Cut the lime in 4 pieces, then, add them to a cup or cocktail shaker
Add the mint leaves, and muddle well
Add the rum*, or the cachaça, to the muddled minty-lime juice, stir and set this mixture aside**
Meanwhile, peel, core then dice a whole fresh pineapple. Discard the skin and the core, then, add the pineapple chunks into a blender with the sugar and pure it. You'll blend this for a few minutes, until there are no chunks left, and you're left with a super creamy, foamy mixture. No need to add water or ice here. We basically want to make a pineapple cream!
After that, add that creamy pineapple juice to a pitcher, or punch bowl, and top it with ice
Remove and discard the limes from the shaker/cup. Then add the minty-lime booze mixture to the pineapple juice. I prefer my drink to be more pineapply, so I try to not add all the mint to the pitcher, to avoid the mint flavor from taking over. But, you can pour all the mint leaves in if you want, or just a few, or none at all. Up to you!
Stir and serve!