Well-typings for Java – The algorithm –

It Coercion IMatrix Ia1 Iɛ1 Iɛ2 Iɛ3 Iɛ4 . . .
0 M ∗ ∗ ∗ ∗ ∗ ∗
1 . . . M ∗ ∗ ∗ ∗ ∗ ∗
1 M ⋖ a1 M M, V ∗ ∗ ∗ ∗ ∗
2 . . . M M, V ∗ ∗ ∗ ∗ ∗
2 a1 ⋖ ɛ1 M M, V M, V ∗ ∗ ∗ ∗
2 ɛ1 ⋖ ɛ2 M M, V M, V M, V ∗ ∗ ∗
2 ɛ2 ⋖ ɛ3 M M, V M, V M, V M, V ∗ ∗
2 ɛ3 ⋖ ɛ4 M M, V M, V M, V M, V M, V ∗
2 . . . M M, V M, V M, V M, V M, V ∗
3 . . . M M, V M, V M, V M, V M, V ∗
Table 1. Example CONSISTENT
The algorithm WTYPE: The presented algorithms are now summerized in
the adaptation of WTYPE. In WTYPE an own well-typing is determined for
each function declaration.
The function WTYPE is given as:
WTYPE: TypeAssumptions × class→ { WellTyping } ∪ { fail }
WTYPE( Ass, Class( cl, extends( τ ′ ), fdecls, ivardecls ) ) =
let
({ f1 : a1, . . . , fn : an }, CoeS) = TYPE( Ass, Class( cl, extends( τ ′ ),
fdecls, ivardecls ) )
(σ, AC) = MATCH( CoeS )
in
if CONSISTENT( AC ) then
{ (AC, Ass ⊢ fi : σ( ai )) | 1�i�n }
else fail
Now we will complete our example.
Example 5. In 2 we declared the Javaλ program
class Matrix extends Vector {
op = #{ m -> #{ f -> f(Matrix.this, m) } } },
determined the set of assumptions Ass1 and the type assumptions op : aop.
In 3 we determined the substition σ and the set of atomic coercions AC and in
4 we showed that AC is consistent.
From this follows, that in WTYPE for op the well-typing
(AC, Ass1 ⊢ op : # # γ ′ 3 (# ɛ ′′
4 (ɛ4, ɛ ′ 4)) (β1)).
is determined.
As in the function CONSISTENT all possible instances are determined, it holds:
ɛ4 = Matrix or Vector.
Furthermore, from AC follows, that it holds β1 ⋖ ɛ ′ 4 and ɛ ′′
4 ⋖ γ ′ 3, which describes
corelations of type variables of the result type.