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A cleave cost is an alternative cost that's paid instead of the spell's mana cost. Casting a spell for its cleave cost doesn't change the spell's mana value.

If an effect allows you to “cast a spell without paying its mana cost,” you can't cast that spell for its cleave cost.

If you cast a spell for its cleave cost, that spell doesn't have any of the text in square brackets while it's on the stack.

If you paid the cleave cost, you must put a card into your hand as Dig Up resolves (assuming there is at least one card in your library).

You can't cast a spell for both its cleave cost and another alternative cost. For example, if an effect gives an Alchemist's Retrieval in your graveyard a flashback cost of {U}, you can't cast it from your graveyard for its cleave cost.