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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 <math>A</math> and a family of types <math>B : A \to Type</math>.

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 <math>circle</math> type, which is defined with two constructors, a basepoint;

<math>base : circle</math>

and a loop;

<math>loop : base = base.</math>

The existence of a new constructor for the identity type makes <math>circle</math> a higher inductive type.

See also

  • Coinduction permits (effectively) infinite structures in type theory.

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