Inaho's world is... MAKEN-KI
Maken-ki! (マケン姫っ!) is a Japanese manga series by Hiromitsu Takeda that later was animated having a 1st season with it's OVAs and now having 2nd season comming soon.
The story starts with Takeru Ohyama, who has enrolled to Tenbi Academy. However, he learns that the school is for students who possess magical and spiritual energies called Elements and who wield crafted weapons known as Makens. The students engage in school sanctioned combat matches that showcase their powers.
Takeru soon finds himself surrounded by girls, such as Haruko, a friend of his since childhood, or Inaho, who claims to be his fiancée and Kodama, who takes him by an enemy and therefore has the need to keep an eye on him all the time.
He joins the madō kenkei kikō ("anti-evil organization"), known as Maken-ki, which supports the student council's operations.
She is the self-proclaimed fiancée of Takeru. She really loves him and even though he doent remember her much, she has decided to wait until he fully does.
Her personallity is basically the Genki type; being most of the time energetic, curious, and having a big fondness towards food (specially snacks, candies...)
At first, Inaho was one of the three students that the Element detector, Keronbo, could not measure. However, she manifests her Maken when she saves Takeru and a cat from a falling boulder. She names the cat Monji and usually wears it on her head. Her Maken, Kamudo, is a gauntlet that greatly increases her strength and speed.