14. Inductive Types and Datatypes

Inductive Types and Datatypes

Inductive types are a fundamental concept in type theory and functional programming languages.

They allow the definition of new types in terms of existing types and constructors.

In Martin-Löf Type Theory (MLTT) and similar systems, inductive types are used to define data structures and algebraic datatypes.

Inductive types are defined by specifying their constructors, which are functions that build elements of the type.

Each constructor specifies how to create an element of the type, potentially with arguments of other types.

For example, in MLTT, we can define the natural numbers as an inductive type with two constructors: zero and successor.

The definition might look like this:

data Nat : Type where
  zero : Nat
  succ : Nat → Nat

This defines the natural numbers as an inductive type called `Nat`, with two constructors `zero` and `succ`.

Each constructor specifies how to build elements of the type `Nat`.

Lab Activity: Implementing Inductive Types and Datatypes

Let's implement a simple example in Python to demonstrate inductive types and datatypes.

class Nat:
    pass

class Zero(Nat):
    pass

class Succ(Nat):
    def __init__(self, pred):
        self.pred = pred

def to_int(n):
    if isinstance(n, Zero):
        return 0
    elif isinstance(n, Succ):
        return 1 + to_int(n.pred)

def main():
    zero = Zero()
    one = Succ(zero)
    two = Succ(one)
    
    print("Zero:", to_int(zero))
    print("One:", to_int(one))
    print("Two:", to_int(two))

if __name__ == "__main__":
    main()
  

In the lab activity, we define a base class `Nat` to represent the natural numbers.

We then define subclasses `Zero` and `Succ` to represent the constructors of the natural numbers.

We implement a conversion function `to_int` to convert natural numbers to integers.

We demonstrate the implementation by creating instances of natural numbers and converting them to integers.