Gemara

130b (Rav Chisda): If one damages or eats Matanos Kehunah (things that must be given to Kohanim), he is exempt.

131a - Question (Beraisa): If the king's officers took Reuven's grain for money that he owed, he must take Ma'aser (from other Tevel, and give the Ma'aseros to the proper recipients. The same applies to one who eats or damages Tevel!)

Answer: That case is different, for he saves money.

Beitzah 9a (Rabah): One may separate Chalah on Yom Tov from a dough kneaded before Yom Tov.

Shmuel's father forbids.

Suggestion: Shmuel argues with hus father;

(Shmuel): One may eat bread of Chutz la'Aretz and separate Chalah afterwards.

Rejection (Rava): Shmuel agrees that if one called it Chalah, Zariim may not eat it.

Gitin 31a (Mishnah): If one designated produce, intending that he can declare some of this produce to be Terumah or Ma'aser when he wants, he may do so. He may assume that the produce or money is intact.

R. Elazar says, if he finds that the produce was lost, he must be concerned for 24 hours.

Rishonim

Rambam (Hilchos Bikurim 5:12): One may eat bread of Chutz la'Aretz and separate Chalah afterwards, for the source of Chalah in Chutz la'Aretz is mid'Rabanan.

Rosh (Beitzah 1:13): Rashi explains that one may eat all the bread and leave over enough for Chalah, and separate it at the end. This is wrong. If so, Shmuel should have said 'he leaves over enough for Chalah.' Rather, he leaves over more than enough for Chalah, in order to separate Chalah on it, for we require "Reishis" (first) from which there is a recognizable remnant. Surely we do not require Min ha'Mukaf (separating the Chalah near what it exempts), for we permit eating before separating.

Mahariyo (48): If Chalah was not separated from a dough, and it was cooked, Bitul does not apply, for the entire dough is Tevel. In any case, one can separate Chalah after it was cooked. He takes a little, so 101 times (as much as the Chalah) will remain in the dough. Even though when he cooked it, it was Tevel, since he separated afterwards, for mid'Rabanan laws we rely on Breirah (to say that it is as if he separate from the beginning - Beitzah 37b). Also, for Chalah of Chutz la'Aretz, we hold like Shmuel, who says that one may eat and separate later. This shows that Yesh Breirah.

Darchei Moshe (YD 323:4, brought in Shach 5): He says to take less than one part in 101, lest the Chalah forbid the dough. This is difficult. One may eat Chalah of Chutz la'Aretz and separate later. We do not distinguish separating more or less than one part in 101! Rather, since Chalah of Chutz la'Aretz is mid'Rabanan, we are not stringent to forbid retroactively. We forbid only from now and onwards if Chalah was separated and then became mixed with the dough, since it was once called Chalah. If it was never called Chalah, we are not stringent. It seems that the Rambam permits a mixture that is half Chalah, and all the more so in this case. Even according to Mahariyo, if it was cooked in 60 times as much water as the Chalah, we ignore Mino (the dough), and the Chalah is Batel in Eino Mino (the water). If so, one may separate any amount up to one part in 60 of the water.

R. Akiva Eiger (on Shach 5): For bread we do not require 101, for roasting forbids only Kedei Netilah. If so, Mahariyo would require leaving over the Shi'ur for Chalah and also Kedei Netilah. (I.e. one must peel off a layer from the bread, since it absorbed from the Chalah.) Even if there is 101 times as much as bread as Chalah, this does not help (exempt from peeling off a layer), since it gets absorbed only in the thickness of Kedei Netilah.

Rambam (Hilchos Ma'aser 12:2): One who bought Demai (Safek Tevel from an ignoramus) may rely on the seller (if he says that it was tithed) to eat it on Shabbos. He may not eat it on Motza'ei Shabbos until he tithes it all like Demai, what he ate and what remains.

Rashi (131a DH She'ani): The Beraisa does not discuss whether Beis Din makes him pay, rather, whether he must tithe. Surely he must tithe, for he saves money. He profits, for if not, he would have had to pay money.

Rashi (Gitin 31a DH Harei): If the Terumah he separated is spoiled (and perhaps it was spoiled when he separated it), he is concerned that his produce is still Tevel. If he did not yet eat it, he must separate from it.

Tosfos Yom Tov (Gitin 3:8): The Bartenura says like Rashi. They connote that if he already ate it, he need not separate. We assume that the Terumah was intact at the time he separated. However, the Rambam obligates separating even if he already ate the Chulin, lest the produce was lost before he separated. I say that they do not argue. Rashi explains the Mishnah, which is R. Elazar's opinion. He is concerned only for the last 24 hours. Similarly, he is concerned only if the Chulin was not eaten yet. The Rambam rules unlike R. Elazar, that one is liable for everything he ate. It is a strong concern, therefore we are concerned even if he already ate. Even according to R. Elazar, we need not say (like Rashi) that he is not concerned if he already ate.

Rashash (Gitin 31a DH Rashi): The Tosfos Yom Tov says that the Rambam holds that he must tithe what he ate. In Chulin, we say that one is exempt. It seems that this is even for one who ate. Rashi explains that Beis Din does not obligate him to pay, but he must tithe. However, this is difficult for Tosfos and the Rambam, who merely cited the Beraisa verbatim. (It connotes that he need not tithe at all.) Also, why is the Rambam stringent about Demai? There, there is no concern for stealing from the Shevet (of Kohanim or Leviyim), for ha'Motzi mi'Chavero Alav ha'Re'ayah! (One need not give unless others can prove that he owes, i.e. it is Tevel.) For Demai there was no enactment to eat and separate afterwards, like we say about Chalah of Chutz la'Aretz.

Ha'arah 6 on Birkei Yosef (YD 323:4): The Rashbatz (Piskei Chalah 7, in Sof Chelek 2 of his Teshuvos) says that b'Di'eved, one may separate Lo Min ha'Mukaf. The Birkei Yosef holds that this helps to fix the Isur. The Tosfos Yom Tov says that the Rambam holds like this. The Sha'ar ha'Melech (Terumos 3:8) disagrees, for if so, how is someone lashed for eating Tevel?! The Birkei Yosef connotes that he fixes it only a little (therefore, he is lashed). Also Yeshu'os Malko asked the Rashash's question. This is not difficult. Indeed, if he wants to fix the Isur that he ate, he separates. The Gemara says that he is exempt from paying. However, there is a clear proof that separating afterwards does not totally fix retroactively. A Mishnah in Demai (7:2) says that one says 'what I will leave over...' The Yerushalmi asked that if he says 'from now', it is all mixed together. If he separates afterwards, he drank Tevel. This shows that separating afterwards does not totally fix the Isur. This is why the Birkei Yosef said that it helps only a little.

Poskim

Rema (YD 323:1): If Chalah became mixed with the dough, if there are not 101 parts for Bitul, if he did not eat the dough, he asks a Chacham to permit (annul) his separation.

Shulchan Aruch: In Chutz la'Aretz permitted eat and afterwards separate Chalah, for the source of Chalah in Chutz la'Aretz is mid'Rabanan.

Rema: He must leave over a little more than the Shi'ur of Chalah that he separates, so that when he separates, the remainder will be recognizable. He can exempt Lo Min ha'Mukaf only this dough that was obligated with this Chalah, but if a dough was kneaded by itself, we may not separate Lo Min ha'Mukaf. One may not eat bread of Eretz Yisrael before separating Chalah. One may not separate it Lo Min ha'Mukaf, just like Terumah.

Beis Yosef (DH u'Mah): Even according to the opinion that nowadays Chalah is mid'Rabanan in Eretz Yisrael, its source is mid'Oraisa.

Beis Yosef (DH v'Chasav): Semag brings from R. Elchanan that the Heter to eat and separate later is when the entire dough was kneaded and obligated together. If Matzos were obligated individually, it is logical that one cannot separate on one to exempt another unless they are together.

Bach (2): The Tur wrote that one may eat and separate afterwards, which connotes that he leaves over more than the Shi'ur of Chalah, in order to separate with a recognizable remainder.

Darchei Moshe (4, brought in Shach 6): Hagahos Maimoniyos (b'Sof Hilchos Zera'im, 6, from Sefer ha'Terumah) says that if the Chalah became mixed with the dough and there are not 101 parts for Bitul, it is good to annul the separation. I say that this is only if the dough is intact, and one can take another Chalah. If it was eaten, one cannot annul the separation, for then retroactively he ate Tevel.

Shach (6): Even though now that he does not annul the separation, retroactively he ate Chalah, which is forbidden to Zarim, many Poskim say that it is Batel is the majority, so it was permitted. All agree that Tevel is forbidden.

Shach (12): One may not separate from another loaf Lo Min ha'Mukaf. He must bring them together (when separating).

Gra (13): Shmuel permitted only regarding Chalah of Chutz la'Aretz. Also, we may not separate Chalah of Eretz Yisrael Lo Min ha'Mukaf.

Birkei Yosef (3): If one forgot to separate Chalah of Eretz Yisrael before Shabbos, one may not mentally separate on Shabbos to permit eating the bread.

Birkei Yosef (4): If one forgot to separate Chalah of Eretz Yisrael and already ate most of the bread b'Shogeg, he separates from what remains on everything, even on what he ate, to fix what is possible. B'Di'eved, one may separate Lo Min ha'Mukaf, like we say regarding Terumah (331:25).

Ha'arah 6: The Birkei Yosef says that he separates from the remainder. It seems that this is not precise. Even if everything was eaten, he can separate from elsewhere. However, perhaps it is precise. If he would separate from elsewhere and intend also for what he ate, this would be like separating from Chayav (what is liable) to exempt Patur (what is exempt), since what he already ate is not totally liable. Separating from a Torah Chiyuv on a Chiyuv mid'Rabanan is like from Chayav on Patur (Menachos 31a). However, perhaps nowadays that for everything we separate any amount, it is not considered from Chayav on Patur, for he does not add anything due to his intent to include what he ate. This requires investigation. R. Yom Tov Algazi (Simchas Yom Tov 3) says that since when we give Terumah to a Kohen one may not separate from Chayav on Patur, the same applies nowadays. We do not distinguish. He says that Teshuvas Maharam connotes like this.