I Problem with Theorem, Lemma and Corollary

[DaTario]

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

 

[andrewkirk]

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.

 

[DaTario]

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

 

[andrewkirk]

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

 

[DaTario]

Thank you very much, andrewkirk.
DaTario

 

[micromass]

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.

 

[disregardthat]

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.

 

[DaTario]

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.

 

[DaTario]

I don't think anybody really cares what you call what.

There is always one referee who cares.

 

[mfb]

There is always one referee who cares.

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).

 

[fresh_42]

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/

 

[dextercioby]

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.

 

[DaTario]

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.

 

[DaTario]

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.