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 " is two, "S " 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 : -> -> ).
This is the expected function for structural recursion for the type "nat".
Implementations
W- and M-types
W-types are well-founded types in intuitionistic type theory. They generalize natural numbers, lists, binary trees, and other "tree-shaped" data types. Let be a universe of types. Given a type : and a dependent family : →, one can form a W-type. The type may be thought of as "labels" for the constructors of the inductive type being defined, whereas indicates the arity of each constructor. W-types may also be understood as well-founded trees with nodes labeled by elements : and where the node labeled by has -many subtrees. Each W-type is isomorphic to the initial algebra of a so-called polynomial functor. Let 0, 1, 2, etc. be finite types with inhabitants 11 : 1, 12, 22:2, etc. One may define the natural numbers as the W-type with : 2 → is defined by = 0, and = 1. One may define lists over a type : as where and 11 is the sole inhabitant of 1. The value of corresponds to the constructor for the empty list, whereas the value of corresponds to the constructor that appends to the beginning of another list. The constructor for elements of a generic W-type has type We can also write this rule in the style of a natural deduction proof, The elimination rule for W-types works similarly to structural induction on trees. If, whenever a property holds for all subtrees of a given tree it also holds for that tree, then it holds for all trees. In extensional type theories, W-types can be defined up to isomorphism as initial algebras for polynomial functors. In this case, the property of initiality corresponds directly to the appropriate induction principle. In intensional type theories with the univalence axiom, this correspondence holds up to homotopy. M-types are dual to W-types, they represent coinductive data such as streams. M-types can be derived from W-types.
Mutually inductive definitions
This technique allows some definitions of multiple types that depend on each other. For example, defining two parity predicates on natural numbers using two mutually inductive types in Coq: Inductive even : nat -> Prop := | zero_is_even : even O | S_of_odd_is_even : with odd : nat -> Prop := | S_of_even_is_odd :.
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
allows definition of a type and a family of types at the same time. So, a type and a family of types.
Higher inductive types
This is a current research area in Homotopy Type Theory. HoTT differs from ITT by its identity type. Higher inductive types not only define a new type with constants and functions that create elements of the type, but also new instances of the identity type that relate them. A simple example is the type, which is defined with two constructors, a basepoint; and a loop; The existence of a new constructor for the identity type makes a higher inductive type.
