Aseh Doche Lo Taaseh

Aseh Doche Lo Taseh is a principle that allows one to violate a negative commandment in order to fulfill a mitzvah. An allegorical reason for this principle is that positive commandments stem from the mitzvah to love Hashem and negative commandments stem from the mitzvah to fear Him. Since loving Him is important important than fearing him the principle is that a positive commandment overrides a negative commandment.

If it can be avoided

1. If the entire situation of the negative commandment can be avoided so that one can do the mitzvah without any prohibition one is obligated to do so and not apply the principle of aseh doche lo taseh. Many hold that this principle is Biblical. Others hold it is rabbinic.
2. If by avoiding the situation of the aveirah in order to fulfill the mitzvah one isn't certain that one will fulfill the mitzvah then some achronim learn that it would be permitted to have the aseh override the lo taseh.

Reasons not to apply aseh doche lo taseh

1. If the lo taseh has the punishment of karet.
2. One aseh can't override both a lo taseh and aseh. This idea is found in Beitzah 8b and Shabbat 133a with respect to Yom Tov.
3. In the bet hamikdash. Rava in Zevachim 97b says that we don’t apply aseh doche lo taseh in the bet hamikdash. His proof is that otherwise we should allow breaking the bones of a korban pesach to eat the meat inside the bones since eating the korban is a mitzvah. Rashba Eruvin 100a s.v. bha and Ritva 100a s.v. amar cites Tosfot as using this rule to explain why aseh doche lo taseh doesn't apply to sprinkling a mixture of bechor and olah blood. Rashi and Tosfot Eruvin 100a seems to ignore this rule and the Turei Even Rosh Hashana 28b s.v. hari points this out.
4. If the mitzvah isn’t fulfilled simultaneously with the prohibition (B’idna). The Gemara Beitzah 8b establishes that one can’t violate an aseh that is overriding a lo taseh if they aren’t simultaneous. The example is that it is forbidden to dig dirt in order to fulfill the mitzvah of kisuy hadam on Yom Tov. Piskei Tosfot Zevachim n. 69 writes that there is no requirement that the mitzvah and sin are simultaneous if by definition there is no other way to fulfill the mitzvah unless a sin is performed. (See Ri Chaver’s question on this Piskei Tosfot.) See Tosfot Bava Batra 13a who writes that a maaseh biyah isn't bidna of the kiyum of pru urevu even though it is necessary for the mitzvah.
5. If the aseh was created because of your peshiya. Tosfot Eruvin 100a writes that if the prohibition is created by your negligence we don’t allow an aseh to override a lav. Tosfot Harosh 100a s.v. rabbi agrees. Rashba Eruvin 100a s.v. bha and Ritva 100a s.v. amar cite this from the Raavad and agree. Ayelet Hashachar Zevachim 80a adds that this is true even if the person who was negligent isn't the kohen who is doing the mitzvah. Rabbi Akiva Eiger Ketubot 40a asks on Tosfot because the gemara Ketubot 40a applies aseh overriding a lo taseh to a man who raped or seduced a single woman who is forbidden to him even though that mitzvah was created by his negligence. He answers that only if the mitzvah by definition is created by negligence we apply aseh doche lo taseh, otherwise we wouldn’t apply aseh doche lo taseh if the aseh could have been created without negligence and in this instance it is. Sfat Emet Zevachim 80a agrees based on Pesachim 83 that applies aseh doche lo taseh to burning notar on Yom Tov even though the mitzvah was created through negligence since by definition notar is always created through negligence.
6. If the aseh is ita b’shayla and the lo taseh isn’t. Turei Even Rosh Hashana 28b writes that the reason that Rabbi Yehoshua didn’t allow applying aseh doche lo taseh to a case of a mixture of bloods is because the aseh wasn’t as serious as the lo taseh. That is, when the blood of an olah mixes with a bechor since the olah could be undone with hatarat nedarim the aseh isn’t as powerful as the kedusha of bechor which couldn’t be undone with hatarat nedarim. The type of logic is also found in Yevamot 5a.
7. If the mitzvah and chefsa are in two chafasim. The Turei Even Rosh Hashana 28b writes that even if the mitzvah is associated with one item and the prohibition is associated with another we can still apply aseh doche lo taseh. His proof is that we apply aseh doche taseh to eating a piece of kodshim that absorbed the taste of another kodshim that is forbidden (Zevachim 97b), even though the taste of the food and the food are separate entities. The same is true of two bloods that are mixed. Even if one is doing a mitzvah with one blood and a prohibition of bal tosif with another blood it is permitted to say aseh doche lo taseh. The Haghot Imrei Baruch argues with the proof of absorbed tastes because that could be viewed as one entity. Ayelet Hashachar Zevachim v. 3 p. 96 agrees with the Turei Even but notes that it is interesting. However, Sfat Emet Zevachim 80a s.v. bgemara amar implies otherwise that one couldn’t apply aseh doche lo taseh to two items where one has the aseh and one the lo taseh.
8. For a part of a mitzvah. Maharal Eruvin 100a writes that we don't apply aseh doche lo taseh to fulfill the last three zerikot of an olah since the primary mitzvah is already done with one zerika. Gilyonei Hashas Eruvin 100a explains that since the four zerikot are one mitzvah we don't say that a part of a mitzvah should allow using aseh doche lo taseh.