W- and M-types
W-types are well-founded types in intuitionistic type theory (ITT).[1] They generalize natural numbers, lists, binary trees, and other "tree-shaped" data types. Let U be a universe of types. Given a type A : U and a dependent family B : A → U, one can form a W-type . The type A may be thought of as "labels" for the (potentially infinitely many) constructors of the inductive type being defined, whereas B indicates the (potentially infinite) arity of each constructor. W-types (resp. M-types) may also be understood as well-founded (resp. non-well-founded) trees with nodes labeled by elements a : A and where the node labeled by a has B(a)-many subtrees.[2] 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
:={\mathsf {W}}_{x:\mathbf {2} }f(x)}
with f : 2 → U is defined by f(12) = 0 (representing the constructor for zero, which takes no arguments), and f(22) = 1 (representing the successor function, which takes one argument).
One may define lists over a type A : U 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 a 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 (under the propositions-as-types interpretation) 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 (resp. M-types) can be defined up to isomorphism as initial algebras (resp. final coalgebras) for polynomial functors. In this case, the property of initiality (res. finality) corresponds directly to the appropriate induction principle.[3] In intensional type theories with the univalence axiom, this correspondence holds up to homotopy (propositional equality).[4][5][6]
M-types are dual to W-types, they represent coinductive (potentially infinite) data such as streams.[7] M-types can be derived from W-types.[8]
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 elements of 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.