Intuitionistic type theory

Intuitionistic type theory is like a language that helps people talk about ideas and things in a special way. It's a way of thinking that helps us understand things better.

Imagine you have a big box of toys. You could sort these toys into different groups based on their similarities. You might put all the cars into one group, all the dolls into another, and so on. Intuitionistic type theory is like doing this with ideas and concepts.

In this type theory, everything has a "type." For example, a car has the type "vehicle," and a doll has the type "toy." These types help us sort and organize our ideas.

Now imagine you have a toy car with wheels that spin. You might describe this toy by saying "it's a vehicle with spinning wheels." In intuitionistic type theory, this would be written as "a vehicle with spinning wheels," and it would be a type of its own.

This type theory is different from other ways of thinking because it focuses on "proofs." A proof is like a way of showing that something is true. For example, you might prove that 2+2=4 by showing two groups of two objects, which makes a total of four objects.

In intuitionistic type theory, we can use proofs to show that a certain idea is true. For example, we can prove that a car is a type of vehicle by showing that it has certain features, like an engine and wheels.

Overall, intuitionistic type theory is a way of thinking and talking about ideas that helps us understand them better. It's like sorting our toys into different groups and using proofs to show why they belong in those groups.