subtypessuch

Subtypessuch is a formal concept used in type theory and data modelling to describe a family of refined subtypes defined by a predicate on a base type. An element x belongs to a subtypessuch S if a predicate P(x) holds. The resulting subtype is typically closed under the operations available for the base type, preserving type safety.

Formally, S = { x : T | P(x) }, where T is the base type and P is a predicate. Subtypessuch can be implemented through refinement types, dependent types, or runtime checks; in practice, language features or external validators enforce the predicate. This approach allows defining precise subsets of data that satisfy additional constraints while reusing the structure and behavior of the base type.

Examples illustrate the idea. The set of even integers can be described as { x in Z | x mod 2 = 0 }. The set of non-empty strings is { s in String | length(s) > 0 }. In database schemas, a subtypessuch might constrain a column to values within a specific range, such as age between 0 and 120. Subtypessuch supports safer interfaces by ensuring that inputs and outputs conform to the specified properties, without requiring separate types for each possible constraint.

Applications of subtypessuch include stronger compile-time guarantees in programming languages, safer data validation in APIs, and more expressive schema design in databases. They intersect with refinement types, dependent types, and traditional subtyping, providing a mechanism to encode and propagate invariants through type systems and data workflows.

See also refinement types, dependent types, subtyping, predicate logic.