Confusion over generics subtyping in java specs?

Question

I'm reading java specs https://docs.oracle.com/javase/specs/jls/se10/html/jls-4.html#jls-4.10.2 and this line confuses me:

D<U1 θ,...,Uk θ> , where D<U1,...,Uk> is a generic type which is a direct supertype of the generic type C and θ is the substitution [F1:=T1,...,Fn:=Tn].

is Uk θ here casting? and after substitution D becomes D<(U1)T1,...,(Uk)Tk> ? If so why does the author omit the brackets in Uk θ as part of casting syntax? Thanks!

Answer 1

Not casting. It's the mathy language used to describe a type.

In code, it's saying that if:

interface D<A, B> {}
interface C<A, B> extends D<A, B> {}

class Y implements D<String, Number> {}
class X implements C<String, Integer> {} // Note Integer is a subtype of Number

Then this compiles:

D d = new X();

Answer 2

Given a generic type declaration C<F1,...,Fn> (n > 0), the direct supertypes of the parameterized type C<T1,...,Tn> , where Ti (1 ≤ i ≤ n) is a type, are all of the following:

D<U1 θ,...,Uk θ> , where D<U1,...,Uk> is a generic type which is a direct supertype of the generic type C<F1,...,Fn> and θ is the substitution [F1:=T1,...,Fn:=Tn].

Let us see an example.

public class D<U1,U2> {
}

//Note the order of F3,F4.
class C<F1,F2,F3,F4> extends D<F4,F3>{

}

class Main{

    public static void main(String[] args) {

        C<Integer,String,Float,Number> obj = new C<>();

        D<Float ,Number> obj2 = obj; // Compilation error

        D<Number ,Float> obj3 = obj; //This is fine


    }
}

What do you think θ is here ?

θ = [F1:= Integer , F2:= String, F3:= Float, F4:= Number ]

Now D<Number,Float> is the correct supertype. But not D<Float,Number> . Why ?

Because of Ui θ . So what does this mean ?

Well , it just means substitute ie Scan U1 , U2 , ... till Uk and if you find any Fi , then replace it by Ti .

In our example , it means Scan F4 and F3 of class D. Did you have F4 and F3 in θ ? Yes. Then F4 , F3 of D have to be as defined in θ .

Note : The answer is as per my understanding. I don't have any concrete reference of what Uk θ means in Java type semantics.