Talk:Set Union Preserves Subsets/Corollary/Proof 1

Renaming. I changed the name because I added another proof and am about to write another proof of the main theorem that relies on this result. --Dfeuer (talk) 22:17, 28 February 2013 (UTC)

I thought "corollary" was for things with only one proof. Also, it seems silly for each to be called a corollary of the other. --Dfeuer (talk) 22:18, 28 February 2013 (UTC)

Corollary is for statements that are, well, a corollary (look it up if you have to). They are bona fide theorems in themselves, but are (usually stated as) a consequence or particular instance of another result. It is perfectly valid to have double subpages, /Corollary/Proof X for suitable X. For example, that is useful when a proof of the general by means of the particular can be given, as you envisage in the present case. — Lord_Farin (talk) 22:22, 28 February 2013 (UTC)

What is the advantage of calling one theorem Theorem A and Theorem B/Corollary, while calling the other Theorem B and Theorem A/Corollary when each can be given a perfectly good name of its own? --Dfeuer (talk) 22:30, 28 February 2013 (UTC)

"...when each can be given a perfectly good name of its own". Yeah. There are cases where that does not apply. Furthermore, it is generally not the case that a corollary is "as strong", in some vague sense, as the main result. If you get the proofs up first (..with adequate spacing, please..), we can worry about names in the future, I reckon? — Lord_Farin (talk) 22:35, 28 February 2013 (UTC)

Yes, but in this case that does apply. Also, if you can't decide whether something is a lemma or a corollary, I think that suggests it should be called neither. I was trying to put proofs up when PM went through and reverted everything. --Dfeuer (talk) 22:40, 28 February 2013 (UTC)

What something is rather depends on what proofs we have, right? Before you crafted/located an alternative proof, there was only the proof of the "corollary/lemma" that made it a corollary to the main page of this bunch. Over time, admittedly such things may change. But without actually having seen your proof I couldn't assess that the status of corollary was no longer adequate, could I? I'm not omniscient and I can't read your thoughts; please keep that in mind. — Lord_Farin (talk) 22:45, 28 February 2013 (UTC)

Well, the situation has changed now. Also, I imagine a very similar technique could be used to prove Constructive Dilemma from theorems whose names I don't know:

$(a \implies b) \implies (a \lor c \implies b \lor c)$

$(c \implies d) \implies (b \lor c \implies b \lor d)$

Since $\implies$ is transitive, or whatever:

$(a \implies b) \land (c \implies d) \implies (a \lor c \implies b \lor d)$ --Dfeuer (talk) 00:43, 1 March 2013 (UTC)