call what.
There is always one referee who cares.:smile:
 
  • #10
DaTario said:
There is always one referee who cares.:smile:
And you cannot predict what that referee will prefer. It is a matter of style, so you can pick what you want, if you submit it for publication and someone tells you to rename it, then rename it - who cares (apart from the person you made happy by renaming it).
 
  • #11
Corollary was derived from the Latin word corollarium and means gift, extra. Something at no extra cost.
Lemma is from the Greek word lemma: something taken for granted, a fact. (cp. dilemma)
Theorem is from Greek theorema: spectacle, sight

Source: http://www.etymonline.com/
 
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  • #12
I've always thought this convention part is clear for the mathematicians. 1. Definition. 2. Axiom. 3. Lemma (if needed). 4 Theorem / Proposition. 5. Corollary.
 
  • #13
fresh_42 said:
Corollary was derived from the Latin word corollarium and means gift, extra. Something at no extra cost.
Lemma is from the Greek word lemma: something taken for granted, a fact. (cp. dilemma)
Theorem is from Greek theorema: spectacle, sight

Source: http://www.etymonline.com/

This " at no extra cost " is where my problem was. If a Lemma is also necessary, together with Theorem A, to prove proposition B, we should not call it Corollary of A.
OBS: the Lemma was not necessary to prove Theorem A.
 
  • #14
mfb said:
And you cannot predict what that referee will prefer. It is a matter of style, so you can pick what you want, if you submit it for publication and someone tells you to rename it, then rename it - who cares (apart from the person you made happy by renaming it).

I guess what you are saying has to do with saying that this subject is full of gray areas, where the division lines are not clearly positioned.
 

FAQ: Problem with Theorem, Lemma and Corollary

1. What is the difference between a theorem, lemma, and corollary?

A theorem is a statement that has been proven to be true using a set of axioms or previously proven theorems. A lemma is a smaller result that is used to prove a theorem. A corollary is a direct consequence of a theorem, typically applying the theorem to a specific situation.

2. How do you prove a theorem, lemma, or corollary?

To prove a theorem, lemma, or corollary, you must use logical reasoning and mathematical techniques to show that the statement is true. This may involve using definitions, axioms, previously proven theorems, and other mathematical tools.

3. Can a corollary be proven without proving its corresponding theorem?

No, a corollary cannot be proven without first proving its corresponding theorem. A corollary is a direct consequence of a theorem and relies on the truth of the theorem in order to be proven.

4. Are theorems, lemmas, and corollaries only used in mathematics?

No, theorems, lemmas, and corollaries are also used in other fields such as science and philosophy. In these fields, they may be used to describe and prove principles, laws, or theories.

5. Can a theorem, lemma, or corollary be rewritten in a different way?

Yes, a theorem, lemma, or corollary can be rewritten in different ways as long as the statement remains logically equivalent. This can often provide a deeper understanding of the statement and its implications.

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