%
% Quasigroup problem 4 Idempotent in MiniZinc.
%
% This model is a translation of the EssencePrime model quasiGroup4Idempotent.eprime
% from the Minion Translator examples.
% """
% The quasiGroup existence problem (CSP lib problem 3)
%
% An m order quasigroup  is an mxm multiplication table of integers 1..m,
% where each element occurrs exactly once in each row and column and certain
% multiplication axioms hold (in this case, we want axiom 4 to hold).
% """
% See
% http://www.dcs.st-and.ac.uk/~ianm/CSPLib/prob/prob003/index.html
% http://www.dcs.st-and.ac.uk/~ianm/CSPLib/prob/prob003/spec.html

%
% This MiniZinc model was created by Hakan Kjellerstrand, hakank@gmail.com
%

% Licenced under CC-BY-4.0 : http://creativecommons.org/licenses/by/4.0/

include "globals.mzn";

int: n = 9; % solutions for n=5, n=9...
set of int: nDomain = 0..n-1;

array[nDomain, nDomain] of var nDomain: quasiGroup;
array[nDomain] of var nDomain: qgDiagonal;

% solve satisfy;
solve :: int_search([quasiGroup[row, col] | row, col in nDomain],
first_fail, indomain_min, complete) satisfy;

constraint

% accessor for diagonal
forall(i in nDomain) (
qgDiagonal[i] = quasiGroup[i,i]
)
/\
% All rows have to be different
forall(row in nDomain) (
all_different([quasiGroup[row,col] | col in nDomain ] )
)
/\
% All columns have to be different
forall(col in nDomain) (
all_different([quasiGroup[row,col] | row in nDomain] )
)
/\
% (j*i)*(i*j) = i
forall(i in nDomain) (
forall(j in nDomain) (
quasiGroup[quasiGroup[j,i],quasiGroup[i,j]] = i
)
)
/\
% Idempotency
forall(i in nDomain) (
quasiGroup[i,i] = i
)
/\
% Implied (from Colton,Miguel 01)
% All-diff diagonal
all_different(qgDiagonal)

/\
% anti-Abelian
forall(i in nDomain) (
forall(j in nDomain) (
(i != j) ->
(quasiGroup[i,j] != quasiGroup[j,i])
)
)
/\
% if (i*i)=j then (j*j) = i
forall(i in nDomain) (
forall(j in nDomain) (
(quasiGroup[i,i]=j) -> (quasiGroup[j,j]=i)
)
)
/\
% Symmetry-breaking constraints
forall(i in nDomain) (
quasiGroup[i,n-1] + 2 >= i
)
;

output [
if col = 0 then "\n" else " " endif ++
show(quasiGroup[row, col])
| row, col in nDomain
] ++ ["\n"];