044: Steiner triple systems

% 
% ECLiPSe sample code - Steiner triplets
%
% The following program computes so-called Steiner triplets.
% These are triplets of numbers from 1 to N such that any
% two triplets have at most one element in common.
%
% Here is an example of running this program:
% 
% ?- steiner(9,X).
% 
% X = [[1, 2, 3], [1, 4, 5], [1, 6, 7], [1, 8, 9],
%      [2, 4, 6], [2, 5, 8], [2, 7, 9], [3, 4, 9],
%      [3, 5, 7], [3, 6, 8], [4, 7, 8], [5, 6, 9]]
% Yes (2.89s cpu, solution 1, maybe more) ? ;
% 

:- lib(ic_sets), lib(ic).

steiner(N, Sets) :-
        NB is N * (N-1) // 6,           % compute number of triplets
        intsets(Sets, NB, 1, N),        % initialise the set variables
        ( foreach(S,Sets) do
            #(S,3)                      % constrain their cardinality to 3
        ),
        ( fromto(Sets,[S1|Ss],Ss,[]) do
            ( foreach(S2,Ss), param(S1) do
                #(S1 /\ S2, C),         % constrain the cardinality
                C #=< 1                 % of pairwise intersections to 1
            )
        ),
        label_sets(Sets).               % search

label_sets([]).
label_sets([S|Ss]) :-
        insetdomain(S,_,_,_),
        label_sets(Ss).