Two objective functions

If we have two objective functions in Eclipse CLP that Cost1 is more important than Cost2, is the following true?

minimize(minimize(labeling(Vars), Cost1), Costs2).

Yes, this works, provided you tell the inner minimize to compute all optimal solutions, rather than just the first one (use the bb_min/3 variant of minimize):

:- lib(ic).
:- lib(branch_and_bound).

minmin(X, Y) :-
    [X,Y] #:: 1..4,
    Cost1 #= -2*X,
    Cost2 #= -Y,
    bb_min(
        bb_min(
            labeling([X,Y]),
            Cost1,
            bb_options{solutions:all}
        ),
        Cost2,
        bb_options{solutions:one}
    ).

The operational behaviour is that first Cost1 is minimized (ignoring Cost2), then Cost2 is minimized (with Cost1 fixed at its minimum):

?- minmin(X, Y).
Found a solution with cost -2
Found a solution with cost -4
Found a solution with cost -6
Found a solution with cost -8
Found a solution with cost -1
Found a solution with cost -2
Found a solution with cost -3
Found a solution with cost -4
X = 4
Y = 4
Yes (0.00s cpu)