finitetype

Finite type, often written finitetype, is a type that has a finite number of distinct values. If a type A has exactly n elements, we say it has cardinality n. In many type-theoretic and programming contexts, there is a canonical finite type Fin n consisting of the numbers 0 through n−1; a type A is finite if there exists a bijection between A and Fin n for some n.

Finite types are closed under finite products and sums: the product A × B has cardinality |A|·|B| and the coproduct A + B has cardinality |A|+|B|. They can be constructed by enumerated data types with finitely many constructors, or by using dependent types to define Fin n and related families.

Examples include the unit type with a single element, the boolean type with two elements, or an enum-like type with three constructors. A generic Fin n type provides a standard finite domain with n elements.

In practice, finite types support exhaustive pattern matching, total functions, and decidable equality. They are central in formal verification, model checking, property-based testing with finite domains, and in libraries of proof assistants and dependently-typed languages. Languages such as Coq and Lean provide a Fintype/Finite typeclass to assert and work with finite representations; Idris and Agda offer similar constructs.

Limitations: not every data structure is finite; many programs rely on infinite or unbounded types such as lists. When using finite types, it is important to enforce explicit bounds or use a Fin n representation to preserve finiteness and tractability.