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Bavli, M'nachos 61:2, quotes the tana Rabbi Shim'on (in my own loose translation):

Three types [of offering] have three commands to them [all told]: each has two, but not the third. They are personal sh'lamim, communal sh'lamim, and a m'tzora's asham. Personal sh'lamim requires leaning while it's alive and waving once it's slaughtered, but not waving while it's alive. Communal sh'lamim requires waving while it's alive and once slaughtered but no leaning. A m'tzora's asham requires leaning and waving while alive, but no waving once slaughtered.

The g'mara then tries to derive that the third rule, the one that doesn't apply to each type of offering, should apply to it. How? By kal vachomer (argument a fortiori). The argument in each case goes like this: Offering A lacks command X but has command Y, so offering B, which has X, should, a fortiori, have Y. For example, communal sh'lamim lacks leaning but has waving while still alive, so personal sh'lamim, which has leaning, should a fortiori have waving while it's alive. And in each case the g'mara refutes the argument.

There are six possible such arguments, for the six possibilities of values for "A" and "B". (Any offering can be "A", and any other can be "B".) Yet the g'mara only goes through three of these: where "A" is the personal sh'lamim and "B" the communal (whose refutation I outline below), where "A" is the communal sh'lamim and "B" the personal (whose refutation is a pasuk that proves live waving can't be extended by derivation from "A" to other offerings), and where "A" is the personal sh'lamim and "B" the m'tzora's asham (whose refutation is a pasuk that proves slaughtered waving can't be extended by derivation from "A" to other offerings). It omits the other three possible arguments.

Now, one of the arguments it omits — where "A" is a m'tzora's asham and "B" is a communal sh'lamim — is dealt with by the argument it gives for another case, where "A" is a personal sh'lamim and "B" is a communal sh'lamim. In the latter case, the g'mara says the argument a fortiori is valid but we have a tradition (halacha l'Moshe miSinay, it seems) that a communal sh'lamim does not have leaning. Very well: once the g'mara's offered that response to the argument from personal sh'lamim, there's no point in trying to make the same argument from the m'tzora's asham.

But why are the other two arguments — where "A" is a communal sh'lamim and "B" is a m'tzora's asham, and where "A" is a m'tzora's asham and "B" is a personal sh'lamim — omitted?

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Okay, so I read through the Gemara, and although I see why it could be a little confusing, I think I also understand how to read it so that it makes perfect sense.

The Gemara first addresses Shalmei Yachid: "If Shalmei Tzibbur, which do not require Semichah, require Tenufas Chayim, Shalmei Yachid, which do require Semichah, should certainly require Tenufas Chayim." To that the Gemara answers that we have a drashah of "Osam" - only Shalmei Tzibbur require Tenufas Chayim, to the exclusion of Shalmei Yachid. Your question had been why we don't turn to Asham Metzora and ask why we don't pose a similar Kal V'chomer from there; presumably the Gemara would say that we've already found a specific exclusion for it, and Asham Metzora wouldn't help.

The Gemara then addresses Shalmei Tzibbur and attempts to learn it from Shalmei Yachid. As you pointed out in your question, this part of the Gemara reads just fine; once we've said it's a Halacha L'Moshe MiSinai, learning from Asham Metzora is a waste of time.

Finally, the Gemara addresses the Asham Metzora and tries learning it from Shalmei Yachid, to which it responded, similar to what it did in its first Kal V'chomer, that we have a drashah of "Oso," to the exclusion of the Asham Metzora. Like above, you could probably answer that once we've found a specific exclusion for it, another Kal V'chomer wouldn't help.

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