# Inductive type

In type theory, a system has inductive types if it has facilities for creating a new type along with constants and functions that create terms of that type. The feature serves a role similar to data structures in a programming language and allows a type theory to add concepts like numbers, relations, and trees. As the name suggests, inductive types can be self-referential, but usually only in a way that permits structural recursion.

The standard example is encoding the natural numbers using Peano's encoding.

Inductive nat : Type :=
| 0 : nat
| S : nat -> nat.


Here, a natural number is created either from the constant "0" or by applying the function "S" to another natural number. "S" is the successor function which represents adding 1 to a number. Thus, "0" is zero, "S 0" is one, "S (S 0)" is two, "S (S (S 0))" is three, and so on.

Since their introduction, inductive types have been extended to encode more and more structures, while still being predicative and supporting structural recursion.

## Elimination

Inductive types usually come with a function to prove properties about them. Thus, "nat" may come with:

nat_elim : (forall P : nat -> Prop, (P 0) -> (forall n, P n -> P (S n)) -> (forall n, P n)).


This is the expected function for structural recursion for the type "nat".

## Implementations

### W Types

W Types were well-founded types in intuitionistic type theory (ITT).

### Mutually Inductive Definitions

This technique allows some definitions of multiple types that depend on each other.

Inductive even : nat -> Prop :=
| zero_is_even 0 : even
| S_of_odd_is_even : (forall n:nat, odd n -> even (S n))
with
Inductive odd : nat -> Prop :=
| S_of_even_is_odd : (forall n:nat, even n -> odd (S n))


### Induction-Recursion

Induction-recursion started as a study into the limits of ITT. Once found, the limits were turned into rules that allowed defining new inductive types. These types could depend upon a function and the function on the type, as long as both were defined simultaneously.

Universe types can be defined using induction-recursion.

### Induction-Induction

Induction-induction allows definition of a type and a family of types at the same time. So, a type $A$ and a family of types $B : A \to Type$.

### Higher Inductive Types

This is a current research area in Homotopy Type Theory (HoTT). HoTT differs from ITT by its identity type (equality). Higher inductive types not only define a new type with constants and functions that create the type, but also new instances of the identity type that relate them.

A simple example is the $circle$ type, which is defined with two constructors, a basepoint;

$base : circle$

and a loop;

$loop : base = base.$

The existence of a new constructor for the identity type makes $circle$ a higher inductive type.