8. Dependent Types and Pi Types

Dependent Types and Pi Types

Dependent types are types that depend on values, allowing types to be parameterized by terms.

They enable richer type systems where types can describe precise properties of values.

One common form of dependent type is the Pi type, denoted as Π, which represents dependent function types.

In Pi types, the type of the result can depend on the value of one or more arguments.

For example, in a dependent type system, we can have a function type Vec : (n : Nat) → Type → Type, where Vec n A represents a vector of length n with elements of type A.

Vec is parameterized by both n and A, and the type of the result depends on the value of n.

Lab Activity: Implementing Dependent Types and Pi Types

Let's implement a basic example of dependent types and Pi types in Python using functions.

def vec(n, A):
    class Vector:
        def __init__(self, elements):
            self.elements = elements

        def __repr__(self):
            return f"Vector({self.elements})"

    return Vector([None] * n)

def main():
    vec2 = vec(2, int)
    vec3 = vec(3, str)
    
    print("Vector of length 2:", vec2)
    print("Vector of length 3:", vec3)

if __name__ == "__main__":
    main()
  

In the lab activity, we define a function `vec` that returns a vector object based on the specified length n and element type A.

We demonstrate dependent types and Pi types by creating vectors of different lengths and element types.