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An increment ability will only trigger if the permanent it's on is a creature. Similarly, if the permanent is no longer a creature when its increment ability resolves, the ability won't do anything.

If increment triggers, the comparison will happen again when the ability tries to resolve. If the amount of mana spent to cast that spell isn't greater than either of that creature's stats anymore, the ability will do nothing.

The increment ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered or otherwise leaves the stack.

When comparing the mana spent to the creature's stats as the increment ability resolves, it's possible that the stat that is less than the mana spent changes from power to toughness or vice versa. For example, say you control a 1/3 creature with increment and you pay two mana to cast a spell. Since the amount of mana you spent is greater than the creature's power, increment will trigger. While the increment ability is on the stack, an effect gives the creature with increment +2/-2, making it a 3/1. When the increment ability resolves, the creature will still get a +1/+1 counter since the amount of mana you spent is greater than that creature's toughness.

Whenever you cast a spell, check the amount of mana you spent to cast it against the power and toughness of the creature with increment. If the amount of mana you spent isn't greater than either of that creature's stats, increment won't trigger at all.