Javac之glb与lub

5.1.10. Capture Conversion

Let G name a generic type declaration (§8.1.2, §9.1.2) with n type parameters A₁,...,A_n with corresponding bounds U₁,...,U_n.

There exists a capture conversion from a parameterized type G<T₁,...,T_n> (§4.5) to a parameterized type G<S₁,...,S_n>, where, for 1 ≤ i ≤ n :

If T_i is a wildcard type argument (§4.5.1) of the form ?, then S_i is a fresh type variable whose upper bound is U_i[A₁:=S₁,...,A_n:=S_n] and whose lower bound is the null type (§4.1).
If T_i is a wildcard type argument of the form ? extends B_i, then S_i is a fresh type variable whose upper bound is glb(B_i, U_i[A₁:=S₁,...,A_n:=S_n]) and whose lower bound is the null type.

glb(V₁,...,V_m) is defined as V₁ & ... & V_m.

It is a compile-time error if, for any two classes (not interfaces) V_i and V_j, V_i is not a subclass of V_j or vice versa.
If T_i is a wildcard type argument of the form ? super B_i, then S_i is a fresh type variable whose upper bound is U_i[A₁:=S₁,...,A_n:=S_n] and whose lower bound is B_i.
Otherwise, S_i = T_i.

Capture conversion on any type other than a parameterized type (§4.5) acts as an identity conversion (§5.1.1).

Capture conversion is not applied recursively.

就是说在G<T1>到G<S1>的过程中可能T1类型也是T1<X>这样的类型。

Capture conversion never requires a special action at run time and therefore never throws an exception at run time.

4.10.4. Least Upper Bound

List<String>

List<Object>

ST(U_i)

the set of supertypes of Ui.

ST(List<String>)=

{

List<String>,

Collection<String>,

Object

}

ST(List<Object>)=

{

List<Object>,

Collection<Object>,

Object

}

EST(U_i)

the set of erased supertypes of U

EST(List<String>)=

{

List,

Collection,

Object

}

EST(List<Object>)=

{

List,

Collection,

Object

}

the intersection of all the sets EST(U_i)

EC =

{

List,

Collection,

Object

}

MEC

the minimal erased candidate set

for U₁ ... U_k

MEC =

{

List

}

Relevant(G)

Relevant(G) =

{

V | 1≤i≤k:

V in ST(U_i) and V=G<...>

}

Relevant(List) =

{

List<String>,

List<Object>

}

The least upper bound, or "lub", of a set of reference types is a shared supertype that is more specific than any other shared supertype (that is, no other shared supertype is a subtype of the least upper bound). This type, lub(U₁, ..., U_k), is determined as follows.

If k = 1, then the lub is the type itself: lub(U) = U.

Otherwise:

For each U_i (1 ≤ i ≤ k):

Let ST(U_i) be the set of supertypes of U_i.

Let EST(U_i), the set of erased supertypes of U_i, be:

EST(U_i) = { |W| | W in ST(U_i) } where |W| is the erasure of W.

The reason for computing the set of erased supertypes is to deal with situations where the set of types includes several distinct parameterizations of a generic type.

For example, given List<String> and List<Object>, simply intersecting the sets ST(List<String>) = { List<String>, Collection<String>, Object } and ST(List<Object>) = { List<Object>, Collection<Object>, Object } would yield a set { Object }, and we would have lost track of the fact that the upper bound can safely be assumed to be a List.

In contrast, intersecting EST(List<String>) = { List, Collection, Object } and EST(List<Object>) = { List, Collection, Object } yields { List, Collection, Object }, which will eventually enable us to produce List<?>.
Let EC, the erased candidate set for U₁ ... U_k, be the intersection of all the sets EST(U_i) (1 ≤ i ≤ k).
Let MEC, the minimal erased candidate set for U₁ ... U_k, be:

MEC = { V | V in EC, and for all W ≠ V in EC, it is not the case that W <: V }

Because we are seeking to infer more precise types, we wish to filter out any candidates that are supertypes of other candidates. This is what computing MEC accomplishes. In our running example, we had EC = { List, Collection, Object }, so MEC = { List }. The next step is to recover type arguments for the erased types in MEC.
For any element G of MEC that is a generic type:

Let the "relevant" parameterizations of G, Relevant(G), be:

Relevant(G) = { V | 1 ≤ i ≤ k: V in ST(U_i) and V = G<...> }

In our running example, the only generic element of MEC is List, and Relevant(List) = { List<String>, List<Object> }. We will now seek to find a type argument for List that contains (§4.5.1) both String and Object.

This is done by means of the least containing parameterization (lcp) operation defined below. The first line defines lcp() on a set, such as Relevant(List), as an operation on a list of the elements of the set. The next line defines the operation on such lists, as a pairwise reduction on the elements of the list. The third line is the definition of lcp() on pairs of parameterized types, which in turn relies on the notion of least containing type argument (lcta). lcta() is defined for all possible cases.

Let the "candidate" parameterization of G, Candidate(G), be the most specific parameterization of the generic type G that contains all the relevant parameterizations of G:

Candidate(G) = lcp(Relevant(G)) // Relevant(G)经过lcp运算后就会得到the most specific parameterization of the generic type G

where lcp(), the least containing invocation, is:
- lcp(S) = lcp(e₁, ..., e_n) where e_i (1 ≤ i ≤ n) in S // 第一行定义了lcp()
- lcp(e₁, ..., e_n) = lcp(lcp(e₁, e₂), e₃, ..., e_n) // 第二行两两来减少列表中的元素
- lcp(G<X₁, ..., X_n>, G<Y₁, ..., Y_n>) = G<lcta(X₁, Y₁), ..., lcta(X_n, Y_n)> // 依赖lcta进行操作，lcta列举出了两个元素的所有情况
- lcp(G<X₁, ..., X_n>) = G<lcta(X₁), ..., lcta(X_n)>
and where lcta(), the least containing type argument, is: (assuming U and V are types)
- lcta(U, V) = U if U = V, otherwise ? extends lub(U, V)
- lcta(U, ? extends V) = ? extends lub(U, V)
- lcta(U, ? super V) = ? super glb(U, V)
- lcta(? extends U, ? extends V) = ? extends lub(U, V)
- lcta(? extends U, ? super V) = U if U = V, otherwise ?
- lcta(? super U, ? super V) = ? super glb(U, V)
- lcta(U) = ? if U's upper bound is Object, otherwise ? extends lub(U,Object)
and where glb() is as defined in §5.1.10.
Let lub(U₁ ... U_k) be:

Best(W₁) & ... & Best(W_r)

where W_i (1 ≤ i ≤ r) are the elements of MEC, the minimal erased candidate set of U₁ ... U_k;

and where, if any of these elements are generic, we use the candidate parameterization (so as to recover type arguments):

Best(X) = Candidate(X) if X is generic; X otherwise.

Strictly speaking, this lub() function only approximates a least upper bound. Formally, there may exist some other type T such that all of U₁ ... U_k are subtypes of T and T is a subtype of lub(U₁, ..., U_k). However, a compiler for the Java programming language must implement lub() as specified above.

It is possible that the lub() function yields an infinite type. This is permissible, and a compiler for the Java programming language must recognize such situations and represent them appropriately using cyclic data structures.

The possibility of an infinite type stems from the recursive calls to lub(). Readers familiar with recursive types should note that an infinite type is not the same as a recursive type.

The declared type of an exception parameter that denotes its type as a union with alternatives D₁ | D₂ | ... | D_n is lub(D₁, D₂, ..., D_n) (§15.12.2.7).

参考：https://docs.oracle.com/javase/specs/jls/se7/html/jls-14.html#jls-14.20

参考：

（1）https://docs.oracle.com/javase/specs/jls/se8/html/jls-4.html

（2）https://docs.oracle.com/javase/specs/jls/se8/html/jls-5.html#jls-5.1.10