Not sure about text displayed behind the main windows; didn't watch that far. But a lot of magic is "free" especially basic spells like Fire and Ice Storm. If you cast a spell that isn't free she'll say something afterwards, like "I'm full of magic power!"
In Madou Monogatari for the SNES (of which Madou Monogatari I for GG is the final dungeon, albiet in a much-simplified form) Arle starts out with the Healing spell, I think. And Bayoen. Both cost magic, and she'll tell you how much magic she's got when you use them.