yap-6.3/library/clpfd.pl

/*  clpfd.pl -- Finite domain constraint solver.

    Copyright (C): 2007, Markus Triska (triska@gmx.at)

    This program is free software; you can redistribute it and/or
    modify it under the terms of the GNU General Public License
    as published by the Free Software Foundation; either version 2
    of the License, or (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public
    License along with this library; if not, write to the Free Software
    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
*/

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Thanks to Tom Schrijvers for his "bounds.pl", the first finite
   domain constraint solver included with SWI-Prolog. I've learned a
   lot from it and could even use some of the code for this solver.
   The propagation queue idea is taken from "prop.pl", a prototype
   solver also written by Tom. Highlights of the present solver:

   Symbolic constants for infinities
   ---------------------------------

   ?- X in 0..5 \/ 10..sup, Y #= -X, Z #= X + Y.
   %@ X = _G109{0..5 \/ 10..sup},
   %@ Y = _G114{inf..-10 \/ -5..0},
   %@ Z = _G120{inf..sup}

   No artificial limits (using GMP)
   ---------------------------------

   ?- N is integer(2**66), X #\= N.
   %@ X = _G1676{inf..73786976294838206463 \/ 73786976294838206465..sup}

   Often stronger propagation
   ---------------------------------

   ?- Y #= abs(X), Y #\= 3, Z * Z #= 4, A in 0..10, A mod 2 #= 0.
   %@ Y = _G1448{0..2 \/ 4..sup},
   %@ X = _G1446{inf..-4 \/ -2..2 \/ 4..sup},
   %@ Z = _G1454{-2 \/ 2},
   %@ A = _G1463{0 \/ 2 \/ 4 \/ 6 \/ 8 \/ 10}

   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

   Many things can be improved; if you want to help, feel free to
   e-mail me. A good starting point is taking a propagation algorithm
   from the literature and adding it.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

:- module(clpfd, [
                  (#>)/2,
                  (#<)/2,
                  (#>=)/2,
                  (#=<)/2,
                  (#=)/2,
                  (#\=)/2,
                  (#\)/1,
                  (#<==>)/2,
                  (#==>)/2,
                  (#<==)/2,
                  (#\/)/2,
                  (#/\)/2,
                  in/2,
                  ins/2,
                  all_different/1,
                  all_distinct/1,
                  sum/3,
                  tuples_in/2,
                  labeling/2,
                  label/1,
                  indomain/1,
                  lex_chain/1,
                  serialized/2
                 ]).


:- use_module(library(swi)).
:- use_module(library(gensym)).

:- op(760, yfx, (#<==>)).
:- op(750, xfy, (#==>)).
:- op(750, yfx, (#<==)).
:- op(740, yfx, (#\/)).
:- op(730, yfx, (#\)).
:- op(720, yfx, (#/\)).
:- op(710,  fy, (#\)).
:- op(700, xfx, (#>)).
:- op(700, xfx, (#<)).
:- op(700, xfx, (#>=)).
:- op(700, xfx, (#=<)).
:- op(700, xfx, (#=)).
:- op(700, xfx, (#\=)).
:- op(700, xfx, in).
:- op(700, xfx, ins).
:- op(450, xfx, (..)).
:- op(700, xfx, cis).
:- op(700, xfx, cis1).
:- op(700, xfx, cis_geq).
:- op(700, xfx, cis_gt).
:- op(700, xfx, cis_leq).
:- op(700, xfx, cis_lt).

/** <module> Constraint Logic Programming over Finite Domains

Constraint programming is a declarative formalism that lets you
describe conditions a solution should satisfy. This library provides
CLP(FD), Constraint Logic Programming over Finite Domains. It can be
used to model and solve various combinatorial problems from diverse
areas such as planning, scheduling, and graph colouring.

As an example, consider the cryptoarithmetic puzzle SEND + MORE =
MONEY, where different letters denote distinct integers between 0 and
9. It can be modeled in CLP(FD) as follows:

==
:- use_module(library(clpfd)).

puzzle([S,E,N,D] + [M,O,R,E] = [M,O,N,E,Y]) :-
        Vars = [S,E,N,D,M,O,R,Y], Vars ins 0..9,
        all_different(Vars),
                  S*1000 + E*100 + N*10 + D +
                  M*1000 + O*100 + R*10 + E #=
        M*10000 + O*1000 + N*100 + E*10 + Y,
        M #> 0, S #> 0,
        label(Vars).

?- puzzle(P).
P = ([9, 5, 6, 7]+[1, 0, 8, 5]=[1, 0, 6, 5, 2])
==

Most predicates of this library are _constraints_: They generalise
arithmetic evaluation of integer expressions in that propagation can
proceed in all directions. This library also provides _enumeration_
_predicates_, which let you systematically search for solutions on
variables whose domains have become finite. A finite domain _expression_
is one of:

    | an integer         | Given value                   |
    | a variable         | Unknown value                 |
    | -Expr              | Unary minus                   |
    | Expr + Expr        | Addition                      |
    | Expr * Expr        | Multiplication                |
    | Expr - Expr.       | Subtraction                   |
    | min(Expr,Expr)     | Minimum of two expressions    |
    | max(Expr,Expr)     | Maximum of two expressions    |
    | Expr mod Expr      | Remainder of integer division |
    | abs(Expr)          | Absolute value                |
    | Expr / Expr        | Integer division              |

The most important finite domain _constraints_ are given in the table
below.

    | Expr1 #>= Expr2  | Expr1 is larger than or equal to Expr2  |
    | Expr1 #=< Expr2  | Expr1 is smaller than or equal to Expr2 |
    | Expr1 #=  Expr2  | Expr1 equals Expr2 |
    | Expr1 #\= Expr2  | Expr1 is not equal to Expr2 |
    | Expr1 #> Expr2   | Expr1 is strictly larger than Expr2 |
    | Expr1 #< Expr2   | Expr1 is strictly smaller than Expr2 |

The constraints #=/2, #\=/2, #</2, #>/2, #=</2, #>=/2 and #\/1 can be
_reified_, which means reflecting their truth values into Boolean 0/1
variables. Let P and Q denote conjunctions (#/\/2) or disjunctions
(#\//2) of reifiable constraints or Boolean variables, then:

    | #\ Q      | True iff Q is false             |
    | P #<==> Q | True iff P and Q are equivalent |
    | P #==> Q  | True iff P implies Q            |
    | P #<== Q  | True iff Q implies P            |

If SWI Prolog is compiled with support for arbitrary precision
integers (using GMP), there is no limit on the size of domains.

@author Markus Triska
*/

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   A bound is either:

   n(N):    integer N
   inf:     infimum of Z (= negative infinity)
   sup:     supremum of Z (= positive infinity)
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

is_bound(n(N)) :- integer(N).
is_bound(inf).
is_bound(sup).

defaulty_to_bound(D, P) :- ( integer(D) -> P = n(D) ; P = D ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Compactified is/2 and predicates for several arithmetic expressions
   with infinities, tailored for the modes needed by this solver.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

% cis_gt only works for terms of depth 0 on both sides
cis_gt(n(N), B) :- cis_gt_numeric(B, N).
cis_gt(sup, B0) :-B0 \== sup.

cis_gt_numeric(n(B), A) :- A > B.
cis_gt_numeric(inf, _).

cis_geq(A, B) :-
        (   cis_gt(A, B) -> true
        ;   A == B -> true
        ).

cis_geq_zero(sup).
cis_geq_zero(n(N)) :- N >= 0.

cis_lt(A, B)  :- cis_gt(B, A).

cis_leq(A, B) :- cis_geq(B, A).

cis_min(inf, _, inf).
cis_min(sup, B, B).
cis_min(n(N), B, Min) :- cis_min_(B, N, Min).

cis_min_(inf, _, inf).
cis_min_(sup, N, n(N)).
cis_min_(n(B), A, n(M)) :- M is min(A,B).

cis_max(sup, _, sup).
cis_max(inf, B, B).
cis_max(n(N), B, Max) :- cis_max_(B, N, Max).

cis_max_(inf, N, n(N)).
cis_max_(sup, _, sup).
cis_max_(n(B), A, n(M)) :- M is max(A,B).

cis_plus(inf, _, inf).
cis_plus(sup, _, sup).
cis_plus(n(A), B, Plus) :- cis_plus_(B, A, Plus).

cis_plus_(sup, _, sup).
cis_plus_(inf, _, inf).
cis_plus_(n(B), A, n(S)) :- S is A + B.

cis_minus(inf, _, inf).
cis_minus(sup, _, sup).
cis_minus(n(A), B, M) :- cis_minus_(B, A, M).

cis_minus_(inf, _, sup).
cis_minus_(sup, _, inf).
cis_minus_(n(B), A, n(M)) :- M is A - B.

cis_uminus(inf, sup).
cis_uminus(sup, inf).
cis_uminus(n(A), n(B)) :- B is -A.

cis_abs(inf, sup).
cis_abs(sup, sup).
cis_abs(n(A), n(B)) :- B is abs(A).

cis_times(inf, B, P) :-
        (   B cis_lt n(0) -> P = sup
        ;   B cis_gt n(0) -> P = inf
        ;   P = n(0)
        ).
cis_times(sup, B, P) :-
        (   B cis_gt n(0) -> P = sup
        ;   B cis_lt n(0) -> P = inf
        ;   B == n(0) -> P = n(0)
        ).
cis_times(n(N), B, P) :- cis_times_(B, N, P).

cis_times_(inf, A, P) :- cis_times(inf, n(A), P).
cis_times_(sup, A, P) :- cis_times(sup, n(A), P).
cis_times_(n(B), A, n(P)) :- P is A * B.

% compactified is/2 for expressions of interest
A cis B :- cis_(B, A).

cis_(n(N), n(N)).
cis_(inf, inf).
cis_(sup, sup).
cis_(A0+B0, E)       :- cis_(A0, A), cis_(B0, B), cis_plus(A, B, E).
cis_(abs(A0), E)     :- cis_(A0, A), cis_abs(A, E).
cis_(min(A0,B0), E)  :- cis_(A0, A), cis_(B0, B), cis_min(A, B, E).
cis_(max(A0,B0), E)  :- cis_(A0, A), cis_(B0, B), cis_max(A, B, E).
cis_(A0-B0, E)       :- cis_(A0, A), cis_(B0, B), cis_minus(A, B, E).
cis_(-A0, E)         :- cis_(A0, A), cis_uminus(A, E).
cis_(A0*B0, E)       :- cis_(A0, A), cis_(B0, B), cis_times(A, B, E).
cis_(floor(A0), E)   :- cis_(A0, A), cis_floor(A, E).
cis_(ceiling(A0), E) :- cis_(A0, A), cis_ceiling(A, E).
cis_(div(A0,B0), E)  :- cis_(A0, A), cis_(B0, B), cis_div(A, B, E).
cis_(A0//B0, E)      :- cis_(A0, A), cis_(B0, B), cis_slash(A, B, E).

% special case for the frequent case of depth 1 expressions

A cis1 B :- cis1_(B, A).

cis1_(n(N), n(N)).
cis1_(inf, inf).
cis1_(sup, sup).
cis1_(A+B, E)        :- cis_plus(A, B, E).
cis1_(abs(A), E)     :- cis_abs(A, E).
cis1_(min(A,B), E)   :- cis_min(A, B, E).
cis1_(max(A,B), E)   :- cis_max(A, B, E).
cis1_(A-B, E)        :- cis_minus(A, B, E).
cis1_(-A, E)         :- cis_uminus(A, E).
cis1_(A*B, E)        :- cis_times(A, B, E).
cis1_(floor(A), E)   :- cis_floor(A, E).
cis1_(ceiling(A), E) :- cis_ceiling(A, E).
cis1_(div(A,B), E)   :- cis_div(A, B, E).
cis1_(A//B, E)       :- cis_slash(A, B, E).

cis_floor(sup, sup).
cis_floor(inf, inf).
cis_floor(n(A), n(B)) :- B is integer(floor(A)).

cis_ceiling(sup, sup).
cis_ceiling(inf, inf).
cis_ceiling(n(A), n(B)) :- B is integer(ceiling(A)).

cis_div(sup, Y, Z) :- ( cis_geq_zero(Y) -> Z = sup ; Z = inf ).
cis_div(inf, Y, Z) :- ( cis_geq_zero(Y) -> Z = inf ; Z = sup ).
cis_div(n(X), Y, Z) :- cis_div_(Y, X, Z).

cis_div_(sup, _, n(0)).
cis_div_(inf, _, n(0)).
cis_div_(n(Y), X, Z) :-
        (   Y =:= 0 -> (  X >= 0 -> Z = sup ; Z = inf )
        ;   Z0 is X // Y, Z = n(Z0)
        ).

cis_slash(sup, _, sup).
cis_slash(inf, _, inf).
cis_slash(n(N), B, S) :- cis_slash_(B, N, S).

cis_slash_(sup, _, n(0)).
cis_slash_(inf, _, n(0)).
cis_slash_(n(B), A, n(S)) :- S is A // B.


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   A domain is a finite set of disjoint intervals. Internally, domains
   are represented as trees. Each node is one of:

   empty: empty domain.

   split(N, Left, Right)
      - split on integer N, with Left and Right domains whose elements are
        all less than and greater than N, respectively. The domain is the
        union of Left and Right, i.e., N is a hole.

   from_to(From, To)
      - interval (From-1, To+1); From and To are bounds

   Desiderata: rebalance domains; singleton intervals.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Type definition and inspection of domains.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

check_domain(D) :-
        (   var(D) -> instantiation_error(D)
        ;   is_domain(D) -> true
        ;   domain_error(clpfd_domain, D)
        ).

is_domain(empty).
is_domain(from_to(From,To)) :-
        is_bound(From), is_bound(To),
        From cis_leq To.
is_domain(split(S, Left, Right)) :-
        integer(S),
        is_domain(Left), is_domain(Right),
        all_less_than(Left, S),
        all_greater_than(Right, S).

all_less_than(empty, _).
all_less_than(from_to(From,To), S) :-
        From cis_lt n(S), To cis_lt n(S).
all_less_than(split(S0,Left,Right), S) :-
        S0 < S,
        all_less_than(Left, S),
        all_less_than(Right, S).

all_greater_than(empty, _).
all_greater_than(from_to(From,To), S) :-
        From cis_gt n(S), To cis_gt n(S).
all_greater_than(split(S0,Left,Right), S) :-
        S0 > S,
        all_greater_than(Left, S),
        all_greater_than(Right, S).

default_domain(from_to(inf,sup)).

domain_infimum(from_to(I, _), I).
domain_infimum(split(_, Left, _), I) :- domain_infimum(Left, I).

domain_supremum(from_to(_, S), S).
domain_supremum(split(_, _, Right), S) :- domain_supremum(Right, S).

domain_num_elements(empty, n(0)).
domain_num_elements(from_to(From,To), Num) :- Num cis To - From + n(1).
domain_num_elements(split(_, Left, Right), Num) :-
        domain_num_elements(Left, NL),
        domain_num_elements(Right, NR),
        Num cis1 NL + NR.

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Test whether domain contains a given integer.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_contains(from_to(From,To), I) :- From cis_leq n(I), n(I) cis_leq To.
domain_contains(split(S, Left, Right), I) :-
        (   I < S -> domain_contains(Left, I)
        ;   I > S -> domain_contains(Right, I)
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Test whether a domain contains another domain.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_subdomain(Dom, Sub) :- domain_subdomain(Dom, Dom, Sub).

domain_subdomain(from_to(_,_), Dom, Sub) :-
        domain_subdomain_fromto(Sub, Dom).
domain_subdomain(split(_, _, _), Dom, Sub) :-
        domain_subdomain_split(Sub, Dom, Sub).

domain_subdomain_split(empty, _, _).
domain_subdomain_split(from_to(From,To), split(S,Left0,Right0), Sub) :-
        (   To cis_lt n(S) -> domain_subdomain(Left0, Left0, Sub)
        ;   From cis_gt n(S) -> domain_subdomain(Right0, Right0, Sub)
        ).
domain_subdomain_split(split(_,Left,Right), Dom, _) :-
        domain_subdomain(Dom, Dom, Left),
        domain_subdomain(Dom, Dom, Right).

domain_subdomain_fromto(empty, _).
domain_subdomain_fromto(from_to(From,To), from_to(From0,To0)) :-
        From0 cis_leq From, To0 cis_geq To.
domain_subdomain_fromto(split(_,Left,Right), Dom) :-
        domain_subdomain_fromto(Left, Dom),
        domain_subdomain_fromto(Right, Dom).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Remove an integer from a domain. The domain is traversed until an
   interval is reached from which the element can be removed, or until
   it is clear that no such interval exists.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_remove(empty, _, empty).
domain_remove(from_to(L0, U0), X, D) :- domain_remove_(L0, U0, X, D).
domain_remove(split(S, Left0, Right0), X, D) :-
        (   X =:= S -> D = split(S, Left0, Right0)
        ;   X < S ->
            domain_remove(Left0, X, Left1),
            (   Left1 == empty -> D = Right0
            ;   D = split(S, Left1, Right0)
            )
        ;   domain_remove(Right0, X, Right1),
            (   Right1 == empty -> D = Left0
            ;   D = split(S, Left0, Right1)
            )
        ).

%?- domain_remove(from_to(n(0),n(5)), 3, D).

domain_remove_(inf, U0, X, D) :-
        (   U0 == n(X) -> U1 cis1 U0 - n(1), D = from_to(inf, U1)
        ;   U0 cis_lt n(X) -> D = from_to(inf,U0)
        ;   L1 cis1 n(X) + n(1), U1 cis1 n(X) - n(1),
            D = split(X, from_to(inf, U1), from_to(L1,U0))
        ).
domain_remove_(n(N), U0, X, D) :- domain_remove_upper(U0, n(N), X, D).

domain_remove_upper(sup, L0, X, D) :-
        (   L0 == n(X) -> L1 cis1 n(X) + n(1), D = from_to(L1,sup)
        ;   L0 cis_gt n(X) -> D = from_to(L0,sup)
        ;   L1 cis1 n(X) + n(1), U1 cis1 n(X) - n(1),
            D = split(X, from_to(L0,U1), from_to(L1,sup))
        ).
domain_remove_upper(n(U00), L0, X, D) :-
        U0 = n(U00),
        (   L0 == U0, n(X) == L0 -> D = empty
        ;   L0 == n(X) -> L1 cis1 n(X) + n(1), D = from_to(L1, U0)
        ;   U0 == n(X) -> U1 cis1 U0 - n(1), D = from_to(L0, U1)
        ;   L0 cis_leq n(X), n(X) cis_leq U0 ->
            U1 cis1 n(X) - n(1), L1 cis1 n(X) + n(1),
            D = split(X, from_to(L0, U1), from_to(L1, U0))
        ;   D = from_to(L0,U0)
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Remove all elements greater than / less than some constant.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_remove_greater_than(empty, _, empty).
domain_remove_greater_than(from_to(From0,To0), G, D) :-
        (   From0 cis_gt n(G) -> D = empty
        ;   To cis1 min(To0,n(G)), D = from_to(From0,To)
        ).
domain_remove_greater_than(split(S,Left0,Right0), G, D) :-
        (   S =< G ->
            domain_remove_greater_than(Right0, G, Right),
            (   Right == empty -> D = Left0
            ;   D = split(S, Left0, Right)
            )
        ;   domain_remove_greater_than(Left0, G, D)
        ).

domain_remove_smaller_than(empty, _, empty).
domain_remove_smaller_than(from_to(From0,To0), V, D) :-
        (   To0 cis_lt n(V) -> D = empty
        ;   From cis1 max(From0,n(V)), D = from_to(From,To0)
        ).
domain_remove_smaller_than(split(S,Left0,Right0), V, D) :-
        (   S >= V ->
            domain_remove_smaller_than(Left0, V, Left),
            (   Left == empty -> D = Right0
            ;   D = split(S, Left, Right0)
            )
        ;   domain_remove_smaller_than(Right0, V, D)
        ).


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Remove a whole domain from another domain. (Set difference.)
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_subtract(Dom0, Sub, Dom) :- domain_subtract(Dom0, Dom0, Sub, Dom).

domain_subtract(empty, _, _, empty).
domain_subtract(from_to(From0,To0), Dom, Sub, D) :-
        (   Sub == empty -> D = Dom
        ;   Sub = from_to(From,To) ->
            (   From == To -> From = n(X), domain_remove(Dom, X, D)
            ;   From cis_gt To0 -> D = Dom
            ;   To cis_lt From0 -> D = Dom
            ;   From cis_leq From0 ->
                (   To cis_geq To0 -> D = empty
                ;   From1 cis1 To + n(1),
                    D = from_to(From1, To0)
                )
            ;   To1 cis1 From - n(1),
                (   To cis_lt To0 ->
                    From = n(S),
                    From2 cis1 To + n(1),
                    D = split(S,from_to(From0,To1),from_to(From2,To0))
                ;   D = from_to(From0,To1)
                )
            )
        ;   Sub = split(S, Left, Right) ->
            (   n(S) cis_gt To0 -> domain_subtract(Dom, Dom, Left, D)
            ;   n(S) cis_lt From0 -> domain_subtract(Dom, Dom, Right, D)
            ;   domain_subtract(Dom, Dom, Left, D1),
                domain_subtract(D1, D1, Right, D)
            )
        ).
domain_subtract(split(S, Left0, Right0), _, Sub, D) :-
        domain_subtract(Left0, Left0, Sub, Left),
        domain_subtract(Right0, Right0, Sub, Right),
        (   Left == empty -> D = Right
        ;   Right == empty -> D = Left
        ;   D = split(S, Left, Right)
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Convert domain to a list of disjoint intervals From-To.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_intervals(D, Is) :- domain_intervals(D, Is, []).

domain_intervals(split(_, Left, Right)) -->
        domain_intervals(Left), domain_intervals(Right).
domain_intervals(empty)                 --> [].
domain_intervals(from_to(From,To))      --> [From-To].

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   To compute the intersection of two domains D1 and D2, we
   (arbitrarily) choose D1 as the reference domain. For each interval
   of D1, we compute how far and to which values D2 lets us extend it.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domains_intersection(D1, D2, Intersection) :-
        domains_intersection_(D1, D2, Intersection),
        Intersection \== empty.

domains_intersection_(empty, _, empty).
domains_intersection_(from_to(L0,U0), D2, Dom) :-
        narrow(D2, L0, U0, Dom).
domains_intersection_(split(S,Left0,Right0), D2, Dom) :-
        domains_intersection_(Left0, D2, Left1),
        domains_intersection_(Right0, D2, Right1),
        (   Left1 == empty, Right1 == empty -> Dom = empty
        ;   Left1 == empty -> Dom = Right1
        ;   Right1 == empty -> Dom = Left1
        ;   Dom = split(S, Left1, Right1)
        ).

narrow(empty, _, _, empty).
narrow(from_to(L0,U0), From0, To0, Dom) :-
        From1 cis1 max(From0,L0), To1 cis1 min(To0,U0),
        (   From1 cis_gt To1 -> Dom = empty
        ;   Dom = from_to(From1,To1)
        ).
narrow(split(S, Left0, Right0), From0, To0, Dom) :-
        (   To0 cis_lt n(S) -> narrow(Left0, From0, To0, Dom)
        ;   From0 cis_gt n(S) -> narrow(Right0, From0, To0, Dom)
        ;   narrow(Left0, From0, To0, Left1),
            narrow(Right0, From0, To0, Right1),
            (   Left1 == empty -> Dom = Right1
            ;   Right1 == empty -> Dom = Left1
            ;   Dom = split(S, Left1, Right1)
            )
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Union of 2 domains.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domains_union(D1, D2, Union) :-
        domain_intervals(D1, Is1),
        domain_intervals(D2, Is2),
        append(Is1, Is2, IsU0),
        merge_intervals(IsU0, IsU1),
        intervals_to_domain(IsU1, Union).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Shift a domain by some offset.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_shift(empty, _, empty).
domain_shift(from_to(From0,To0), O, from_to(From,To)) :-
        From cis1 From0 + n(O), To cis1 To0 + n(O).
domain_shift(split(S0, Left0, Right0), O, split(S, Left, Right)) :-
        S is S0 + O,
        domain_shift(Left0, O, Left),
        domain_shift(Right0, O, Right).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   The new domain contains all values of the old domain,
   multiplied by a constant multiplier.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_expand(D0, M, D) :-
        (   M < 0 -> domain_negate(D0, D1), M1 is abs(M)
        ;   D1 = D0, M1 = M
        ),
        domain_expand_(D1, M1, D).

domain_expand_(empty, _, empty).
domain_expand_(from_to(From0, To0), M, D) :-
        (   M =:= 1 -> D = from_to(From0, To0)
        ;   From0 == inf -> To cis1 To0*n(M), D = from_to(inf, To)
        ;   To0 == sup -> From cis1 From0*n(M), D = from_to(From, sup)
        ;   % domain is bounded
            all_multiples(From0, To0, M, D)
        ).
domain_expand_(split(S0, Left0, Right0), M, split(S, Left, Right)) :-
        S is M*S0,
        domain_expand_(Left0, M, Left),
        domain_expand_(Right0, M, Right).

all_multiples(From, To, M, D) :-
        Mid cis (From + To) // n(2),
        Mult cis1 Mid * n(M),
        (   From == To -> D = from_to(Mult,Mult)
        ;   Left cis1 Mid - n(1),
            Right cis1 Mid + n(1),
            n(S1) cis1 Mult +  n(1),
            n(S2) cis1 Mult - n(1),
            (   Mid == From ->
                D = split(S1, from_to(Mult,Mult), RightD),
                all_multiples(Right, To, M, RightD)
            ;   Mid == To ->
                D = split(S2, LeftD, from_to(Mult,Mult)),
                all_multiples(From, Left, M, LeftD)
            ;   D = split(S2, LeftD, split(S1, from_to(Mult,Mult), RightD)),
                all_multiples(From, Left, M, LeftD),
                all_multiples(Right, To, M, RightD)
            )
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   similar to domain_expand/3, tailored for division: an interval
   [From,To] is extended to [From*M, ((To+1)*M - 1)], i.e., to all
   values that integer-divided by M yield a value from interval.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_expand_more(D0, M, D) :-
        %format("expanding ~w by ~w\n", [D0,M]),
        (   M < 0 -> domain_negate(D0, D1), M1 is abs(M)
        ;   D1 = D0, M1 = M
        ),
        domain_expand_more_(D1, M1, D).
        %format("yield: ~w\n", [D]).

domain_expand_more_(empty, _, empty).
domain_expand_more_(from_to(From0, To0), M, from_to(From,To)) :-
        (   From0 cis_lt n(0) ->
            From cis (From0-n(1))*n(M) + n(1)
        ;   From cis From0*n(M)
        ),
        (   To0 cis_lt n(0) ->
            To cis To0*n(M)
        ;   To cis (To0+n(1))*n(M) - n(1)
        ).
domain_expand_more_(split(S0, Left0, Right0), M, D) :-
        S is M*S0,
        domain_expand_more_(Left0, M, Left),
        domain_expand_more_(Right0, M, Right),
        domain_supremum(Left, LeftSup),
        domain_infimum(Right, RightInf),
        (   LeftSup cis_lt n(S), n(S) cis_lt RightInf ->
            D = split(S, Left, Right)
        ;   domain_infimum(Left, Inf),
            domain_supremum(Right, Sup),
            D = from_to(Inf, Sup)
        ).


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Scale a domain down by a constant multiplier. Assuming (//)/2.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_contract(D0, M, D) :-
        %format("contracting ~w by ~w\n", [D0,M]),
        (   M < 0 -> domain_negate(D0, D1), M1 is abs(M)
        ;   D1 = D0, M1 = M
        ),
        domain_contract_(D1, M1, D).

domain_contract_(empty, _, empty).
domain_contract_(from_to(From0, To0), M, from_to(From,To)) :-
        From cis1 From0 // n(M), To cis1 To0 // n(M).
domain_contract_(split(S0,Left0,Right0), M, D) :-
        S is S0 // M,
        %  Scaled down domains do not necessarily retain any holes of
        %  the original domain.
        domain_contract_(Left0, M, Left),
        domain_contract_(Right0, M, Right),
        domain_supremum(Left, LeftSup),
        domain_infimum(Right, RightInf),
        (   LeftSup cis_lt n(S), n(S) cis_lt RightInf ->
            D = split(S, Left, Right)
        ;   domain_infimum(Left0, Inf),
            % TODO: this is not necessarily an interval
            domain_supremum(Right0, Sup),
            min_divide(Inf, Sup, n(M), n(M), From0),
            max_divide(Inf, Sup, n(M), n(M), To0),
            domain_infimum(Left, LeftInf),
            domain_supremum(Right, RightSup),
            From cis max(LeftInf, ceiling(From0)),
            To cis min(RightSup, floor(To0)),
            D = from_to(From, To)
        ).
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Similar to domain_contract, tailored for division, i.e.,
   {21,23} contracted by 4 is 5. It contracts "less".
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_contract_less(D0, M, D) :-
        (   M < 0 -> domain_negate(D0, D1), M1 is abs(M)
        ;   D1 = D0, M1 = M
        ),
        (   domain_infimum(D1, n(_)), domain_supremum(D1, n(_)) ->
            % bounded domain
            domain_intervals(D1, Is),
            intervals_contract_less(Is, M1, Cs, []),
            list_to_domain(Cs, D)
        ;   domain_contract_less_(D1, M1, D)
        ).

intervals_contract_less([], _)               --> [].
intervals_contract_less([n(A0)-n(B0)|Is], M) -->
        { A is A0 // M, B is B0 // M, numlist(A, B, Ns) },
        dlist(Ns),
        intervals_contract_less(Is, M).

domain_contract_less_(empty, _, empty).
domain_contract_less_(from_to(From0, To0), M, from_to(From,To)) :-
        From cis1 From0 // n(M), To cis1 To0 // n(M).
domain_contract_less_(split(S0,Left0,Right0), M, D) :-
        S is S0 // M,
        %  Scaled down domains do not necessarily retain any holes of
        %  the original domain.
        domain_contract_less_(Left0, M, Left),
        domain_contract_less_(Right0, M, Right),
        domain_supremum(Left, LeftSup),
        domain_infimum(Right, RightInf),
        (   LeftSup cis_lt n(S), n(S) cis_lt RightInf ->
            D = split(S, Left, Right)
        ;   domain_infimum(Left0, Inf),
            % TODO: this is not necessarily an interval
            domain_supremum(Right0, Sup),
            min_divide(Inf, Sup, n(M), n(M), From0),
            max_divide(Inf, Sup, n(M), n(M), To0),
            domain_infimum(Left, LeftInf),
            domain_supremum(Right, RightSup),
            From cis max(LeftInf, floor(From0)),
            To cis min(RightSup, ceiling(To0)),
            D = from_to(From, To)
            %format("got: ~w\n", [D])
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Negate the domain. Left and Right sub-domains and bounds switch sides.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_negate(empty, empty).
domain_negate(from_to(From0, To0), from_to(To,From)) :-
        From cis1 -From0, To cis1 -To0.
domain_negate(split(S0, Left0, Right0), split(S, Left, Right)) :-
        S is -S0,
        domain_negate(Left0, Right),
        domain_negate(Right0, Left).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Construct a domain from a list of integers. Try to balance it.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

list_to_disjoint_intervals([], []).
list_to_disjoint_intervals([N|Ns], Is) :-
        list_to_disjoint_intervals(Ns, N, N, Is).

list_to_disjoint_intervals([], M, N, [n(M)-n(N)]).
list_to_disjoint_intervals([B|Bs], M, N, Is) :-
        (   B =:= N + 1 ->
            list_to_disjoint_intervals(Bs, M, B, Is)
        ;   Is = [n(M)-n(N)|Rest],
            list_to_disjoint_intervals(Bs, B, B, Rest)
        ).

list_to_domain(List0, D) :-
        (   List0 == [] -> D = empty
        ;   sort(List0, List),
            list_to_disjoint_intervals(List, Is),
            intervals_to_domain(Is, D)
        ).

intervals_to_domain([], empty) :- !.
intervals_to_domain([M-N], from_to(M,N)) :- !.
intervals_to_domain(Is, D) :-
        length(Is, L),
        FL is L // 2,
        length(Front, FL),
        append(Front, Tail, Is),
        Tail = [n(Start)-_|_],
        Hole is Start - 1,
        intervals_to_domain(Front, Left),
        intervals_to_domain(Tail, Right),
        D = split(Hole, Left, Right).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%% in(+Var, +Domain)
%
%  Constrain Var to elements of Domain. Domain is one of:
%
%         * Lower..Upper
%           All integers _I_ such that _Lower_ =< _I_ =< _Upper_. The atoms
%           *inf* and *sup* denote negative and positive infinity,
%           respectively.
%         * Domain1 \/ Domain2
%           The union of Domain1 and Domain2.

type_error(Type, Term)     :- throw(error(type_error(Type, Term), _)).
domain_error(Type, Term)   :- throw(error(domain_error(Type, Term), _)).
instantiation_error(_Term) :- throw(error(instantiation_error, _)).

must_be(_, _).

V in D :-
        fd_variable(V),
        (   is_drep(D) -> true
        ;   domain_error(clpfd_domain, D)
        ),
        drep_to_domain(D, Dom),
        domain(V, Dom).

fd_variable(V) :-
        (   var(V) -> true
        ;   integer(V) -> true
        ;   type_error(integer, V)
        ).

%% ins(?Vars, +Domain)
%
%  Constrain the variables or integers in Vars to Domain.

Vs ins D :-
        (   var(Vs) -> true
        ;   must_be(list, Vs),
            maplist(fd_variable, Vs)
        ),
        (   is_drep(D) -> true
        ;   domain_error(clpfd_domain, D)
        ),
        drep_to_domain(D, Dom),
        domains(Vs, Dom).

%% indomain(?Var)
%
% Bind Var to all feasible values of its domain on backtracking. The
% domain of Var must be finite.

indomain(Var) :-
        (   var(Var) ->
            finite_domain(Var),
            indomain_(up, Var)
        ;   must_be(integer, Var)
        ).

indomain_(Order, Var) :-
        (   var(Var) ->
            get(Var, Dom, _),
            order_dom_next(Order, Dom, Next),
            (   Var = Next
            ;   Var #\= Next,
                indomain_(Order, Var)
            )
        ;   must_be(integer, Var)
        ).

order_dom_next(up, Dom, Next)   :- domain_infimum(Dom, n(Next)).
order_dom_next(down, Dom, Next) :- domain_supremum(Dom, n(Next)).


%% label(+Vars)
%
% Equivalent to labeling([], Vars).

label(Vs) :- labeling([], Vs).

%% labeling(+Options, +Vars)
%
% Labeling means systematically trying out values for the finite
% domain variables Vars until all of them are ground. The domain of
% each variable in Vars must be finite. Options is a list of options
% that let you exhibit some control over the search process. Two sets
% of options exist: One for variable selection strategy, and one for
% optimisation. The variable selection strategy lets you specify which
% variable should be labeled next and is one of:
%
%   * leftmost
%   Label the variables in the order they occur in Vars. This is the
%   default.
%
%   * ff
%   _|First fail|_. Label the leftmost variable with smallest domain next,
%   in order to detect infeasibility early. This is often a good
%   strategy.
%
%   * min
%   Label the leftmost variable whose lower bound is the lowest next.
%
%   * max
%   Label the leftmost variable whose upper bound is the highest next.
%
% The second set of options lets you influence the order of solutions:
%
%   * min(Expr)
%   * max(Expr)
%
% This generates solutions in ascending/descending order with respect
% to the evaluation of the arithmetic expression Expr. All variables
% of Expr must also be contained in Vars.
%
% If more than one option of a category is specified, the one
% occurring rightmost in the option list takes precedence over all
% others of that category. Labeling is always complete and always
% terminates.
%

labeling(Options, Vars) :-
        must_be(list, Options),
        must_be(list, Vars),
        maplist(finite_domain, Vars),
        label(Options, leftmost, up, none, Vars).

finite_domain(Var) :-
        (   var(Var) ->
            get(Var, Dom, _),
            (   domain_infimum(Dom, n(_)), domain_supremum(Dom, n(_)) -> true
            ;   instantiation_error(Var)
            )
        ;   integer(Var) -> true
        ;   must_be(integer, Var)
        ).


label([Option|Options], Selection, Order, Optimisation, Vars) :-
        (   var(Option)-> instantiation_error(Option)
        ;   selection(Option) ->
            label(Options, Option, Order, Optimisation, Vars)
        ;   order(Option) ->
            label(Options, Selection, Option, Optimisation, Vars)
        ;   optimization(Option) ->
            label(Options, Selection, Order, Option, Vars)
        ;   domain_error(labeling_option, Option)
        ).
label([], Selection, Order, Optimisation, Vars) :-
        ( Optimisation == none ->
            label(Vars, Selection, Order)
        ;   optimize(Vars, Selection, Order, Optimisation)
        ).


label([], _, _) :- !.
label(Vars, Selection, Order) :-
        select_var(Selection, Vars, Var, RVars),
        (   var(Var) ->
            get(Var, Dom, _),
            order_dom_next(Order, Dom, Next),
            (   Var = Next, label(RVars, Selection, Order)
            ;   Var #\= Next, label(Vars, Selection, Order)
            )
        ;   must_be(integer, Var),
            label(RVars, Selection, Order)
        ).

selection(ff).
selection(ffc).
selection(min).
selection(max).
selection(leftmost).

order(up).
order(down).

optimization(min(_)).
optimization(max(_)).

select_var(leftmost, [Var|Vars], Var, Vars).
select_var(min, [V|Vs], Var, RVars) :-
        find_min(Vs, V, Var),
        delete_eq([V|Vs], Var, RVars).
select_var(max, [V|Vs], Var, RVars) :-
        find_max(Vs, V, Var),
        delete_eq([V|Vs], Var, RVars).
select_var(ff, [V|Vs], Var, RVars) :-
        find_ff(Vs, V, Var),
        delete_eq([V|Vs], Var, RVars).
select_var(ffc, Vars0, Var, Vars) :-
        find_ffc(Vars0, Var),
        delete_eq(Vars0, Var, Vars).

find_min([], Var, Var).
find_min([V|Vs], CM, Min) :-
        (   min_lt(V, CM) ->
            find_min(Vs, V, Min)
        ;   find_min(Vs, CM, Min)
        ).

find_max([], Var, Var).
find_max([V|Vs], CM, Max) :-
        (   max_gt(V, CM) ->
            find_max(Vs, V, Max)
        ;   find_max(Vs, CM, Max)
        ).

find_ff([], Var, Var).
find_ff([V|Vs], CM, FF) :-
        (   ff_lt(V, CM) ->
            find_ff(Vs, V, FF)
        ;   find_ff(Vs, CM, FF)
        ).

find_ffc(Vars0, Var) :-
        find_ff(Vars0, _, SD),
        (   var(SD) ->
            find_ffc(Vars0, SD, Var)
        ;   Var = SD
        ).

find_ffc([], Var, Var).
find_ffc([V|Vs], Prev, FF) :-
        (   ffc_lt(V, Prev) ->
            find_ffc(Vs, V, FF)
        ;   find_ffc(Vs, Prev, FF)
        ).

ff_lt(X, Y) :-
        (   var(X) ->
            get(X, DX, _),
            domain_num_elements(DX, NX)
        ;   NX = n(1)
        ),
        (   var(Y) ->
            get(Y, DY, _),
            domain_num_elements(DY, NY)
        ;   NY = n(1)
        ),
        NX cis_lt NY.

ffc_lt(X, Y) :-
        (   var(X) ->
            get(X, XD, XPs),
            domain_num_elements(XD, NXD),
            length(XPs, NXPs)
        ;   NXD = n(0), NXPs = n(0)
        ),
        (   var(Y) ->
            get(Y, YD, YPs),
            domain_num_elements(YD, NYD),
            length(YPs, NYPs)
        ;   NYD = n(0), NYPs = n(0)
        ),
        NXD == NYD,
        NXPs > NYPs.

min_lt(X,Y) :- bounds(X,LX,_), bounds(Y,LY,_), LX cis_lt LY.

max_gt(X,Y) :- bounds(X,_,UX), bounds(Y,_,UY), UX cis_gt UY.

bounds(X, L, U) :-
        (   var(X) ->
            get(X, Dom, _),
            domain_infimum(Dom, L),
            domain_supremum(Dom, U)
        ;   L = n(X), U = L
        ).

delete_eq([],_,[]).
delete_eq([X|Xs],Y,List) :-
        (   X == Y -> List = Xs
        ;   List = [X|Tail],
            delete_eq(Xs,Y,Tail)
        ).

optimize(Vars, Selection, Order, Opt) :-
        copy_term(Vars-Opt, Vars1-Opt1),
        once(label(Vars1, Selection, Order)),
        functor(Opt1, Direction, _),
        maplist(arg(1), [Opt,Opt1], [Expr,Expr1]),
        optimize(Direction, Selection, Order, Vars, Expr1, Expr).

optimize(Direction, Selection, Order, Vars, Expr0, Expr) :-
        Val0 is Expr0,
        copy_term(Vars-Expr, Vars1-Expr1),
        (   Direction == min -> Tighten = (Expr1 #< Val0)
        ;   Tighten = (Expr1 #> Val0) % max
        ),
        (   Tighten, label(Vars1, Selection, Order) ->
            optimize(Direction, Selection, Order, Vars, Expr1, Expr)
        ;
              (   Expr #= Val0, label(Vars, Selection, Order)
              ;   Expr #\= Val0,
                  Opt =.. [Direction,Expr],
                  optimize(Vars, Selection, Order, Opt)
              )
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% all_different(+Vars)
%
% Constrain Vars to be pairwise distinct.

all_different(Ls) :-
        length(Ls, _),
        all_different(Ls, []),
        do_queue.

all_different([], _).
all_different([X|Right], Left) :-
        (   var(X) ->
            Prop = propagator(pdifferent(Left,Right,X), mutable(passive)),
            init_propagator(X, Prop),
            trigger_prop(Prop)
        ;   exclude_fire(Left, Right, X)
        ),
        all_different(Right, [X|Left]).

%%      sum(+Vars, +Op, +Expr)
%
%       Constrain the sum of a list.  The sum/3 constraint demands that
%       "sumlist(Vars) Op Expr" hold, e.g.:
%
%       ==
%               sum(List, #=< 100)
%       ==

sum(Ls, Op, Value) :- sum(Ls, 0, Op, Value).

sum([], Sum, Op, Value) :- call(Op, Sum, Value).
sum([X|Xs], Acc, Op, Value) :-
        NAcc #= Acc + X,
        sum(Xs, NAcc, Op, Value).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Constraint propagation proceeds as follows: Each CLP(FD) variable
   has an attribute that stores its associated domain and contains a
   list of all associated constraints. Whenever the domain of a
   variable is changed, all constraints it participates in are
   triggered: They are stored in a global structure that contains a
   list of triggered constraints. do_queue/0 works off all triggered
   constraints, possible triggering new ones, until fixpoint.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

% % LIFO queue
% make_queue :- nb_setval('$propagator_queue',[]).

% push_queue(E) :-
%         b_getval('$propagator_queue',Q),
%         b_setval('$propagator_queue',[E|Q]).
% pop_queue(E) :-
%         b_getval('$propagator_queue',[E|Q]),
%         b_setval('$propagator_queue',Q).


% FIFO queue
make_queue :- nb_setval('$clpfd_queue', Q-Q).
push_queue(E) :-
        b_getval('$clpfd_queue', H-[E|T]), b_setval('$clpfd_queue', H-T).
pop_queue(E) :-
        b_getval('$clpfd_queue', H-T),
        nonvar(H), H = [E|NH], b_setval('$clpfd_queue', NH-T).

fetch_constraint_(C) :-
        pop_queue(ps(C0,State)),
        (   State == dead -> fetch_constraint_(C)
        ;   C = C0
        ).

:- make_queue.

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Parsing a CLP(FD) expression has two important side-effects: First,
   it constrains the variables occurring in the expression to
   integers. Second, it constrains some of them even more: For
   example, in X/Y and X mod Y, Y is constrained to be #\= 0.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

parse_clpfd(Expr, Result) :-
        (   var(Expr) ->
            get(Expr, ED, EPs),
            put(Expr, ED, EPs), % constrain to integers
            Result = Expr
        ;   integer(Expr) -> Result = Expr
        ;   Expr = (L + R) ->
            parse_clpfd(L, RL), parse_clpfd(R, RR),
            myplus(RL, RR, Result)
        ;   Expr = (L * R) ->
            parse_clpfd(L, RL), parse_clpfd(R, RR),
            mytimes(RL, RR, Result)
        ;   Expr = (L - R) ->
            parse_clpfd(L, RL), parse_clpfd(R, RR),
            mytimes(-1, RR, RRR),
            myplus(RL, RRR, Result)
        ;   Expr = (- E) ->
            parse_clpfd(E, RE),
            mytimes(-1, RE, Result)
        ;   Expr = max(L, R) ->
            parse_clpfd(L, RL), parse_clpfd(R, RR),
            mymax(RL, RR, Result)
        ;   Expr = min(L,R) ->
            parse_clpfd(L, RL), parse_clpfd(R, RR),
            mymin(RL, RR, Result)
        ;   Expr = L mod R ->
            parse_clpfd(L, RL), parse_clpfd(R, RR),
            RR #\= 0,
            mymod(RL, RR, Result)
        ;   Expr = abs(E) ->
            parse_clpfd(E, RE),
            myabs(RE, Result),
            Result #>= 0
        ;   Expr = (L / R) ->
            parse_clpfd(L, RL), parse_clpfd(R, RR), RR #\= 0,
            mydiv(RL, RR, Result)
        ;   Expr = (L ^ R) ->
            parse_clpfd(L, RL), parse_clpfd(R, RR),
            myexp(RL, RR, Result)
        ;   domain_error(clpfd_expression, Expr)
        ).

trigger_twice(Prop) :-
        % two invocations necessary for fixpoint when posting initially
        trigger_prop(Prop), do_queue,
        trigger_prop(Prop), do_queue.

neq(A, B) :-
        Prop = propagator(pneq(A, B), mutable(passive)),
        init_propagator(A, Prop), init_propagator(B, Prop),
        trigger_twice(Prop).

geq(A, B) :-
        (   get(A, AD, APs) ->
            domain_infimum(AD, AI),
            (   get(B, BD, _) ->
                domain_supremum(BD, BS),
                (   AI cis_geq BS -> true
                ;   Prop = propagator(pgeq(A,B), mutable(passive)),
                    init_propagator(A, Prop),
                    init_propagator(B, Prop),
                    trigger_twice(Prop)
                )
            ;   domain_remove_smaller_than(AD, B, AD1),
                put(A, AD1, APs),
                do_queue
            )
        ;   get(B, BD, BPs) ->
            domain_remove_greater_than(BD, A, BD1),
            put(B, BD1, BPs),
            do_queue
        ;   A >= B
        ).

myplus(X, Y, Z) :-
        Prop = propagator(pplus(X,Y,Z),mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop), trigger_twice(Prop).

mytimes(X, Y, Z) :-
        Prop = propagator(ptimes(X,Y,Z),mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop), trigger_twice(Prop).

mydiv(X, Y, Z) :-
        Prop = propagator(pdiv(X,Y,Z), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop), trigger_twice(Prop).

myexp(X, Y, Z) :-
        Prop = propagator(pexp(X,Y,Z), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop), trigger_twice(Prop).

myabs(X, Y) :-
        Prop = propagator(pabs(X,Y), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        trigger_prop(Prop), trigger_twice(Prop).

mymod(X, M, K) :-
        Prop = propagator(pmod(X,M,K), mutable(passive)),
        init_propagator(X, Prop), init_propagator(M, Prop),
        init_propagator(K, Prop), trigger_twice(Prop).

mymax(X, Y, Z) :-
        X #=< Z, Y #=< Z,
        Prop = propagator(pmax(X,Y,Z), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop), trigger_twice(Prop).

mymin(X, Y, Z) :-
        X #>= Z, Y #>= Z,
        Prop = propagator(pmin(X,Y,Z), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop), trigger_twice(Prop).

%% ?X #>= ?Y
%
% X is greater than or equal to Y.

X #>= Y :- parse_clpfd(X,RX), parse_clpfd(Y,RY), geq(RX,RY), reinforce(RX).

%% ?X #=< ?Y
%
% X is less than or equal to Y.

X #=< Y :- Y #>= X.

%% ?X #= ?Y
%
% X equals Y.

X #= Y  :- parse_clpfd(X,RX), parse_clpfd(Y,RX), reinforce(RX).

%% ?X #\= ?Y
%
% X is not Y.

X #\= Y :-
        (   var(X), integer(Y) ->
            neq_num(X, Y),
            do_queue,
            reinforce(X)
        ;   parse_clpfd(X, RX), parse_clpfd(Y, RY), neq(RX, RY)
        ).

% X is distinct from the number N. This is used internally in some
% propagators, and does not reinforce other constraints.

neq_num(X, N) :-
        (   get(X, XD, XPs) ->
            domain_remove(XD, N, XD1),
            put(X, XD1, XPs)
        ;   true
        ).

%% ?X #> ?Y
%
% X is greater than Y.

X #> Y  :- X #>= Y + 1.

%% #<(?X, ?Y)
%
% X is less than Y.

X #< Y  :- Y #> X.

%% #\ +Q
%
% The reifiable constraint Q does _not_ hold.

#\ Q       :- reify(Q, 0), do_queue.

%% ?P #<==> ?Q
%
% P and Q are equivalent.

L #<==> R  :- reify(L, B), reify(R, B), do_queue.

%% ?P #==> ?Q
%
% P implies Q.

L #==> R   :- reify(L, BL), reify(R, BR), myimpl(BL, BR), do_queue.

%% ?P #<== ?Q
%
% ?Q implies ?P.

L #<== R   :- reify(L, BL), reify(R, BR), myimpl(BR, BL), do_queue.

%% ?P #/\ ?Q
%
% P and Q hold.

L #/\ R    :- reify(L, 1), reify(R, 1), do_queue.

%% ?P #\/ ?Q
%
% P or Q holds.

L #\/ R    :- reify(L, BL), reify(R, BR), myor(BL, BR, 1), do_queue.

myor(X, Y, Z) :-
        Prop = propagator(por(X,Y,Z), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop),
        trigger_prop(Prop).

myimpl(X, Y) :-
        Prop = propagator(pimpl(X,Y), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        trigger_prop(Prop).

mydefined(X, Y, Z) :-
        Prop = propagator(defined(X,Y,Z), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop),
        trigger_prop(Prop).

my_reified_div(X, Y, D, Z) :-
        Prop = propagator(reified_div(X,Y,D,Z), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop), init_propagator(D, Prop),
        trigger_twice(Prop).

my_reified_mod(X, Y, D, Z) :-
        Prop = propagator(reified_mod(X,Y,D,Z), mutable(passive)),
        init_propagator(X, Prop), init_propagator(Y, Prop),
        init_propagator(Z, Prop), init_propagator(D, Prop),
        trigger_twice(Prop).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   A constraint that is being reified need not hold. Therefore, in
   X/Y, Y can as well be 0, for example. Note that it is OK to
   constrain the *result* of an expression (which does not appear
   explicitly in the expression and is not visible to the outside),
   but not the operands, except for requiring that they be integers.
   In contrast to parse_clpfd/2, the result of an expression can now
   also be undefined, in which case the constraint cannot hold.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

parse_reified_clpfd(Expr, Result, Defined) :-
        (   var(Expr) ->
            get(Expr, ED, EPs),
            put(Expr, ED, EPs), % constrain to integers
            Result = Expr, Defined = 1
        ;   integer(Expr) -> Result = Expr, Defined = 1
        ;   Expr = (L + R) ->
            parse_reified_clpfd(L, RL, DL), parse_reified_clpfd(R, RR, DR),
            myplus(RL, RR, Result), mydefined(DL, DR, Defined)
        ;   Expr = (L * R) ->
            parse_reified_clpfd(L, RL, DL), parse_reified_clpfd(R, RR, DR),
            mytimes(RL, RR, Result), mydefined(DL, DR, Defined)
        ;   Expr = (L - R) ->
            parse_reified_clpfd(L, RL, DL), parse_reified_clpfd(R, RR, DR),
            mytimes(-1, RR, RRR),
            myplus(RL, RRR, Result), mydefined(DL, DR, Defined)
        ;   Expr = (- E) ->
            parse_reified_clpfd(E, RE, Defined),
            mytimes(-1, RE, Result)
        ;   Expr = max(L, R) ->
            parse_reified_clpfd(L, RL, DL), parse_reified_clpfd(R, RR, DR),
            mymax(RL, RR, Result), mydefined(DL, DR, Defined)
        ;   Expr = min(L,R) ->
            parse_reified_clpfd(L, RL, DL), parse_reified_clpfd(R, RR, DR),
            mymin(RL, RR, Result), mydefined(DL, DR, Defined)
        ;   Expr = L mod R ->
            parse_reified_clpfd(L, RL, DL), parse_reified_clpfd(R, RR, DR),
            mydefined(DL, DR, Defined1),
            my_reified_mod(RL, RR, Defined2, Result),
            mydefined(Defined1, Defined2, Defined)
        ;   Expr = abs(E) ->
            parse_reified_clpfd(E, RE, Defined),
            myabs(RE, Result),
            Result #>= 0
        ;   Expr = (L / R) ->
            parse_reified_clpfd(L, RL, DL), parse_reified_clpfd(R, RR, DR),
            mydefined(DL, DR, Defined1),
            my_reified_div(RL, RR, Defined2, Result),
            mydefined(Defined1, Defined2, Defined)
        ;   Expr = (L ^ R) ->
            parse_reified_clpfd(L, RL, DL), parse_reified_clpfd(R, RR, DR),
            mydefined(DL, DR, Defined),
            myexp(RL, RR, Result)
        ;   domain_error(clpfd_expression, Expr)
        ).

reify(Expr, B) :-
        B in 0..1,
        (   var(Expr) -> B = Expr
        ;   integer(Expr) -> B = Expr
        ;   Expr = (L #>= R) ->
            parse_reified_clpfd(L, LR, LD), parse_reified_clpfd(R, RR, RD),
            Prop = propagator(reified_geq(LD,LR,RD,RR,B), mutable(passive)),
            init_propagator(LR, Prop), init_propagator(RR, Prop),
            init_propagator(B, Prop), init_propagator(LD, Prop),
            init_propagator(RD, Prop), trigger_prop(Prop)
        ;   Expr = (L #> R)  -> reify(L #>= (R+1), B)
        ;   Expr = (L #=< R) -> reify(R #>= L, B)
        ;   Expr = (L #< R)  -> reify(R #>= (L+1), B)
        ;   Expr = (L #= R)  ->
            parse_reified_clpfd(L, LR, LD), parse_reified_clpfd(R, RR, RD),
            Prop = propagator(reified_eq(LD,LR,RD,RR,B), mutable(passive)),
            init_propagator(LR, Prop), init_propagator(RR, Prop),
            init_propagator(B, Prop), init_propagator(LD, Prop),
            init_propagator(RD, Prop), trigger_prop(Prop)
        ;   Expr = (L #\= R) ->
            parse_reified_clpfd(L, LR, LD), parse_reified_clpfd(R, RR, RD),
            Prop = propagator(reified_neq(LD,LR,RD,RR,B), mutable(passive)),
            init_propagator(LR, Prop), init_propagator(RR, Prop),
            init_propagator(B, Prop), init_propagator(LD, Prop),
            init_propagator(RD, Prop), trigger_prop(Prop)
        ;   Expr = (L #==> R) -> reify((#\ L) #\/ R, B)
        ;   Expr = (L #<== R) -> reify(R #==> L, B)
        ;   Expr = (L #<==> R) -> reify((L #==> R) #/\ (R #==> L), B)
        ;   Expr = (L #/\ R) ->
            reify(L, LR), reify(R, RR),
            Prop = propagator(reified_and(LR,RR,B), mutable(passive)),
            init_propagator(LR, Prop), init_propagator(RR, Prop),
            init_propagator(B, Prop),
            trigger_prop(Prop)
        ;   Expr = (L #\/ R) ->
            reify(L, LR), reify(R, RR),
            Prop = propagator(reified_or(LR,RR,B), mutable(passive)),
            init_propagator(LR, Prop), init_propagator(RR, Prop),
            init_propagator(B, Prop),
            trigger_prop(Prop)
        ;   Expr = (#\ Q) ->
            reify(Q, QR),
            Prop = propagator(reified_not(QR,B), mutable(passive)),
            init_propagator(QR, Prop), init_propagator(B, Prop),
            trigger_prop(Prop)
        ;   domain_error(clpfd_reifiable_expression, Expr)
        ).


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   A drep is a user-accessible and visible domain representation. N,
   N..M, and D1 \/ D2 are dreps, if D1 and D2 are dreps.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

is_drep(V)      :- var(V), !, instantiation_error(V).
is_drep(N)      :- integer(N), !.
is_drep(N..M)   :- !, drep_bound(N), drep_bound(M).
is_drep(D1\/D2) :- !, is_drep(D1), is_drep(D2).

drep_bound(V)   :- var(V), !, instantiation_error(V).
drep_bound(sup) :- !. % should infinities be accessible?
drep_bound(inf) :- !.
drep_bound(I)   :- integer(I), !.

drep_to_intervals(I)        --> { integer(I) }, !, [n(I)-n(I)].
drep_to_intervals(N..M)     -->
        (   { defaulty_to_bound(N, N1), defaulty_to_bound(M, M1),
              N1 cis_leq M1} -> [N1-M1]
        ;   []
        ).
drep_to_intervals(D1 \/ D2) -->
        drep_to_intervals(D1), drep_to_intervals(D2).

drep_to_domain(DR, D) :-
        drep_to_intervals(DR, Is0, []),
        merge_intervals(Is0, Is1),
        intervals_to_domain(Is1, D).

merge_intervals(Is0, Is) :-
        keysort(Is0, Is1),
        merge_overlapping(Is1, Is).

merge_overlapping([], []).
merge_overlapping([A-B0|ABs0], [A-B|ABs]) :-
        merge_remaining(ABs0, B0, B, Rest),
        merge_overlapping(Rest, ABs).

merge_remaining([], B, B, []).
merge_remaining([N-M|NMs], B0, B, Rest) :-
        Next cis1 B0 + n(1),
        (   N cis_gt Next -> B = B0, Rest = [N-M|NMs]
        ;   B1 cis1 max(B0,M),
            merge_remaining(NMs, B1, B, Rest)
        ).

domain(V, Dom) :-
        (   var(V) ->
            get(V, Dom0, VPs),
            domains_intersection(Dom, Dom0, Dom1),
            %format("intersected\n: ~w\n ~w\n==> ~w\n\n", [Dom,Dom0,Dom1]),
            put(V, Dom1, VPs),
            do_queue,
            reinforce(V)
        ;   domain_contains(Dom, V)
        ).

domains([], _).
domains([V|Vs], D) :- domain(V, D), domains(Vs, D).


get(X, Dom, Ps) :-
        (   get_attr(X, clpfd, Attr) -> Attr = clpfd(_,_,_,Dom,Ps)
        ;   var(X) -> default_domain(Dom), Ps = []
        ).

get(X, Dom, Inf, Sup, Ps) :-
        get(X, Dom, Ps),
        domain_infimum(Dom, Inf),
        domain_supremum(Dom, Sup).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Set the experimental flag 'clpfd_propagation' to 'terminating' to
   make propagation terminating. Currently, this is done by allowing
   the left and right boundaries, as well as the distance between the
   smallest and largest number occurring in the domain representation
   to be changed at most once after a constraint is posted, unless the
   domain is bounded.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

put(X, Dom, Pos) :-
        (   current_prolog_flag(clpfd_propagation, terminating) ->
            put_terminating(X, Dom, Pos)
        ;   put_full(X, Dom, Pos)
        ).

put_terminating(X, Dom, Ps) :-
        Dom \== empty,
        (   Dom = from_to(F, F) -> F = n(X)
        ;   (   get_attr(X, clpfd, Attr) ->
                Attr = clpfd(Left,Right,Spread,OldDom, _OldPs),
                put_attr(X, clpfd, clpfd(Left,Right,Spread,Dom,Ps)),
                (   OldDom == Dom -> true
                ;   domain_intervals(Dom, Is),
                    domain_intervals(OldDom, Is) -> true
                ;   domain_infimum(Dom, Inf), domain_supremum(Dom, Sup),
                    (   Inf = n(_), Sup = n(_) ->
                        put_attr(X, clpfd, clpfd(.,.,.,Dom,Ps)),
                        trigger_props(Ps)
                    ;   % infinite domain; consider border and spread changes
                        domain_infimum(OldDom, OldInf),
                        (   Inf == OldInf -> LeftP = Left
                        ;   LeftP = yes
                        ),
                        domain_supremum(OldDom, OldSup),
                        (   Sup == OldSup -> RightP = Right
                        ;   RightP = yes
                        ),
                        domain_spread(OldDom, OldSpread),
                        domain_spread(Dom, NewSpread),
                        (   NewSpread == OldSpread -> SpreadP = Spread
                        ;   SpreadP = yes
                        ),
                        put_attr(X, clpfd, clpfd(LeftP,RightP,SpreadP,Dom,Ps)),
                        (   RightP == yes, Right = yes -> true
                        ;   LeftP == yes, Left = yes -> true
                        ;   SpreadP == yes, Spread = yes -> true
                        ;   trigger_props(Ps)
                        )
                    )
                )
            ;   var(X) ->
                put_attr(X, clpfd, clpfd(no,no,no,Dom, Ps))
            ;   true
            )
        ).

domain_spread(Dom, Spread) :-
        domain_smallest_finite(Dom, S),
        domain_largest_finite(Dom, L),
        Spread cis1 L - S.

smallest_finite(inf, Y, Y).
smallest_finite(n(N), _, n(N)).

domain_smallest_finite(from_to(F,T), S) :- smallest_finite(F, T, S).
domain_smallest_finite(split(_, L, _), S) :- domain_smallest_finite(L, S).

largest_finite(sup, Y, Y).
largest_finite(n(N), _, n(N)).

domain_largest_finite(from_to(F,T), L) :- largest_finite(T, F, L).
domain_largest_finite(split(_, _, R), L) :- domain_largest_finite(R, L).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   With terminating propagation, all relevant constraints get a
   propagation opportunity whenever a new constraint is posted.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

reinforce(X) :-
        (   current_prolog_flag(clpfd_propagation, terminating) ->
            % reinforce([], X), do_queue
            collect_variables(X, [], Vs),
            maplist(reinforce_, Vs),
            do_queue
        ;   % full propagation reduces to fixpoint anyway
            true
        ).

collect_variables(X, Vs0, Vs) :-
        (   get(X, _, Ps) ->
            term_variables(Ps, Vs1),
            all_collect(Vs1, [X|Vs0], Vs)
        ;   Vs = Vs0
        ).

all_collect([], Vs, Vs).
all_collect([X|Xs], Vs0, Vs) :-
        (   member(V, Vs0), X == V -> all_collect(Xs, Vs0, Vs)
        ;   collect_variables(X, Vs0, Vs1),
            all_collect(Xs, Vs1, Vs)
        ).

reinforce_(X) :-
        (   get(X, Dom, Ps) ->
            put_full(X, Dom, Ps)
        ;   true
        ).

put_full(X, Dom, Ps) :-
        Dom \== empty,
        (   Dom = from_to(F, F) -> F = n(X)
        ;   (   get_attr(X, clpfd, Attr) ->
                Attr = clpfd(_,_,_,OldDom, _OldPs),
                put_attr(X, clpfd, clpfd(no,no,no,Dom, Ps)),
                %format("putting dom: ~w\n", [Dom]),
                (   OldDom == Dom -> true
                ;   domain_intervals(Dom, Is),
                    domain_intervals(OldDom, Is) -> true
                ;   trigger_props(Ps)
                )
            ;   var(X) -> %format('\t~w in ~w .. ~w\n',[X,L,U]),
                put_attr(X, clpfd, clpfd(no,no,no,Dom, Ps))
            ;   true
            )
        ).

trigger_props([]).
trigger_props([P|Ps]) :- trigger_prop(P), trigger_props(Ps).

trigger_prop(Propagator) :-
        arg(2, Propagator, MState),
        MState = mutable(State),
        (   State == dead -> true
        ;   State == queued -> true
        ;   % passive
            % format("triggering: ~w\n", [Propagator]),
            setarg(1, MState, queued),
            push_queue(ps(Propagator,MState))
        ).

kill(MState) :- setarg(1, MState, dead).

activate_propagator(propagator(P,MState)) :-
        MState = mutable(State),
        (   State == dead -> true
        ;   %format("running: ~w\n", [P]),
            setarg(1, MState, passive),
            run_propagator(P, MState)
        ).

:- nb_setval('$clpfd_queue_status', enabled).

disable_queue :- b_setval('$clpfd_queue_status', disabled).
enable_queue  :- b_setval('$clpfd_queue_status', enabled), do_queue.

do_queue :-
        (   b_getval('$clpfd_queue_status', enabled) ->
            (   fetch_constraint_(C) ->
                activate_propagator(C),
                %C = propagator(Prop,_),
                %functor(Prop, FP, _),
                %format("\n\ngot: ~w\n\n", [FP]),
                do_queue
            ;   true
            )
        ;   true
        ).

init_propagator(Var, Prop) :-
        (   var(Var) -> get(Var, Dom, Ps), put(Var, Dom, [Prop|Ps])
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%% lex_chain(+Lists)
%
% Constrains Lists to be lexicographically non-decreasing.

lex_chain([]).
lex_chain([Ls|Lss]) :-
        lex_chain_lag(Lss, Ls),
        lex_chain(Lss).

lex_chain_lag([], _).
lex_chain_lag([Ls|Lss], Ls0) :-
        lex_le(Ls0, Ls),
        lex_chain_lag(Lss, Ls).

lex_le([], []).
lex_le([V1|V1s], [V2|V2s]) :-
        V1 #=< V2,
        (   integer(V1) ->
            (   integer(V2) ->
                (   V1 =:= V2 -> lex_le(V1s, V2s) ;  true )
            ;   freeze(V2, lex_le([V1|V1s], [V2|V2s]))
            )
        ;   freeze(V1, lex_le([V1|V1s], [V2|V2s]))
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%% tuples_in(+Tuples, +Relation).
%
% Relation is a list of ground lists of integers. Tuples is a list of
% lists containing integers and finite domain variables. Tuples are
% constrained to be elements of Relation.

tuples_in(Tuples, Relation) :-
        ground(Relation),
        tuples_domain(Tuples, Relation),
        do_queue.

tuples_domain([], _).
tuples_domain([Tuple|Tuples], Relation) :-
        relation_unifiable(Relation, Tuple, Us, 0, _),
        (   ground(Tuple) -> memberchk(Tuple, Relation)
        ;   tuple_domain(Tuple, Us),
            tuple_freeze(Tuple, Us)
        ),
        tuples_domain(Tuples, Relation).

tuple_domain([], _).
tuple_domain([T|Ts], Relation0) :-
        take_firsts(Relation0, Firsts, Relation1),
        ( var(T) ->
            (   Firsts = [Unique] -> T = Unique
            ;   list_to_domain(Firsts, FDom),
                get(T, TDom, TPs),
                domains_intersection(FDom, TDom, TDom1),
                put(T, TDom1, TPs)
            )
        ;   true
        ),
        tuple_domain(Ts, Relation1).

take_firsts([], [], []).
take_firsts([[F|Os]|Rest], [F|Fs], [Os|Oss]) :-
        take_firsts(Rest, Fs, Oss).

tuple_freeze(Tuple, Relation) :-
        gensym('$clpfd_rel_', RID),
        nb_setval(RID, Relation),
        Prop = propagator(rel_tuple(RID,Tuple), mutable(passive)),
        tuple_freeze(Tuple, Tuple, Prop).

tuple_freeze([],  _, _).
tuple_freeze([T|Ts], Tuple, Prop) :-
        ( var(T) ->
            %Prop = propagator(rel_tuple(RID,Tuple), mutable(passive)),
            init_propagator(T, Prop),
            trigger_prop(Prop)
        ;   true
        ),
        tuple_freeze(Ts, Tuple, Prop).

relation_unifiable([], _, [], Changed, Changed).
relation_unifiable([R|Rs], Tuple, Us, Changed0, Changed) :-
        (   all_in_domain(R, Tuple) ->
            Us = [R|Rest],
            relation_unifiable(Rs, Tuple, Rest, Changed0, Changed)
        ;   relation_unifiable(Rs, Tuple, Us, 1, Changed)
        ).

all_in_domain([], []).
all_in_domain([A|As], [T|Ts]) :-
        ( var(T) ->
            get(T, Dom, _),
            domain_contains(Dom, A)
        ;   T =:= A
        ),
        all_in_domain(As, Ts).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

run_propagator(pdifferent(Left,Right,X), _MState) :-
        (   ground(X) ->
            disable_queue,
            exclude_fire(Left, Right, X),
            enable_queue
        ;   true
        ).

run_propagator(pdistinct(Left,Right,X), _MState) :-
        (   ground(X) ->
            disable_queue,
            exclude_fire(Left, Right, X),
            enable_queue
        ;   %outof_reducer(Left, Right, X)
            %(   var(X) -> kill_if_isolated(Left, Right, X, MState)
            %;   true
            %),
            true
        ).

run_propagator(check_distinct(Left,Right,X), _) :-
        \+ list_contains(Left, X),
        \+ list_contains(Right, X).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(pneq(A, B), MState) :-
        (   nonvar(A) ->
            (   nonvar(B) -> A =\= B, kill(MState)
            ;   get(B, BD0, BExp0),
                domain_remove(BD0, A, BD1),
                kill(MState),
                put(B, BD1, BExp0)
            )
        ;   nonvar(B) -> run_propagator(pneq(B, A), MState)
        ;   A \== B,
            get(A, _, AI, AS, _), get(B, _, BI, BS, _),
            (   AS cis_lt BI -> kill(MState)
            ;   AI cis_gt BS -> kill(MState)
            ;   true
            )
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(pgeq(A,B), MState) :-
        (   A == B -> true
        ;   nonvar(A) ->
            (   nonvar(B) -> kill(MState), A >= B
            ;   get(B, BD, BPs),
                domain_remove_greater_than(BD, A, BD1),
                kill(MState),
                put(B, BD1, BPs)
            )
        ;   nonvar(B) ->
            get(A, AD, APs),
            domain_remove_smaller_than(AD, B, AD1),
            kill(MState),
            put(A, AD1, APs)
        ;   get(A, AD, AL, AU, APs),
            get(B, _, BL, BU, _),
            AU cis_geq BL,
            (   AL cis_gt BU -> kill(MState)
            ;   AU == BL -> A = B
            ;   NAL cis1 max(AL,BL),
                domains_intersection(from_to(NAL,AU), AD, NAD),
                put(A, NAD, APs),
                (   get(B, BD2, BL2, BU2, BPs2) ->
                    NBU cis1 min(BU2, AU),
                    domains_intersection(from_to(BL2,NBU), BD2, NBD),
                    put(B, NBD, BPs2)
                ;   true
                )
            )
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

run_propagator(rel_tuple(RID, Tuple), MState) :-
        b_getval(RID, Relation),
        (   ground(Tuple) -> kill(MState), memberchk(Tuple, Relation)
        ;   relation_unifiable(Relation, Tuple, Us, 0, Changed),
            Us = [_|_],
            (   Us = [Single] -> kill(MState), Single = Tuple
            ;   Changed =:= 0 -> true
            ;   b_setval(RID, Us),
                disable_queue,
                tuple_domain(Tuple, Us),
                enable_queue
            )
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

run_propagator(pserialized(Var,Duration,Left,SDs), _MState) :-
        myserialized(Duration, Left, SDs, Var).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% X + Y = Z
run_propagator(pplus(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Z is X + Y
            ;   nonvar(Z) -> kill(MState), Y is Z - X
            ;   get(Z, ZD, ZPs),
                get(Y, YD, YPs),
                domain_shift(YD, X, Shifted_YD),
                domains_intersection(Shifted_YD, ZD, ZD1),
                put(Z, ZD1, ZPs),
                (   var(Y) ->
                    O is -X,
                    domain_shift(ZD1, O, YD1),
                    put(Y, YD1, YPs)
                ;   true
                )
            )
        ;   nonvar(Y) -> run_propagator(pplus(Y,X,Z), MState)
        ;   nonvar(Z) ->
            get(X, XD, XPs),
            get(Y, YD, YPs),
            domain_negate(XD, XDN),
            domain_shift(XDN, Z, YD1),
            domains_intersection(YD, YD1, YD2),
            domain_negate(YD2, YD2N),
            domain_shift(YD2N, Z, XD1),
            domains_intersection(XD, XD1, XD2),
            put(X, XD2, XPs),
            put(Y, YD2, YPs)
        ;   (   X == Y, get(Z, ZD, _), \+ domain_contains(ZD, 0) ->
                neq_num(X, 0)
            ;   true
            ),
            (   get(X, XD, XL, XU, XPs), get(Y, YD, YL, YU, YPs),
                get(Z, ZD, ZL, ZU, _) ->
                NXL cis max(XL, ZL-YU),
                NXU cis min(XU, ZU-YL),
                (   NXL == XL, NXU == XU -> true
                ;   domains_intersection(XD, from_to(NXL, NXU), NXD),
                    put(X, NXD, XPs)
                ),
                (   get(Y, YD2, YL2, YU2, YPs2) ->
                    NYL cis max(YL2, ZL-NXU),
                    NYU cis min(YU2, ZU-NXL),
                    (   NYL == YL2, NYU == YU2 -> true
                    ;   domains_intersection(YD2, from_to(NYL, NYU), NYD),
                        put(Y, NYD, YPs2)
                    )
                ;   NYL = Y, NYU = Y
                ),
                (   get(Z, ZD2, ZL2, ZU2, ZPs2) ->
                    NZL cis max(ZL2,NXL+NYL),
                    NZU cis min(ZU2,NXU+NYU),
                    (   NZL == ZL2, NZU == ZU2 -> true
                    ;   domains_intersection(ZD2, from_to(NZL,NZU), NZD),
                        put(Z, NZD, ZPs2)
                    )
                ;   true
                )
            ;   true
            )

%             (   XI \== inf, XS \== sup,
%                 YI \== inf, YS \== sup ->
%                 To1 is To + 1,
%                 Low is min(XI,YI), Up is max(XS,YS),
%                 pplus_remove_impossibles(From, To1, Low, Up, XD, YD, ZD1, ZD2)
%             ;   ZD2 = ZD1
%             ),
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

run_propagator(ptimes(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Z is X * Y
            ;   X =:= 0 -> kill(MState), Z = 0
            ;   nonvar(Z) -> kill(MState), 0 =:= Z mod X, Y is Z // X
            ;   get(Y, YD, _),
                get(Z, ZD, ZPs),
                domain_expand(YD, X, Scaled_YD),
                domains_intersection(ZD, Scaled_YD, ZD1),
                put(Z, ZD1, ZPs),
                (   get(Y, YDom2, YPs2) ->
                    domain_contract(ZD1, X, Contract),
                    domains_intersection(Contract, YDom2, NYDom),
                    put(Y, NYDom, YPs2)
                ;   kill(MState), Z is X * Y
                )
            )
        ;   nonvar(Y) -> mytimes(Y,X,Z)
        ;   nonvar(Z) ->
            (   X == Y ->
                Z >= 0,
                catch(PRoot is integer(floor(sqrt(Z))),error(evaluation_error(float_overflow), _), true),
                (   nonvar(PRoot), PRoot**2 =:= Z ->
                    kill(MState),
                    NRoot is -PRoot,
                    get(X, TXD, TXPs), % temporary variables for this section
                    (   PRoot =:= 0 -> TXD1 = from_to(n(0),n(0))
                    ;   TXD1 = split(0, from_to(n(NRoot),n(NRoot)),
                                     from_to(n(PRoot),n(PRoot)))
                    ),
                    domains_intersection(TXD, TXD1, TXD2),
                    put(X, TXD2, TXPs)
                ;   % be more tolerant until GMP integer sqrt is available
                    true
                )
            ;   true
            ),
            (   get(X, XD, XL, XU, XPs) ->
                get(Y, YD, YL, YU, _),
                min_divide(n(Z),n(Z),YL,YU,TNXL),
                max_divide(n(Z),n(Z),YL,YU,TNXU),
                NXL cis max(XL,ceiling(TNXL)),
                NXU cis min(XU,floor(TNXU)),
                (   NXL == XL, NXU == XU -> true
                ;   domains_intersection(from_to(NXL,NXU), XD, XD1),
                    put(X, XD1, XPs)
                ),
                (   get(Y, YD2, YL2,YU2,YExp2) ->
                    min_divide(n(Z),n(Z),NXL,NXU,NYLT),
                    max_divide(n(Z),n(Z),NXL,NXU,NYUT),
                    NYL cis max(YL2,ceiling(NYLT)),
                    NYU cis min(YU2,floor(NYUT)),
                    (   NYL == YL2, NYU == YU2 -> true
                    ;   domains_intersection(from_to(NYL,NYU), YD2, YD3),
                        put(Y, YD3, YExp2)
                    )
                ;   (   Y \== 0 -> 0 =:= Z mod Y, kill(MState), X is Z // Y
                    ;   kill(MState), Z = 0
                    )
                )
            ;   true
            ),
            (   Z =\= 0 -> neq_num(X, 0), neq_num(Y, 0)
            ;   true
            )
        ;   (   X == Y -> geq(Z, 0) ; true ),
            (   get(X,XD,XL,XU,XExp), get(Y,YD,YL,YU,_), get(Z,ZD,ZL,ZU,_) ->
                min_divide(ZL,ZU,YL,YU,TXL),
                NXL cis max(XL,ceiling(TXL)),
                max_divide(ZL,ZU,YL,YU,TXU),
                NXU cis min(XU,floor(TXU)),
                domains_intersection(from_to(NXL,NXU), XD, XD1),
                put(X, XD1, XExp),
                (   get(Y,YD2,YL2,YU2,YExp2) ->
                    min_divide(ZL,ZU,XL,XU,TYL),
                    NYL cis max(YL2,ceiling(TYL)),
                    max_divide(ZL,ZU,XL,XU,TYU),
                    NYU cis min(YU2,floor(TYU)),
                    (   NYL == YL2, NYU == YU2 -> true
                    ;   domains_intersection(from_to(NYL,NYU), YD2, YD3),
                        put(Y, YD3, YExp2)
                    )
                ;   NYL = Y, NYU = Y
                ),
                (   get(Z, ZD2, ZL2, ZU2, ZExp2) ->
                    min_times(NXL,NXU,NYL,NYU,TZL),
                    NZL cis1 max(ZL2,TZL),
                    max_times(NXL,NXU,NYL,NYU,TZU),
                    NZU cis1 min(ZU2,TZU),
                    (   NZL == ZL2, NZU == ZU2 -> true
                    ;   domains_intersection(from_to(NZL,NZU), ZD2, ZD3),
                        put(Z, ZD3, ZExp2)
                    )
                ;   true
                )
            ;   true
            )
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% X / Y = Z

run_propagator(pdiv(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Y =\= 0, Z is X // Y
            ;   get(Y, YD, YL, YU, YPs),
                (   nonvar(Z) -> true
                    % TODO: cover this
                ;   get(Z, ZD, ZL, ZU, ZPs),
                    (   YL cis_leq n(0), YU cis_geq n(0) ->
                        NZL cis max(-abs(n(X)), ZL),
                        NZU cis min(abs(n(X)), ZU)
                    ;   X >= 0, YL cis_gt n(0) ->
                        NZL cis max(n(X)//YU, ZL),
                        NZU cis min(n(X)//YL, ZU)
                    ;   % TODO: cover more cases
                        NZL = ZL, NZU = ZU
                    ),
                    (   NZL = ZL, NZU = ZU -> true
                    ;   domains_intersection(from_to(NZL,NZU), ZD, NZD),
                        put(Z, NZD, ZPs)
                    )
                )
            )
        ;   nonvar(Y) ->
            Y =\= 0,
            get(X, XD, XL, XU, XPs),
            (   nonvar(Z) ->
                (   Z > 0, Y > 0 ->
                    NXL cis max(n(Z)*n(Y), XL),
                    NXU cis min((n(Z)+n(1))*n(Y)-n(1), XU)
                ;   Z =:= 0 ->
                    NXL cis max(-abs(n(Y)) + n(1), XL),
                    NXU cis min(abs(n(Y)) - n(1), XU)
                ;   % TODO: cover more cases
                    NXL = XL, NXU = XU
                ),
                (   NXL == XL, NXU == XU -> true
                ;   domains_intersection(from_to(NXL,NXU), XD, NXD),
                    put(X, NXD, XPs)
                )
            ;   get(Z, ZD, ZPs),
                domain_contract_less(XD, Y, Contracted),
                domains_intersection(Contracted, ZD, NZD),
                put(Z, NZD, ZPs),
                (   \+ domain_contains(NZD, 0), get(X, XD2, XPs2) ->
                    domain_expand_more(NZD, Y, Expanded),
                    domains_intersection(Expanded, XD2, NXD2),
                    put(X, NXD2, XPs2)
                ;   true
                )
            )
        ;   nonvar(Z) ->
            get(X, XD, XL, XU, XPs),
            get(Y, YD, YL, YU, YPs),
            (   YL cis_geq n(0), XL cis_geq n(0) ->
                NXL cis max(YL*n(Z), XL),
                NXU cis min(YU*(n(Z)+n(1))-n(1), XU)
            ;   %TODO: cover more cases
                NXL = XL, NXU = XU
            ),
            (   NXL == XL, NXU == XU -> true
            ;   domains_intersection(from_to(NXL,NXU), XD, NXD),
                put(X, NXD, XPs)
            )
        ;   (   X == Y -> Z = 1 ; true )
        ).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Y = abs(X)

run_propagator(pabs(X,Y), MState) :-
        (   nonvar(X) -> kill(MState), Y is abs(X)
        ;   nonvar(Y) ->
            Y >= 0,
            (   Y =:= 0 -> X = 0
            ;   get(X, XD, XPs),
                YN is -Y,
                domains_intersection(split(0, from_to(n(YN),n(YN)),
                                           from_to(n(Y),n(Y))), XD, XD1),
                put(X, XD1, XPs)
            )
        ;   get(X, XD, XPs),
            get(Y, YD, _),
            domain_negate(YD, YDNegative),
            domains_union(YD, YDNegative, XD1),
            domains_intersection(XD, XD1, XD2),
            put(X, XD2, XPs),
            (   get(Y, YD1, YPs1) ->
                domain_negate(XD2, XD2Neg),
                domains_union(XD2, XD2Neg, YD2),
                domain_remove_smaller_than(YD2, 0, YD3),
                domains_intersection(YD1, YD3, YD4),
                put(Y, YD4, YPs1)
            ;   true
            )
        ).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% K =_ X   (mod M)

run_propagator(pmod(X,M,K), MState) :-
        (   nonvar(X) ->
            (   nonvar(M) -> kill(MState), M =\= 0, K is X mod M
            ;   true
            )
        ;   nonvar(M) ->
            M =\= 0,
            (   M =:= 1 -> K = 0
            ;   get(K, KD, KPs) ->
                MP is abs(M) - 1,
                MN is -MP,
                get(K, KD, KPs),
                domains_intersection(from_to(n(MN), n(MP)), KD, KD1),
                put(K, KD1, KPs)
            ;   get(X, XD, XPs),
                (   fail, domain_supremum(XD, n(_)), domain_infimum(XD, n(_)) ->
                    % bounded domain (propagation currently disabled)
                    kill(MState),
                    findall(E, (domain_to_list(XD, XLs),
                                   member(E, XLs), E mod M =:= K), Es),
                    list_to_domain(Es, XD1),
                    domains_intersection(XD, XD1, XD2),
                    put(X, XD2, XPs)
                ;   % if possible, propagate at the boundaries
                    (   nonvar(K), domain_infimum(XD, n(Min)) ->
                        (   Min mod M =:= K -> true
                        ;   neq_num(X, Min)
                        )
                    ;   true
                    ),
                    (   nonvar(K), domain_supremum(XD, n(Max)) ->
                        (   Max mod M =:= K -> true
                        ;   neq_num(X, Max)
                        )
                    ;   true
                    )
                )
            )
        ;   true % TODO: propagate more
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Z = max(X,Y)

run_propagator(pmax(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Z is max(X,Y)
            ;   nonvar(Z) ->
                (   Z == X -> kill(MState), X #>= Y
                ;   Z > X -> Z = Y
                ;   fail % Z < X
                )
            ;   get(Y, YD, YInf, YSup, _),
                (   YInf cis_gt n(X) -> Z = Y
                ;   YSup cis_lt n(X) -> Z = X
                ;   YSup = n(M) ->
                    get(Z, ZD, ZPs),
                    domain_remove_greater_than(ZD, M, ZD1),
                    put(Z, ZD1, ZPs)
                ;   true
                )
            )
        ;   nonvar(Y) -> run_propagator(pmax(Y,X,Z), MState)
        ;   get(Z, ZD, ZPs) ->
            get(X, _, XInf, XSup, _),
            get(Y, YD, YInf, YSup, _),
            (   YInf cis_gt YSup -> Z = Y
            ;   YSup cis_lt XInf -> Z = X
            ;   n(M) cis1 max(XSup, YSup) ->
                domain_remove_greater_than(ZD, M, ZD1),
                put(Z, ZD1, ZPs)
            ;   true
            )
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Z = min(X,Y)

run_propagator(pmin(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Z is min(X,Y)
            ;   nonvar(Z) ->
                (   Z == X -> X #=< Y
                ;   Z < X -> Z = Y
                ;   fail % Z > X
                )
            ;   get(Y, YD, YInf, YSup, _),
                (   YSup cis_lt n(X) -> Z = Y
                ;   YInf cis_gt n(X) -> Z = X
                ;   YInf = n(M) ->
                    get(Z, ZD, ZPs),
                    domain_remove_smaller_than(ZD, M, ZD1),
                    put(Z, ZD1, ZPs)
                ;   true
                )
            )
        ;   nonvar(Y) -> run_propagator(pmin(Y,X,Z), MState)
        ;   get(Z, ZD, ZPs) ->
            get(X, _, XInf, XSup, _),
            get(Y, YD, YInf, YSup, _),
            (   YSup cis_lt YInf -> Z = Y
            ;   YInf cis_gt XSup -> Z = X
            ;   n(M) cis1 min(XInf, YInf) ->
                domain_remove_smaller_than(ZD, M, ZD1),
                put(Z, ZD1, ZPs)
            ;   true
            )
        ;   true
        ).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Z = X ^ Y

run_propagator(pexp(X,Y,Z), MState) :-
        (   X == 1 -> kill(MState), Z = 1
        ;   X == 0 -> kill(MState), Z #<==> Y #= 0
        ;   Y == 1 -> kill(MState), Z = X
        ;   Y == 0 -> kill(MState), Z = 1
        ;   nonvar(X), nonvar(Y) ->
            ( Y >= 0 -> true ; X =:= -1 ),
            kill(MState),
            Z is X**Y
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% reified constraints

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% propagate definedness of expressions - 0 means undefined, 1 defined.
% An expression is defined iff all its subexpressions are defined.

run_propagator(defined(X,Y,Z), MState) :-
        (   X == 1, Y == 1 -> kill(MState), Z = 1
        ;   Z == 1 -> kill(MState), X = 1, Y = 1
        ;   X == 0 -> kill(MState), Z = 0
        ;   Y == 0 -> kill(MState), Z = 0
        ;   true
        ).

% The result of X/Y and X mod Y is undefined iff Y is 0.

run_propagator(reified_div(X,Y,D,Z), MState) :-
        (   Y == 0 -> kill(MState), D = 0
        ;   D == 1 -> kill(MState), Z #= X / Y
        ;   integer(Y), Y =\= 0 -> kill(MState), D = 1, Z #= X / Y
        ;   get(Y, YD, _), \+ domain_contains(YD, 0) ->
            D = 1, Z #= X / Y
        ;   true
        ).

run_propagator(reified_mod(X,Y,D,Z), MState) :-
        (   Y == 0 -> kill(MState), D = 0
        ;   D == 1 -> kill(MState), Z #= X mod Y
        ;   integer(Y), Y =\= 0 -> kill(MState), D = 1, Z #= X mod Y
        ;   get(Y, YD, _), \+ domain_contains(YD, 0) ->
            kill(MState),
            D = 1, Z #= X mod Y
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

run_propagator(reified_geq(DX,X,DY,Y,B), MState) :-
        (   DX == 0 -> kill(MState), B = 0
        ;   DY == 0 -> kill(MState), B = 0
        ;   B == 1 -> kill(MState), DX = 1, DY = 1, X #>= Y
        ;   DX == 1, DY == 1 ->
            (   var(B) ->
                (   nonvar(X) ->
                    (   nonvar(Y) ->
                        kill(MState),
                        (   X >= Y -> B = 1 ; B = 0 )
                    ;   get(Y, _, YL, YU, _),
                        (   n(X) cis_geq YU -> B = 1
                        ;   n(X) cis_lt YL -> B = 0
                        ;   true
                        )
                    )
                ;   nonvar(Y) ->
                    get(X, _, XL, XU, _),
                    (   XL cis_geq n(Y) -> B = 1
                    ;   XU cis_lt n(Y) -> B = 0
                    ;   B in 0..1
                    )
                ;   get(X, _, XL, XU, _),
                    get(Y, _, YL, YU, _),
                    (   XL cis_geq YU -> B = 1
                    ;   XU cis_lt YL -> B = 0
                    ;   true
                    )
                )
            ;   B =:= 0 -> kill(MState), X #< Y
            ;   true
            )
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(reified_eq(DX,X,DY,Y,B), MState) :-
        (   DX == 0 -> kill(MState), B = 0
        ;   DY == 0 -> kill(MState), B = 0
        ;   B == 1 -> kill(MState), DX = 1, DY = 1, X = Y
        ;   DX == 1, DY == 1 ->
            (   var(B) ->
                (   nonvar(X) ->
                    (   nonvar(Y) ->
                        kill(MState),
                        (   X =:= Y -> B = 1 ; B = 0)
                    ;   get(Y, YD, _),
                        (   domain_contains(YD, X) -> true
                        ;   B = 0
                        )
                    )
                ;   nonvar(Y) -> run_propagator(reified_eq(DY,Y,DX,X,B), MState)
                ;   X == Y -> B = 1
                ;   get(X, _, XL, XU, _),
                    get(Y, _, YL, YU, _),
                    (   XL cis_gt YU -> B = 0
                    ;   YL cis_gt XU -> B = 0
                    ;   true
                    )
                )
            ;   B =:= 0 -> kill(MState), X #\= Y
            ;   true
            )
        ;   true
        ).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(reified_neq(DX,X,DY,Y,B), MState) :-
        (   DX == 0 -> kill(MState), B = 0
        ;   DY == 0 -> kill(MState), B = 0
        ;   B == 1 -> kill(MState), DX = 1, DY = 1, X #\= Y
        ;   DX == 1, DY == 1 ->
            (   var(B) ->
                (   nonvar(X) ->
                    (   nonvar(Y) ->
                        kill(MState),
                        (   X =\= Y -> B = 1 ; B = 0)
                    ;   get(Y, YD, _),
                        (   domain_contains(YD, X) -> true
                        ;   B = 1
                        )
                    )
                ;   nonvar(Y) -> run_propagator(reified_neq(DY,Y,DX,X,B), MState)
                ;   X == Y -> B = 0
                ;   get(X, _, XL, XU, _),
                    get(Y, _, YL, YU, _),
                    (   XL cis_gt YU -> B = 1
                    ;   YL cis_gt XU -> B = 1
                    ;   true
                    )
                )
            ;   B =:= 0 -> kill(MState), X = Y
            ;   true
            )
        ;   true
        ).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(reified_and(X,Y,B), MState) :-
        (   var(B) ->
            (   nonvar(X) ->
                (   X =:= 0 -> B = 0
                ;   X =:= 1 -> B = Y
                )
            ;   nonvar(Y) -> run_propagator(reified_and(Y,X,B), MState)
            ;   true
            )
        ;   B =:= 0 ->
            (   X == 1 -> kill(MState), Y = 0
            ;   Y == 1 -> kill(MState), X = 0
            ;   true
            )
        ;   B =:= 1 -> kill(MState), X = 1, Y = 1
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(reified_or(X,Y,B), MState) :-
        (   var(B) ->
            (   nonvar(X) ->
                (   X =:= 1 -> B = 1
                ;   X =:= 0 -> B = Y
                )
            ;   nonvar(Y) -> run_propagator(reified_or(Y,X,B), MState)
            ;   true
            )
        ;   B =:= 0 -> kill(MState), X = 0, Y = 0
        ;   B =:= 1 ->
            (   X == 0 -> Y = 1
            ;   Y == 0 -> X = 1
            ;   true
            )
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(reified_not(X,Y), MState) :-
        (   X == 0 -> kill(MState), Y = 1
        ;   X == 1 -> kill(MState), Y = 0
        ;   Y == 0 -> kill(MState), X = 1
        ;   Y == 1 -> kill(MState), X = 0
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(pimpl(X, Y), MState) :-
        (   nonvar(X) ->
            (   X =:= 1 -> kill(MState), Y = 1
            ;   true
            )
        ;   nonvar(Y) ->
            (   Y =:= 0 -> kill(MState), X = 0
            ;   true
            )
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

run_propagator(por(X, Y, Z), MState) :-
        (   nonvar(X) ->
            (   X =:= 0 -> Y = Z
            ;   X =:= 1 -> Z = 1
            )
        ;   nonvar(Y) -> run_propagator(por(Y,X,Z), MState)
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

max_times(L1,U1,L2,U2,Max) :-
        Max cis max(max(L1*L2,L1*U2),max(U1*L2,U1*U2)).
min_times(L1,U1,L2,U2,Min) :-
        Min cis min(min(L1*L2,L1*U2),min(U1*L2,U1*U2)).

max_divide(L1,U1,L2,U2,Max) :-
        (   L2 cis_leq n(0) , U2 cis_geq n(0) -> Max = sup
        ;   Max cis max(max(div(L1,L2),div(L1,U2)),max(div(U1,L2),div(U1,U2)))
        ).
min_divide(L1,U1,L2,U2,Min) :-
        (   L2 cis_leq n(0) , U2 cis_geq n(0) -> Min = inf
        ;   Min cis min(min(div(L1,L2),div(L1,U2)),min(div(U1,L2),div(U1,U2)))
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Weak arc consistent all_distinct/1 constraint.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

%% all_distinct(+Ls).
%
% Like all_different/1, with stronger propagation.

%all_distinct(Ls) :- all_different(Ls).
all_distinct(Ls) :-
        length(Ls, _),
        MState = mutable(passive),
        all_distinct(Ls, [], MState),
        do_queue.

all_distinct([], _, _).
all_distinct([X|Right], Left, MState) :-
        %\+ list_contains(Right, X),
        (   var(X) ->
            Prop = propagator(pdistinct(Left,Right,X), mutable(passive)),
            init_propagator(X, Prop),
            trigger_prop(Prop)
%             Prop2 = propagator(check_distinct(Left,Right,X), mutable(passive)),
%             init_propagator(X, Prop2),
%             trigger_prop(Prop2)
        ;   exclude_fire(Left, Right, X)
        ),
        outof_reducer(Left, Right, X),
        all_distinct(Right, [X|Left], MState).

exclude_fire(Left, Right, E) :-
        exclude_list(Left, E),
        exclude_list(Right, E).

exclude_list([], _).
exclude_list([V|Vs], Val) :-
        (   var(V) ->
            get(V, VD, VPs),
            domain_remove(VD, Val, VD1),
            put(V, VD1, VPs)
        ;   V =\= Val
        ),
        exclude_list(Vs, Val).

list_contains([X|Xs], Y) :-
        (   X == Y -> true
        ;   list_contains(Xs, Y)
        ).

kill_if_isolated(Left, Right, X, MState) :-
        append(Left, Right, Others),
        get(X, XDom, _),
        (   all_empty_intersection(Others, XDom) -> kill(MState)
        ;   true
        ).

all_empty_intersection([], _).
all_empty_intersection([V|Vs], XDom) :-
        (   var(V) ->
            get(V, VDom, _),
            domains_intersection_(VDom, XDom, empty),
            all_empty_intersection(Vs, XDom)
        ;   all_empty_intersection(Vs, XDom)
        ).

outof_reducer(Left, Right, Var) :-
        (   var(Var) ->
            append(Left, Right, Others),
            get(Var, Dom, _),
            domain_num_elements(Dom, N),
            num_subsets(Others, Dom, 0, Num, NonSubs),
            (   n(Num) cis_geq N -> fail
            ;   n(Num) cis1 N - n(1) ->
                reduce_from_others(NonSubs, Dom)
            ;   true
            )
        ;   %\+ list_contains(Right, Var),
            %\+ list_contains(Left, Var)
            true
        ).

reduce_from_others([], _).
reduce_from_others([X|Xs], Dom) :-
        (   var(X) ->
            get(X, XDom, XPs),
            domain_subtract(XDom, Dom, NXDom),
            put(X, NXDom, XPs)
        ;   true
        ),
        reduce_from_others(Xs, Dom).

num_subsets([], _Dom, Num, Num, []).
num_subsets([S|Ss], Dom, Num0, Num, NonSubs) :-
        (   var(S) ->
            get(S, SDom, _),
            (   domain_subdomain(Dom, SDom) ->
                Num1 is Num0 + 1,
                num_subsets(Ss, Dom, Num1, Num, NonSubs)
            ;   NonSubs = [S|Rest],
                num_subsets(Ss, Dom, Num0, Num, Rest)
            )
        ;   num_subsets(Ss, Dom, Num0, Num, NonSubs)
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%      serialized(+Starts, +Durations)
%
%       Constrain a set of intervals to a non-overlapping sequence.
%       Starts = [S_1,...,S_n], is a list of variables or integers,
%       Durations = [D_1,...,D_n] is a list of non-negative integers.
%       Constrains Starts and Durations to denote a set of
%       non-overlapping tasks, i.e.: S_i + D_i =< S_j or S_j + D_j =<
%       S_i for all 1 =< i < j =< n.
%
%  @see Dorndorf et al. 2000, "Constraint Propagation Techniques for the
%       Disjunctive Scheduling Problem"

serialized(Starts, Durations) :-
        pair_up(Starts, Durations, SDs),
        serialize(SDs, []),
        do_queue.

pair_up([], [], []).
pair_up([A|As], [B|Bs], [A-n(B)|ABs]) :- pair_up(As, Bs, ABs).

% attribute: pserialized(Var, Duration, Left, Right)
%   Left and Right are lists of Start-Duration pairs representing
%   other tasks occupying the same resource

serialize([], _).
serialize([Start-D|SDs], Left) :-
        (   var(Start) ->
            Prop = propagator(pserialized(Start,D,Left,SDs), mutable(passive)),
            init_propagator(Start, Prop),
            trigger_prop(Prop)
        ;   true
        ),
        myserialized(D, Left, SDs, Start),
        serialize(SDs, [Start-D|Left]).

% consistency check / propagation
% Currently implements 2-b-consistency

myserialized(Duration, Left, Right, Start) :-
        myserialized(Left, Start, Duration),
        myserialized(Right, Start, Duration).


earliest_start_time(Start, EST) :-
        (   get(Start, D, _) ->
            domain_infimum(D, EST)
        ;   EST = n(Start)
        ).

latest_start_time(Start, LST) :-
        (   get(Start, D, _) ->
            domain_supremum(D, LST)
        ;   LST = n(Start)
        ).

myserialized([], _, _).
myserialized([S_I-D_I|SDs], S_J, D_J) :-
        (   var(S_I) ->
            serialize_lower_bound(S_I, D_I, Start, D_J),
            (   var(S_I) -> serialize_upper_bound(S_I, D_I, Start, D_J)
            ;   true
            )
        ;   var(S_J) ->
            serialize_lower_bound(S_J, D_J, S, D_I),
            (   var(S_J) -> serialize_upper_bound(S_J, D_J, S, D_I)
            ;   true
            )
        ;   D_I = n(D_II), D_J = n(D_JJ),
            (   S_I + D_II =< S_J -> true
            ;   S_J + D_JJ =< S_I -> true
            ;   fail
            )
        ),
        myserialized(SDs, S_J, D_J).

serialize_lower_bound(I, D_I, J, D_J) :-
        get(I, DomI, Ps),
        domain_infimum(DomI, EST_I),
        latest_start_time(J, LST_J),
        (   Sum cis EST_I + D_I, Sum cis_gt LST_J ->
            earliest_start_time(J, EST_J),
            EST cis EST_J+D_J,
            domain_remove_smaller_than(DomI, EST, DomI1),
            put(I, DomI1, Ps)
        ;   true
        ).

serialize_upper_bound(I, D_I, J, D_J) :-
        get(I, DomI, Ps),
        domain_supremum(DomI, LST_I),
        earliest_start_time(J, EST_J),
        (   Sum cis EST_J + D_J, Sum cis_gt LST_I ->
            latest_start_time(J, LST_J),
            LST cis LST_J-D_I,
            domain_remove_greater_than(DomI, LST, DomI1),
            put(I, DomI1, Ps)
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Hooks
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

attr_unify_hook(clpfd(_,_,_,Dom,Ps), Other) :-
        (   nonvar(Other) ->
            (   integer(Other) -> true
            ;   type_error(integer, Other)
            ),
            domain_contains(Dom, Other),
            trigger_props(Ps),
            do_queue
        ;   get(Other, OD, OPs),
            domains_intersection(Dom, OD, Dom1),
            append(Ps, OPs, Ps1),
            put(Other, Dom1, Ps1),
            trigger_props(Ps1),
            do_queue
        ).

bound_portray(inf, inf).
bound_portray(sup, sup).
bound_portray(n(N), N).

domain_to_drep(Dom, Drep) :-
        domain_intervals(Dom, [A0-B0|Rest]),
        bound_portray(A0, A),
        bound_portray(B0, B),
        (   A == B -> Drep0 = A
        ;   Drep0 = A..B
        ),
        intervals_to_drep(Rest, Drep0, Drep).

intervals_to_drep([], Drep, Drep).
intervals_to_drep([A0-B0|Rest], Drep0, Drep) :-
        bound_portray(A0, A),
        bound_portray(B0, B),
        (   A == B -> D1 = A
        ;   D1 = A..B
        ),
        intervals_to_drep(Rest, Drep0 \/ D1, Drep).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

numlist(F, T, Ls) :- T1 is T + 1, numlist_(F, T1, Ls).

numlist_(F, F, []) :- !.
numlist_(F0, T, [F0|Rest]) :- F1 is F0 + 1, numlist_(F1, T, Rest).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

attribute_goal(X, Goal) :-
        get_attr(X, clpfd, clpfd(_,_,_,Dom,Ps)),
        domain_to_drep(Dom, Drep),
        attributes_goals(Ps, X in Drep, Goal).

attributes_goals([], Goal, Goal).
attributes_goals([propagator(P, State)|As], Goal0, Goal) :-
        (   State = mutable(dead) -> Goal1 = Goal0
        ;   State = mutable(processed) -> Goal1 = Goal0
        ;   attribute_goal_(P, G) ->
            setarg(1, State, processed),
            Goal1 = (Goal0,G)
        ;   Goal1 = Goal0 % no conversion for P yet
        ),
        attributes_goals(As, Goal1, Goal).

attribute_goal_(pgeq(A,B), A #>= B).
attribute_goal_(pplus(X,Y,Z), X + Y #= Z).
attribute_goal_(pneq(A,B), A #\= B).
attribute_goal_(ptimes(X,Y,Z), X*Y #= Z).
attribute_goal_(pdiv(X,Y,Z), X/Y #= Z).
attribute_goal_(pabs(X,Y), Y #= abs(X)).
attribute_goal_(pmod(X,M,K), X mod M #= K).
attribute_goal_(pmax(X,Y,Z), Z #= max(X,Y)).
attribute_goal_(pmin(X,Y,Z), Z #= min(X,Y)).
attribute_goal_(por(X,Y,Z), X #\/ Y #<==> Z).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
domain_to_list(Domain, List) :- domain_to_list(Domain, List, []).

domain_to_list(split(_, Left, Right)) -->
        domain_to_list(Left), domain_to_list(Right).
domain_to_list(empty)                 --> [].
domain_to_list(from_to(n(F),n(T)))    --> { numlist(F, T, Ns) }, dlist(Ns).

dlist([])     --> [].
dlist([L|Ls]) --> [L], dlist(Ls).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Testing
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

%?- test_intersection([1,2,3,4,5], [1,5], I).

%?- test_intersection([1,2,3,4,5], [], I).

test_intersection(List1, List2, Is) :-
        list_to_domain(List1, D1),
        list_to_domain(List2, D2),
        domains_intersection(D1, D2, I),
        domain_to_list(I, Is).

test_subdomain(L1, L2) :-
        list_to_domain(L1, D1),
        list_to_domain(L2, D2),
        domain_subdomain(D1, D2).