hasType

HasType is a formal relation used in type theory and programming language semantics to express that a term e has a type T within a given typing context Γ. It is commonly written as HasType Γ e T and is the central judgment used by type checkers and proof systems to certify well-typed terms.

In a typical simply typed or dependently typed setting, HasType is defined by a small set of inference rules. Common rules include: Var: if a variable x : T appears in Γ, then HasType Γ x T. Abs: if HasType (Γ, x : T1) e T2, then HasType Γ (λx. e) (T1 -> T2). App: if HasType Γ e1 (T1 -> T2) and HasType Γ e2 T1, then HasType Γ (e1 e2) T2. More complex systems extend these with additional rules for constants, let bindings, or polymorphism.

HasType plays a key role in establishing type safety, since proofs of preservation (types are maintained under evaluation) and progress (well-typed terms do not get stuck) are usually phrased in terms of HasType judgments. In practical type systems, HasType is complemented by a separate mechanism for type synthesis or bidirectional checking, where some terms infer their type while others are checked against a given type.

Examples illustrate its use: in a context Γ that assigns a type to a variable x, x has that type; the identity function λx. x has type α -> α in a context that assigns a fresh type variable α; and the application e1 e2 has type T2 if e1 has a function type T1 -> T2 and e2 has type T1.

See also typing judgment, type system, and type safety.