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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

If you cast a card “without paying its mana cost,” you can’t pay any alternative costs. On the other hand, if the card has optional additional costs (such as kicker or multikicker), you may pay those when you cast the card. If the card has mandatory additional costs (such as Momentous Fall does), you must pay those when you cast the card.

Any cards you can’t cast (because they’re lands or they have no legal targets, perhaps) or choose not to cast simply remain exiled.

You cast nonland cards from exile as part of the resolution of Hellcarver Demon’s ability. You may cast those cards in any order. Timing restrictions based on the card’s type (such as creature or sorcery) are ignored. Other restrictions are not (such as “Cast [this card] only before the combat damage step”). You cast all of the cards you like, putting them onto the stack, then Hellcarver Demon’s ability finishes resolving. The spells you cast this way will then resolve as normal, one at a time, in the opposite order that they were put on the stack.

The cards from your library are exiled face up.

If you control two Hellcarver Demons that deal combat damage to a player at the same time, each Demon’s ability triggers. When the first one resolves, you’ll sacrifice all permanents except that Hellcarver Demon (meaning you’ll sacrifice the other one). Then you’ll exile six cards from your library and may cast nonland cards exiled that way. After those spells resolve, the other Demon’s ability resolves. You’ll sacrifice all permanents you now control (including the first Demon and any permanents put on the battlefield as a result of the spells you just cast), exile six more cards from your library, and cast any number of them.

The triggered ability is mandatory. All the permanents you control, including all lands, will be sacrificed.