%%  Schur numbers
%%  The range of integers to be partitioned is n and the number of parts 
%%  is k. 
%%  The main predicate is "inpart". Atom "inpart(X,P)" represents the fact 
%%  that number X is included in part P.
%%  The goal is to find a partition of integers {1,2,...,n} into k parts
%%  such that each part is sum free. 

% Domain predicates
number(1..n). part(1..p).

% Assign each integer to exactly one part. 
1 { inpart(X,P):part(P) } 1 :- number(X).

% X and X+X cannot be in the same part
:- number(X),  part(P), inpart(X,P), inpart(X+X,P).

% X, Y, and X+Y cannot be in the same part
:- number(X;Y), X != Y, part(P), inpart(X,P), inpart(Y,P), inpart((X+Y),P).


% Remove some symmetries by taking always the lowest free part
:- number(X), part(P), inpart(X,P), part(P1), P1 < P, not occupied(X,P1). 

% occupied(X,P) iff there is a number Y < X in P
occupied(X,P) :- number(X;Y), part(P),  Y < X, inpart(Y,P).