15122 Compile-Time Step 2: Determine Method Signature

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Restricting the analysis to the unary case, we have the constraint A >> F = G< super U> A must be a supertype of the generic type G However, since A is not a parameterized type, it cannot depend upon U in any way It is a supertype of the type G< super X> for every X such that super X is a valid type argument to G No meaningful constraint on U can be derived from A

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If A is an invocation of a generic type declaration H, where H is either G or superclass or superinterface of G, then:

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If H G , then let S1, , Sn be the formal type parameters of G, and let H<U1, , Ul> be the unique invocation of H that is a supertype of G<S1 , , Sn>, and let V = H< super U1, , super Ul>[Sk = U] Then this algorithm is applied recursively to the constraint A >> V

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The treatment here is analogous to the case where A = G< extends U> Here our example would produce an invocation H< super U1>[S = U]

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We have A = G< super W> >> F = G< super U> for some type expression W It must be the case that W << U, by the subtyping rules for wildcard types

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Otherwise, if A is of the form G<, Xk-1, super W, , Xk+1, >, this algorithm is applied recursively to the constraint W << U

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Otherwise, no constraint is implied on Tj

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Compile-Time Step 2: Determine Method Signature 15122

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Next, for each type variable Tj, 1 j n , the implied equality constraints are resolved as follows: For each implied equality constraint Tj = U or U = Tj:

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Otherwise, if U is Tj, then this constraint carries no information and may be discarded Otherwise, the constraint is of the form Tj = Tk for k j Then all constraints involving Tj are rewritten such that Tj is replaced with Tk, and processing continues with the next type variable Then, for each remaining type variable Tj, the constraints Tj :> U are considered Given that these constraints are Tj :> U1 Tj :> Uk, the type of Tj is inferred as lub(U1 Uk), computed as follows: For a type U, we write ST(U) for the set of supertypes of U, and de ne the erased supertype set of U, EST(U) = { V | W in ST(U) and V = |W| } where|W| is the erasure ( 46) of W

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The reason for computing the set of erased supertypes is to deal with situations where a type variable is constrained to be a supertype of several distinct invocations of a generic

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If U is not one of the type parameters of the method, then U is the type inferred for Tj Then all remaining constraints involving Tj are rewritten such that Tj is replaced with U There are necessarily no further equality constraints involving Tj, and processing continues with the next type parameter, if any

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This concludes the process of determining constraints on the formal type parameters of a method Note that this process does not impose any constraints on the type parameters based on their declared bounds Once the actual type arguments are inferred, they will be tested against the declared bounds of the formal type parameters as part of applicability testing Note also that type inference does not affect soundness in any way If the types inferred are nonsensical, the invocation will yield a type error The type inference algorithm should be viewed as a heuristic, designed to perfdorm well in practice If it fails to infer the desired result, explicit type paramneters may be used instead