Well-typings for Java – The algorithm –

– It holds C ≤∗ C if for each θi and θ ′ i , respectively,
one of the following conditions is valid3 • θi = ?θi, θ ′ i = ?θ ′
i and θi ≤∗ θ ′
i.
• θi = ? θi, θ ′ i = ? θ ′
i and θ ′
i ≤∗ θi.
• θi, θ ′ i ∈ STypeTS ( BT V ) and θi = θ ′ i
• θ ′ i = ?θi
• θ ′ i = ? θi
(cp. [GJSB05] §4.5.1.1 type argument containment)
– Let C be the capture conversions of C and
C ≤∗ C then holds
– θ ≤ ∗ θ|T
C ≤ ∗ C.
Definition 4 (Subtyping relation ≤ ∗ on TypeTS ( BT V )). Let ≤ ∗ be a
subtyping relation on simple types STypeTS ( BT V ). Then, the extension to
TypeTS ( BT V ) is defined as:
# θ (θ ′ 1, . . . , θ ′ n) ≤ ∗ # θ ′ (θ1, . . . , θn) iff θ ≤ ∗ θ ′ and θi ≤ ∗ θ ′ i.
Example 1. Let the following Javaλ program be given.
class Matrix extends Vector {
}
##Matrix(#Matrix(Matrix, Matrix))(Matrix)
op = #{ Matrix m -> #{ #Matrix(Matrix, Matrix) f -> f(Matrix.this, m)}}
#Matrix(Matrix, Matrix)
mul = #{ (Matrix m1, Matrix m2) ->
Matrix ret = new Matrix ();
for(int i = 0; i < size(); i++) {
Vector