Variance and Generalized Constraints for C $^{\sharp}$ Generics

Java Wildcards Meet Definition-Site Variance

“…In general, we want for var the following property, which is a generalization of the subtype lifting lemma of Emir et al's modeling of definition-site variance [8]:…”

“…A "monotonicity" judgment of the syntactic form "vX T mono" appears originally in Emir et al's definition-site variance treatment [8] and later in Kennedy and Pierce [13] as "vX T ok". The semantics of these judgments in the aforementioned sources are similar to its definition here but differs in that they had no function similar to var nor a variance lattice.…”

“…The semantics of these judgments in the aforementioned sources are similar to its definition here but differs in that they had no function similar to var nor a variance lattice. The negation operator ¬ also appears in [8] and [13] and is used to transform a variance by contravariance. Using the implications in Section 4.1, it is easy to show the following properties, which are important for type checking class definitions: Figure 5 contains rules for checking class and method definitions and the definition of the override predicate.…”

“…Since covariance, +, is the identity element for the ⊗ operator (+ ⊗ v = v), the variances v do not need to be transformed by +. By (8), negating the def-site variance annotations in the judgment "¬vX T mono" reflects that T is assumed to be in a contravariant position. We need to reverse the subtype relationship order for argument types and ranges in method type signatures.…”

“…7 Negating the range of a method type signature ensures the range is wider for the subtype. For code examples motivating why ranges need to be widened for the subtype, see Section 2.4 of [8]. More generally, if (1) e.<T>m() type checks implying the type actual T is within the type bounds for m's type argument and (2) typeof(e ) <: typeof(e), then e .<T>m() should type check as well even if m is overridden in the subclass.…”