typesincludingT1

Types including T1 is a concept used in type theory and programming language design to denote the class of types that contain a distinguished type T1 as a component, parameter, or substructure. The term is used to discuss how T1 participates in the construction of more complex types and to reason about properties that are preserved when building new types from T1.

Formally, a type T is said to include T1 if there exists a type expression built from T1 and other type constructors such that T is equal to that expression. In languages with algebraic data types and higher-order type constructors, this often means T may be formed by applying type constructors like List, Optional, Pair, or function arrows to T1, with T1 appearing in one or more positions. Examples: if T1 is Int, then List<Int>, Optional<Int>, Pair<Int, Bool>, and Int -> String all include T1.

The notion is useful for reasoning about type relationships, substitutions, and generic programming. For instance, properties like monotonicity under certain constructors can be discussed: if a type constructor preserves inclusion of T1, then applying it to a type that includes T1 yields another type that includes T1. This idea also relates to subtyping in languages with structural typing, and to embedding or containment in dependent or higher-kinded types.

Limitations include that not all languages support a uniform notion of type inclusion, and the exact meaning can vary with the type system. In practice, types including T1 are a convenient abstraction for discussing how a base type propagates through complex type expressions.