Gemara

(Mishnah): If a Zevach became Pasul, if there is another Zevach without Nesachim, the Nesachim of the Pasul Zevach are offered for it.

Question Rav Chisda taught that if oil was designated for a Minchah, it may not be used for another Minchah!

Answer (R. Yanai): Beis Din (of Kohanim) stipulates that Nesachim (for Korbanos Tzibur) are Kodesh on condition that if they are not needed, they may be used for a different Zevach.

Ta'anis 10b (Beraisa): If one was fasting for a Choleh, and he recovered, or due to an affliction, and it passed, he completes the fast.

25b (Beraisa - R. Meir): If they were fasting and rain fell, they complete the fast only if it fell after sunrise;

R. Yehudah says, they complete the fast only if it fell after midday;

R. Yosi says, they complete the fast only if it fell after nine hours of the day.

R. Yehudah Nesiah decreed a fast. Rain fell after sunrise. R. Ami told him that we finish only if it fell after midday.

(Mishnah): A case occurred in Lud (in which the Tzibur was fasting for rain. Rain came before midday. R. Tarfon told them to eat, drink and make a Yom Tov. They did so, and towards evening they said Hallel ha'Gadol.)

Question: Why didn't they say Halled immediately?

Answer (Abaye and Rava): We say Hallel only when satiated, with a full stomach.

Rishonim

Rambam (Hilchos Ta'anis 1:15): If an individual was fasting for a Choleh, and he recovered, or due to an affliction, and it passed, he must complete his fast.

Rambam (16): If the Tzibur was fasting for rain and rain came before midday, they do not complete the fast. Rather, they eat, drink and gather and say Hallel ha'Gadol. We say it only when satiated, with a full stomach. If rain came after midday, since the majority of the day was in Kedushah, they complete the fast. Similarly, if the Tzibur was fasting due to an affliction and it passed, or due to a decree (against Torah) and it was nullified, they complete the fast only if they were saved after midday.

Ra'avad: The Rambam distinguishes between an individual and a Tzibur. I say that there is no difference. In every case they finish, except for fasts for rain, for the only affliction that totally goes away is (lack of) rain, which is temporary.

Rosh (Ta'anis 1:6): If one was fasting for a Choleh, and he recovered, or due to an affliction, and it passed, he must complete his fast. The Ra'avad asked why this is unlike fasting for rain. If rain comes before midday, we cease fasting. He answered that if enough rain came for all their needs, they have no more need for rain now. We always need mercy for other afflictions, that they should not return. This is wrong. Even if they did not receive enough rain for all their needs, they eat, drink and rejoice and say Hallel, for their prayers were answered, and they trust that Hash-m will give them all their needs! We could say so also about other afflictions! Rather, a Tzibur is unlike an individual. The Tzibur must praise and thank amidst a full stomach. An individual finishes his fast. The Rambam connotes like this. (Regarding a Tzibur, he does not distinguish between rain and other afflictions. Rather, he must distinguish between a Tzibur and an individual. An individual completes all the fasts he accepted. It seems that the same applies if the Choleh died. Since he accepted fasts Stam without stipulating, he intended that through this acceptance his prayers will be accepted. The Gemara discussed when the Choleh recovered, for then a Tzibur is unlike an individual.

Ran (Ta'anis 2b DH Harei): Presumably, he finishes not only that day's fast, rather, all the fasts he accepted. The Ra'avad explained why we distinguish fasts for rain to fasts for other afflictions. The Rambam connotes that an individual completes fasting for all afflictions. We exert a Tzibur (to fast) only according to what is needed.

Mordechai (620, citing Avi ha'Ezri):The Beraisa teaches when one must complete, based on when the Choleh recovered or the affliction passed. It seems that the Halachah follows R. Yehudah. However, Rashi says that it refers to the Choleh recovering. Likewise, if the Choleh died, the one who vowed must fulfill his vow.

Poskim

Shulchan Aruch (OC 569:1): If an individual was fasting due to an affliction, and it passed, or for a Choleh, and he recovered, he must complete all the fasts that he accepted on himself. However, if the Tzibur was fasting for anything, and they were answered before midday, they do not finish. If Chachamim and most of the Tzibur want to complete the fast, an individual may not separate from them.

Beis Yosef (DH Al): The Tur says that if the affliction passed, one must complete the fast. If not, he looks like one who stipulates with his Maker. This connotes even if the affliction passed before midday.

Beis Yosef (DH v'Gam): The Magid Mishneh says that a Tzibur does not finish the fast, for we minimize exertion of the Tzibur. Alternatively, when Beis Din decrees a fast, their heart (intent) stipulates about this. For an individual, Devarim sheb'Lev (unspoken intents) are invalid. Since the Rambam and Rosh agree, we rule like them.

Terumas ha'Deshen (280): If Chachamim and a Tzibur decreed a fast for another Tzibur in dire affliction of a severe Tefisah (royal decree), and after they fasted more than half the day, they heard that they were exempted from the Tefisah yesterday, before they accepted the fast, letter of the law they need not complete the fast. Even if it were a proper vow, since retroactively it is found that it was mistake, it need not be permitted (Nedarim 65a). Therefore, even an individual need not complete the fast. Or Zaru'a says that if a Chacham and most of the Tzibur want to complete the fast, an individual may not separate from them. Here, we would never be stringent on the Tzibur, since it was a mistake from the beginning. Once, in such a case, at the end of the day (just before Minchah), people heard that the Tefisah was annulled before the Tzibur accepted to fast. Even so, a big Chacham commanded that the Tzibur finish the fast and read in the Torah and Haftorah like on other fasts. Perhaps this was because it was so late in the day.

Shulchan Aruch (565:3): Some say that an individual says Aneinu (on a fast day) only at Minchah, lest Bulmus seize him (and he will need to eat due to mortal danger), and retroactively he was a liar in his prayer (he said 'this fast day').

Taz (3): The Rosh asked, why can one borrow a fast and repay? He makes himself a liar in his prayer! When Rav suggested to R. Yehoshua brei d'Rav Idi that he borrow his fast and repay, why didn't he answer 'I already prayed Aneinu. I do not want to make myself a liar!'? Rather, since he intended to fast, he is not a liar if he eats afterwards. Tosfos and the Beis Yosef answered that R. Yehoshua had not yet prayed Aneinu. Semak brought a proof that we are not concerned for becoming liars from the case when rain fell before midday. We eat, even though we already said Aneinu! I answer that here he is a liar (if he will need to eat), because he attributes Hash-m's answer to his prayer due to the fast. This implies that if he will not fast, Hash-m will not answer him. How can he do so? Perhaps Bulmus will seize him, and he will forfeit being answered, or he will be a liar in his prayer, for he will still ask to be answered (but earlier, he attributed being answered to the fast)! Rain is different, for (if he eats after rain comes,) he needs no other answer. His request was already fulfilled. Also, only a Tzibur need not complete the fast, for Lev Beis Din stipulates when they must fast. Devarim sheb'Lev do not help for an individual.

Mishbetzos Zahav (1): Also regarding Tzedakah, we distinguish a Tzibur from an individual. If (they pledged Tzedakah to be a merit for a Choleh, and) the Choleh died, a Tzibur is exempt, for Lev Beis Din stipulates. An individual is liable, unless he stipulated.

Magen Avraham (1): We do not exert a Tzibur. Also, Lev Beis Din stipulates for them. If so, even if the Choleh died, they need not complete the fast, according to the Ran. However, the Rosh exempts from completing the fast only if they were answered, so that they can say Hallel amidst satiation. The Shulchan Aruch says 'they were answered.' This connotes like the Rosh. However, the Levush connotes like the Ran.

Eshel Avraham: If they vowed Tzedakah for a Choleh and he recovered, according to all reasons, they must fulfill the vow, both a Tzibur and an individual. If he died, the Ran exempts, for Lev Beis Din stipulates, but the Rosh obligates even to finish a fast, even if he died before midday.

Nehar Shalom (1): We hold like the Rosh. Firstly, the Ra'avad exempts from completing the fast only regarding a fast for rain. For other matters we finish, even if we were answered. We do not hold like the Ra'avad, for the Rosh and Ran are a majority against him. However, if the Choleh died, the Rosh and Ra'avad are a majority against the Ran. Also, the Rambam distinguishes between a Tzibur and individual regarding an affliction that passed or a decree that was annulled. This implies that if a Choleh recovered, a Tzibur is like an individual, like the Rosh. Also, the Rosh explicitly said that the Gemara discussed a Choleh who recovered, for then a Tzibur is unlike an individual. The Ran did not clarify his opinion. Perhaps Lev Beis Din stipulates only for when they are answered, but if the Choleh dies, they must complete the fast. Also, the Ran permits completing the fast, just he does not obligate this. Since the Rosh and those with him obligate completing the fast, why shouldn't we follow them?! Therefore, the Shulchan Aruch's ruling is solid.

Sha'ar ha'Tziyon (8): The Magen Avraham sides to be lenient, like the Levush. The proofs of Nehar Shalom are not solid. It is a mere reasoning to say that Lev Beis Din stipulates only when they are answered, for they become happy, and there is no fast, but not when he died, and they were not answered at all.

Mishnah Berurah (4): Also if one vowed fasts or Tzedakah so that Ploni will live, even if Ploni died, he must fulfill them, unless he vowed 'if Ploni will live...'

Mishnah Berurah (6): Before midday, the fast did not begin, for midday is the normal time to eat (the first meal of the day).

Note: Most people eat at four hours of the day (Shabbos 10a). At midday, the fast began for everyone, even Chachamim, who eat at six hours of the day.

Birkei Yosef (575:3): If individuals accepted the first three fasts due to lack of rain, according to Beis Din, and rain came, they need not complete them, for Lev Beis Din stipulates.