:- encoding(utf8).

/*  Part of SWI-Prolog

    Author:        Markus Triska
    E-mail:        triska@metalevel.at
    WWW:           http://www.swi-prolog.org
    Copyright (C): 2007-2017 Markus Triska
    All rights reserved.

    Redistribution and use in source and binary forms, with or without
    modification, are permitted provided that the following conditions
    are met:

    1. Redistributions of source code must retain the above copyright
       notice, this list of conditions and the following disclaimer.

    2. Redistributions in binary form must reproduce the above copyright
       notice, this list of conditions and the following disclaimer in
       the documentation and/or other materials provided with the
       distribution.

    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
    "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
    FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
    COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
    INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
    BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
    LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
    CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
    LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
    ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
    POSSIBILITY OF SUCH DAMAGE.
*/

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   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 #>= 0, Y #=< 0.
   %@ X in 0..sup,
   %@ Y in inf..0.

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

   ?- N #= 2^66, X #\= N.
   %@ N = 73786976294838206464,
   %@ X in inf..73786976294838206463\/73786976294838206465..sup.

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

   ?- Y #= abs(X), Y #\= 3, Z * Z #= 4.
   %@ Y in 0..2\/4..sup,
   %@ Y#=abs(X),
   %@ X in inf.. -4\/ -2..2\/4..sup,
   %@ Z in -2\/2.

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

   Development of this library has moved to SICStus Prolog. If you
   need any additional features or want to help, please file an issue at:

                    https://github.com/triska/clpz
                    ==============================

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

:- module(clpfd, [
                  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(700, xfx, in_set),
                  op(450, xfx, ..), % should bind more tightly than \/
                  (#>)/2,
                  (#<)/2,
                  (#>=)/2,
                  (#=<)/2,
                  (#=)/2,
                  (#\=)/2,
                  (#\)/1,
                  (#<==>)/2,
                  (#==>)/2,
                  (#<==)/2,
                  (#\/)/2,
                  (#\)/2,
                  (#/\)/2,
                  (in)/2,
                  (ins)/2,
                  all_different/1,
                  all_distinct/1,
                  sum/3,
                  scalar_product/4,
                  tuples_in/2,
                  labeling/2,
                  label/1,
                  indomain/1,
                  lex_chain/1,
                  serialized/2,
                  global_cardinality/2,
                  global_cardinality/3,
                  circuit/1,
                  cumulative/1,
                  cumulative/2,
                  disjoint2/1,
                  element/3,
                  automaton/3,
                  automaton/8,
                  transpose/2,
                  zcompare/3,
                  chain/2,
                  fd_var/1,
                  fd_inf/2,
                  fd_sup/2,
                  fd_size/2,
                  fd_dom/2,
                  fd_degree/2,
                                        % SICStus compatible fd_set API
                  (in_set)/2,
                  fd_set/2,
                  is_fdset/1,
                  empty_fdset/1,
                  fdset_parts/4,
                  empty_interval/2,
                  fdset_interval/3,
                  fdset_singleton/2,
                  fdset_min/2,
                  fdset_max/2,
                  fdset_size/2,
                  list_to_fdset/2,
                  fdset_to_list/2,
                  range_to_fdset/2,
                  fdset_to_range/2,
                  fdset_add_element/3,
                  fdset_del_element/3,
                  fdset_disjoint/2,
                  fdset_intersect/2,
                  fdset_intersection/3,
                  fdset_member/2,
                  fdset_eq/2,
                  fdset_subset/2,
                  fdset_subtract/3,
                  fdset_union/3,
                  fdset_union/2,
                  fdset_complement/2
                 ]).

:- meta_predicate with_local_attributes(?, ?, 0, ?).
:- public                               % called from goal_expansion
        clpfd_equal/2,
        clpfd_geq/2.

:- use_module(library(apply)).
:- use_module(library(apply_macros)).
:- use_module(library(assoc)).
:- use_module(library(error)).
:- use_module(library(lists)).
:- use_module(library(pairs)).

:- set_prolog_flag(generate_debug_info, false).

:- create_prolog_flag(optimise_clpfd, true, [keep(true)]).
:- create_prolog_flag(clpfd_monotonic, false, [keep(true)]).
:- create_prolog_flag(clpfd_propagation, default, [keep(true)]). % oneof([default,full])

:- op(700, xfx, cis).
:- op(700, xfx, cis_geq).
:- op(700, xfx, cis_gt).
:- op(700, xfx, cis_leq).
:- op(700, xfx, cis_lt).

/** <module> CLP(FD): Constraint Logic Programming over Finite Domains

**Development of this library has moved to SICStus Prolog.**

Please see [**CLP(Z)**](https://github.com/triska/clpz) for more
information.

## Introduction			{#clpfd-intro}

This library provides CLP(FD): Constraint Logic Programming over
Finite Domains. This is an instance of the general [CLP(_X_)
scheme](<#clp>), extending logic programming with reasoning over
specialised domains. CLP(FD) lets us reason about **integers** in a
way that honors the relational nature of Prolog.

Read [**The Power of Prolog**](https://www.metalevel.at/prolog) to
understand how this library is meant to be used in practice.

There are two major use cases of CLP(FD) constraints:

    1. [**declarative integer arithmetic**](<#clpfd-integer-arith>)
    2. solving **combinatorial problems** such as planning, scheduling
       and allocation tasks.

The predicates of this library can be classified as:

    * _arithmetic_ constraints like #=/2, #>/2 and #\=/2 [](<#clpfd-arithmetic>)
    * the _membership_ constraints in/2 and ins/2 [](<#clpfd-membership>)
    * the _enumeration_ predicates indomain/1, label/1 and labeling/2 [](<#clpfd-enumeration>)
    * _combinatorial_ constraints like all_distinct/1 and global_cardinality/2 [](<#clpfd-global>)
    * _reification_ predicates such as #<==>/2 [](<#clpfd-reification-predicates>)
    * _reflection_ predicates such as fd_dom/2 [](<#clpfd-reflection-predicates>)

In most cases, [_arithmetic constraints_](<#clpfd-arith-constraints>)
are the only predicates you will ever need from this library. When
reasoning over integers, simply replace low-level arithmetic
predicates like `(is)/2` and `(>)/2` by the corresponding CLP(FD)
constraints like #=/2 and #>/2 to honor and preserve declarative
properties of your programs. For satisfactory performance, arithmetic
constraints are implicitly rewritten at compilation time so that
low-level fallback predicates are automatically used whenever
possible.

Almost all Prolog programs also reason about integers. Therefore, it
is highly advisable that you make CLP(FD) constraints available in all
your programs. One way to do this is to put the following directive in
your =|<config>/init.pl|= initialisation file:

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

All example programs that appear in the CLP(FD) documentation assume
that you have done this.

Important concepts and principles of this library are illustrated by
means of usage examples that are available in a public git repository:
[**github.com/triska/clpfd**](https://github.com/triska/clpfd)

If you are used to the complicated operational considerations that
low-level arithmetic primitives necessitate, then moving to CLP(FD)
constraints may, due to their power and convenience, at first feel to
you excessive and almost like cheating. It _isn't_. Constraints are an
integral part of all popular Prolog systems, and they are designed
to help you eliminate and avoid the use of low-level and less general
primitives by providing declarative alternatives that are meant to be
used instead.

When teaching Prolog, CLP(FD) constraints should be introduced
_before_ explaining low-level arithmetic predicates and their
procedural idiosyncrasies. This is because constraints are easy to
explain, understand and use due to their purely relational nature. In
contrast, the modedness and directionality of low-level arithmetic
primitives are impure limitations that are better deferred to more
advanced lectures.

We recommend the following reference (PDF:
[metalevel.at/swiclpfd.pdf](https://www.metalevel.at/swiclpfd.pdf)) for
citing this library in scientific publications:

==
@inproceedings{Triska12,
  author    = {Markus Triska},
  title     = {The Finite Domain Constraint Solver of {SWI-Prolog}},
  booktitle = {FLOPS},
  series    = {LNCS},
  volume    = {7294},
  year      = {2012},
  pages     = {307-316}
}
==

More information about CLP(FD) constraints and their implementation is
contained in: [**metalevel.at/drt.pdf**](https://www.metalevel.at/drt.pdf)

The best way to discuss applying, improving and extending CLP(FD)
constraints is to use the dedicated `clpfd` tag on
[stackoverflow.com](http://stackoverflow.com). Several of the world's
foremost CLP(FD) experts regularly participate in these discussions
and will help you for free on this platform.

## Arithmetic constraints		{#clpfd-arith-constraints}

In modern Prolog systems, *arithmetic constraints* subsume and
supersede low-level predicates over integers. The main advantage of
arithmetic constraints is that they are true _relations_ and can be
used in all directions. For most programs, arithmetic constraints are
the only predicates you will ever need from this library.

The most important arithmetic constraint is #=/2, which subsumes both
`(is)/2` and `(=:=)/2` over integers. Use #=/2 to make your programs
more general. See [declarative integer
arithmetic](<#clpfd-integer-arith>).

In total, the arithmetic constraints are:

    | Expr1 `#=`  Expr2  | Expr1 equals Expr2                       |
    | Expr1 `#\=` Expr2  | Expr1 is not equal to Expr2              |
    | Expr1 `#>=` Expr2  | Expr1 is greater than or equal to Expr2  |
    | Expr1 `#=<` Expr2  | Expr1 is less than or equal to Expr2     |
    | Expr1 `#>` Expr2   | Expr1 is greater than Expr2              |
    | Expr1 `#<` Expr2   | Expr1 is less than Expr2                 |

`Expr1` and `Expr2` denote *arithmetic expressions*, which are:

    | _integer_          | Given value                          |
    | _variable_         | Unknown integer                      |
    | ?(_variable_)      | Unknown integer                      |
    | -Expr              | Unary minus                          |
    | Expr + Expr        | Addition                             |
    | Expr * Expr        | Multiplication                       |
    | Expr - Expr        | Subtraction                          |
    | Expr ^ Expr        | Exponentiation                       |
    | min(Expr,Expr)     | Minimum of two expressions           |
    | max(Expr,Expr)     | Maximum of two expressions           |
    | Expr `mod` Expr    | Modulo induced by floored division   |
    | Expr `rem` Expr    | Modulo induced by truncated division |
    | abs(Expr)          | Absolute value                       |
    | Expr // Expr       | Truncated integer division           |
    | Expr div Expr      | Floored integer division             |

where `Expr` again denotes an arithmetic expression.

The bitwise operations `(\)/1`, `(/\)/2`, `(\/)/2`, `(>>)/2`,
`(<<)/2`, `lsb/1`, `msb/1`, `popcount/1` and `(xor)/2` are also
supported.

## Declarative integer arithmetic		{#clpfd-integer-arith}

The [_arithmetic constraints_](<#clpfd-arith-constraints>)   #=/2,  #>/2
etc. are meant  to  be  used   _instead_  of  the  primitives  `(is)/2`,
`(=:=)/2`, `(>)/2` etc. over integers. Almost   all Prolog programs also
reason about integers. Therefore, it  is   recommended  that you put the
following directive in your =|<config>/init.pl|=  initialisation file to
make CLP(FD) constraints available in all your programs:

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

Throughout the following, it is assumed that you have done this.

The most basic use of CLP(FD) constraints is _evaluation_ of
arithmetic expressions involving integers. For example:

==
?- X #= 1+2.
X = 3.
==

This could in principle also be achieved with the lower-level
predicate `(is)/2`. However, an important advantage of arithmetic
constraints is their purely relational nature: Constraints can be used
in _all directions_, also if one or more of their arguments are only
partially instantiated. For example:

==
?- 3 #= Y+2.
Y = 1.
==

This relational nature makes CLP(FD) constraints easy to explain and
use, and well suited for beginners and experienced Prolog programmers
alike. In contrast, when using low-level integer arithmetic, we get:

==
?- 3 is Y+2.
ERROR: is/2: Arguments are not sufficiently instantiated

?- 3 =:= Y+2.
ERROR: =:=/2: Arguments are not sufficiently instantiated
==

Due to the necessary operational considerations, the use of these
low-level arithmetic predicates is considerably harder to understand
and should therefore be deferred to more advanced lectures.

For supported expressions, CLP(FD) constraints are drop-in
replacements of these low-level arithmetic predicates, often yielding
more general programs. See [`n_factorial/2`](<#clpfd-factorial>) for an
example.

This library uses goal_expansion/2 to automatically rewrite
constraints at compilation time so that low-level arithmetic
predicates are _automatically_ used whenever possible. For example,
the predicate:

==
positive_integer(N) :- N #>= 1.
==

is executed as if it were written as:

==
positive_integer(N) :-
        (   integer(N)
        ->  N >= 1
        ;   N #>= 1
        ).
==

This illustrates why the performance of CLP(FD) constraints is almost
always completely satisfactory when they are used in modes that can be
handled by low-level arithmetic. To disable the automatic rewriting,
set the Prolog flag `optimise_clpfd` to `false`.

If you are used to the complicated operational considerations that
low-level arithmetic primitives necessitate, then moving to CLP(FD)
constraints may, due to their power and convenience, at first feel to
you excessive and almost like cheating. It _isn't_. Constraints are an
integral part of all popular Prolog systems, and they are designed
to help you eliminate and avoid the use of low-level and less general
primitives by providing declarative alternatives that are meant to be
used instead.


## Example: Factorial relation {#clpfd-factorial}

We illustrate the benefit of using #=/2 for more generality with a
simple example.

Consider first a rather conventional definition of `n_factorial/2`,
relating each natural number _N_ to its factorial _F_:

==
n_factorial(0, 1).
n_factorial(N, F) :-
        N #> 0,
        N1 #= N - 1,
        n_factorial(N1, F1),
        F #= N * F1.
==

This program uses CLP(FD) constraints _instead_ of low-level
arithmetic throughout, and everything that _would have worked_ with
low-level arithmetic _also_ works with CLP(FD) constraints, retaining
roughly the same performance. For example:

==
?- n_factorial(47, F).
F = 258623241511168180642964355153611979969197632389120000000000 ;
false.
==

Now the point: Due to the increased flexibility and generality of
CLP(FD) constraints, we are free to _reorder_ the goals as follows:

==
n_factorial(0, 1).
n_factorial(N, F) :-
        N #> 0,
        N1 #= N - 1,
        F #= N * F1,
        n_factorial(N1, F1).
==

In this concrete case, _termination_ properties of the predicate are
improved. For example, the following queries now both terminate:

==
?- n_factorial(N, 1).
N = 0 ;
N = 1 ;
false.

?- n_factorial(N, 3).
false.
==

To make the predicate terminate if _any_ argument is instantiated, add
the (implied) constraint `F #\= 0` before the recursive call.
Otherwise, the query `n_factorial(N, 0)` is the only non-terminating
case of this kind.

The value of CLP(FD) constraints does _not_ lie in completely freeing
us from _all_ procedural phenomena. For example, the two programs do
not even have the same _termination properties_ in all cases.
Instead, the primary benefit of CLP(FD) constraints is that they allow
you to try different execution orders and apply [**declarative
debugging**](https://www.metalevel.at/prolog/debugging)
techniques _at all_!  Reordering goals (and clauses) can significantly
impact the performance of Prolog programs, and you are free to try
different variants if you use declarative approaches. Moreover, since
all CLP(FD) constraints _always terminate_, placing them earlier can
at most _improve_, never worsen, the termination properties of your
programs. An additional benefit of CLP(FD) constraints is that they
eliminate the complexity of introducing `(is)/2` and `(=:=)/2` to
beginners, since _both_ predicates are subsumed by #=/2 when reasoning
over integers.

In the case above, the clauses are mutually exclusive _if_ the first
argument is sufficiently instantiated. To make the predicate
deterministic in such cases while retaining its generality, you can
use zcompare/3 to _reify_ a comparison, making the different cases
distinguishable by pattern matching. For example, in this concrete
case and others like it, you can use `zcompare(Comp, 0, N)` to obtain as
`Comp` the symbolic outcome (`<`, `=`, `>`) of 0 compared to N.

## Combinatorial constraints  {#clpfd-combinatorial}

In addition to subsuming and replacing low-level arithmetic
predicates, CLP(FD) constraints are often used to solve combinatorial
problems such as planning, scheduling and allocation tasks. Among the
most frequently used *combinatorial constraints* are all_distinct/1,
global_cardinality/2 and cumulative/2. This library also provides
several other constraints like disjoint2/1 and automaton/8, which are
useful in more specialized applications.

## Domains                             {#clpfd-domains}

Each CLP(FD) variable has an associated set of admissible integers,
which we call the variable's *domain*. Initially, the domain of each
CLP(FD) variable is the set of _all_ integers. CLP(FD) constraints
like #=/2, #>/2 and #\=/2 can at most reduce, and never extend, the
domains of their arguments. The constraints in/2 and ins/2 let us
explicitly state domains of CLP(FD) variables. The process of
determining and adjusting domains of variables is called constraint
*propagation*, and it is performed automatically by this library. When
the domain of a variable contains only one element, then the variable
is automatically unified to that element.

Domains are taken into account when further constraints are stated,
and by enumeration predicates like labeling/2.

## Example: Sudoku {#clpfd-sudoku}

As another example, consider _Sudoku_: It is a popular puzzle
over integers that can be easily solved with CLP(FD) constraints.

==
sudoku(Rows) :-
        length(Rows, 9), maplist(same_length(Rows), Rows),
        append(Rows, Vs), Vs ins 1..9,
        maplist(all_distinct, Rows),
        transpose(Rows, Columns),
        maplist(all_distinct, Columns),
        Rows = [As,Bs,Cs,Ds,Es,Fs,Gs,Hs,Is],
        blocks(As, Bs, Cs),
        blocks(Ds, Es, Fs),
        blocks(Gs, Hs, Is).

blocks([], [], []).
blocks([N1,N2,N3|Ns1], [N4,N5,N6|Ns2], [N7,N8,N9|Ns3]) :-
        all_distinct([N1,N2,N3,N4,N5,N6,N7,N8,N9]),
        blocks(Ns1, Ns2, Ns3).

problem(1, [[_,_,_,_,_,_,_,_,_],
            [_,_,_,_,_,3,_,8,5],
            [_,_,1,_,2,_,_,_,_],
            [_,_,_,5,_,7,_,_,_],
            [_,_,4,_,_,_,1,_,_],
            [_,9,_,_,_,_,_,_,_],
            [5,_,_,_,_,_,_,7,3],
            [_,_,2,_,1,_,_,_,_],
            [_,_,_,_,4,_,_,_,9]]).
==

Sample query:

==
?- problem(1, Rows), sudoku(Rows), maplist(writeln, Rows).
[9,8,7,6,5,4,3,2,1]
[2,4,6,1,7,3,9,8,5]
[3,5,1,9,2,8,7,4,6]
[1,2,8,5,3,7,6,9,4]
[6,3,4,8,9,2,1,5,7]
[7,9,5,4,6,1,8,3,2]
[5,1,9,2,8,6,4,7,3]
[4,7,2,3,1,9,5,6,8]
[8,6,3,7,4,5,2,1,9]
Rows = [[9, 8, 7, 6, 5, 4, 3, 2|...], ... , [...|...]].
==

In this concrete case, the constraint solver is strong enough to find
the unique solution without any search. For the general case, see
[search](<#clpfd-search>).


## Residual goals				{#clpfd-residual-goals}

Here is an example session with a few queries and their answers:

==
?- X #> 3.
X in 4..sup.

?- X #\= 20.
X in inf..19\/21..sup.

?- 2*X #= 10.
X = 5.

?- X*X #= 144.
X in -12\/12.

?- 4*X + 2*Y #= 24, X + Y #= 9, [X,Y] ins 0..sup.
X = 3,
Y = 6.

?- X #= Y #<==> B, X in 0..3, Y in 4..5.
B = 0,
X in 0..3,
Y in 4..5.
==

The answers emitted by the toplevel are called _residual programs_,
and the goals that comprise each answer are called **residual goals**.
In each case above, and as for all pure programs, the residual program
is declaratively equivalent to the original query. From the residual
goals, it is clear that the constraint solver has deduced additional
domain restrictions in many cases.

To inspect residual goals, it is best to let the toplevel display them
for us. Wrap the call of your predicate into call_residue_vars/2 to
make sure that all constrained variables are displayed. To make the
constraints a variable is involved in available as a Prolog term for
further reasoning within your program, use copy_term/3. For example:

==
?- X #= Y + Z, X in 0..5, copy_term([X,Y,Z], [X,Y,Z], Gs).
Gs = [clpfd: (X in 0..5), clpfd: (Y+Z#=X)],
X in 0..5,
Y+Z#=X.
==

This library also provides _reflection_ predicates (like fd_dom/2,
fd_size/2 etc.) with which we can inspect a variable's current
domain. These predicates can be useful if you want to implement your
own labeling strategies.

## Core relations and search    {#clpfd-search}

Using CLP(FD) constraints to solve combinatorial tasks typically
consists of two phases:

    1. **Modeling**. In this phase, all relevant constraints are stated.
    2. **Search**. In this phase, _enumeration predicates_ are used
       to search for concrete solutions.

It is good practice to keep the modeling part, via a dedicated
predicate called the *core relation*, separate from the actual
search for solutions. This lets us observe termination and
determinism properties of the core relation in isolation from the
search, and more easily try different search strategies.

As an example of a constraint satisfaction problem, 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:

==
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.
==

Notice that we are _not_ using labeling/2 in this predicate, so that
we can first execute and observe the modeling part in isolation.
Sample query and its result (actual variables replaced for
readability):

==
?- puzzle(As+Bs=Cs).
As = [9, A2, A3, A4],
Bs = [1, 0, B3, A2],
Cs = [1, 0, A3, A2, C5],
A2 in 4..7,
all_different([9, A2, A3, A4, 1, 0, B3, C5]),
91*A2+A4+10*B3#=90*A3+C5,
A3 in 5..8,
A4 in 2..8,
B3 in 2..8,
C5 in 2..8.
==

From this answer, we see that this core relation _terminates_ and is in
fact _deterministic_. Moreover, we see from the residual goals that
the constraint solver has deduced more stringent bounds for all
variables. Such observations are only possible if modeling and search
parts are cleanly separated.

Labeling can then be used to search for solutions in a separate
predicate or goal:

==
?- puzzle(As+Bs=Cs), label(As).
As = [9, 5, 6, 7],
Bs = [1, 0, 8, 5],
Cs = [1, 0, 6, 5, 2] ;
false.
==

In this case, it suffices to label a subset of variables to find the
puzzle's unique solution, since the constraint solver is strong enough
to reduce the domains of remaining variables to singleton sets. In
general though, it is necessary to label all variables to obtain
ground solutions.

## Example: Eight queens puzzle {#clpfd-n-queens}

We illustrate the concepts of the preceding sections by means of the
so-called _eight queens puzzle_. The task is to place 8 queens on an
8x8 chessboard such that none of the queens is under attack. This
means that no two queens share the same row, column or diagonal.

To express this puzzle via CLP(FD) constraints, we must first pick a
suitable representation. Since CLP(FD) constraints reason over
_integers_, we must find a way to map the positions of queens to
integers. Several such mappings are conceivable, and it is not
immediately obvious which we should use. On top of that, different
constraints can be used to express the desired relations. For such
reasons, _modeling_ combinatorial problems via CLP(FD) constraints
often necessitates some creativity and has been described as more of
an art than a science.

In our concrete case, we observe that there must be exactly one queen
per column. The following representation therefore suggests itself: We
are looking for 8 integers, one for each column, where each integer
denotes the _row_ of the queen that is placed in the respective
column, and which are subject to certain constraints.

In fact, let us now generalize the task to the so-called _N queens
puzzle_, which is obtained by replacing 8 by _N_ everywhere it occurs
in the above description. We implement the above considerations in the
**core relation** `n_queens/2`, where the first argument is the number
of queens (which is identical to the number of rows and columns of the
generalized chessboard), and the second argument is a list of _N_
integers that represents a solution in the form described above.

==
n_queens(N, Qs) :-
        length(Qs, N),
        Qs ins 1..N,
        safe_queens(Qs).

safe_queens([]).
safe_queens([Q|Qs]) :- safe_queens(Qs, Q, 1), safe_queens(Qs).

safe_queens([], _, _).
safe_queens([Q|Qs], Q0, D0) :-
        Q0 #\= Q,
        abs(Q0 - Q) #\= D0,
        D1 #= D0 + 1,
        safe_queens(Qs, Q0, D1).
==

Note that all these predicates can be used in _all directions_: We
can use them to _find_ solutions, _test_ solutions and _complete_
partially instantiated solutions.

The original task can be readily solved with the following query:

==
?- n_queens(8, Qs), label(Qs).
Qs = [1, 5, 8, 6, 3, 7, 2, 4] .
==

Using suitable labeling strategies, we can easily find solutions with
80 queens and more:

==
?- n_queens(80, Qs), labeling([ff], Qs).
Qs = [1, 3, 5, 44, 42, 4, 50, 7, 68|...] .

?- time((n_queens(90, Qs), labeling([ff], Qs))).
% 5,904,401 inferences, 0.722 CPU in 0.737 seconds (98% CPU)
Qs = [1, 3, 5, 50, 42, 4, 49, 7, 59|...] .
==

Experimenting with different search strategies is easy because we have
separated the core relation from the actual search.



## Optimisation    {#clpfd-optimisation}

We can use labeling/2 to minimize or maximize the value of a CLP(FD)
expression, and generate solutions in increasing or decreasing order
of the value. See the labeling options `min(Expr)` and `max(Expr)`,
respectively.

Again, to easily try different labeling options in connection with
optimisation, we recommend to introduce a dedicated predicate for
posting constraints, and to use `labeling/2` in a separate goal. This
way, we can observe properties of the core relation in isolation,
and try different labeling options without recompiling our code.

If necessary, we can use `once/1` to commit to the first optimal
solution. However, it is often very valuable to see alternative
solutions that are _also_ optimal, so that we can choose among optimal
solutions by other criteria. For the sake of
[**purity**](https://www.metalevel.at/prolog/purity) and
completeness, we recommend to avoid `once/1` and other constructs that
lead to impurities in CLP(FD) programs.

Related to optimisation with CLP(FD) constraints are
[`library(simplex)`](http://eu.swi-prolog.org/man/simplex.html) and
CLP(Q) which reason about _linear_ constraints over rational numbers.

## Reification				{#clpfd-reification}

The constraints in/2, in_set/2, #=/2, #\=/2, #</2, #>/2, #=</2, and
#>=/2 can be _reified_, which means reflecting their truth values into
Boolean values represented by the integers 0 and 1. Let P and Q denote
reifiable constraints or Boolean variables, then:

    | #\ Q      | True iff Q is false                  |
    | P #\/ Q   | True iff either P or Q               |
    | P #/\ Q   | True iff both P and Q                |
    | P #\ Q    | True iff either P or Q, but not both |
    | P #<==> Q | True iff P and Q are equivalent      |
    | P #==> Q  | True iff P implies Q                 |
    | P #<== Q  | True iff Q implies P                 |

The constraints of this table are reifiable as well.

When reasoning over Boolean variables, also consider using
CLP(B) constraints as provided by
[`library(clpb)`](http://eu.swi-prolog.org/man/clpb.html).

## Enabling monotonic CLP(FD)		{#clpfd-monotonicity}

In the default execution mode, CLP(FD) constraints still exhibit some
non-relational properties. For example, _adding_ constraints can yield
new solutions:

==
?-          X #= 2, X = 1+1.
false.

?- X = 1+1, X #= 2, X = 1+1.
X = 1+1.
==

This behaviour is highly problematic from a logical point of view, and
it may render declarative debugging techniques inapplicable.

Set the Prolog flag `clpfd_monotonic` to `true` to make CLP(FD)
**monotonic**: This means that _adding_ new constraints _cannot_ yield
new solutions. When this flag is `true`, we must wrap variables that
occur in arithmetic expressions with the functor `(?)/1` or `(#)/1`. For
example:

==
?- set_prolog_flag(clpfd_monotonic, true).
true.

?- #(X) #= #(Y) + #(Z).
#(Y)+ #(Z)#= #(X).

?-          X #= 2, X = 1+1.
ERROR: Arguments are not sufficiently instantiated
==

The wrapper can be omitted for variables that are already constrained
to integers.

## Custom constraints			{#clpfd-custom-constraints}

We can define custom constraints. The mechanism to do this is not yet
finalised, and we welcome suggestions and descriptions of use cases
that are important to you.

As an example of how it can be done currently, let us define a new
custom constraint `oneground(X,Y,Z)`, where Z shall be 1 if at least
one of X and Y is instantiated:

==
:- multifile clpfd:run_propagator/2.

oneground(X, Y, Z) :-
        clpfd:make_propagator(oneground(X, Y, Z), Prop),
        clpfd:init_propagator(X, Prop),
        clpfd:init_propagator(Y, Prop),
        clpfd:trigger_once(Prop).

clpfd:run_propagator(oneground(X, Y, Z), MState) :-
        (   integer(X) -> clpfd:kill(MState), Z = 1
        ;   integer(Y) -> clpfd:kill(MState), Z = 1
        ;   true
        ).
==

First, clpfd:make_propagator/2 is used to transform a user-defined
representation of the new constraint to an internal form. With
clpfd:init_propagator/2, this internal form is then attached to X and
Y. From now on, the propagator will be invoked whenever the domains of
X or Y are changed. Then, clpfd:trigger_once/1 is used to give the
propagator its first chance for propagation even though the variables'
domains have not yet changed. Finally, clpfd:run_propagator/2 is
extended to define the actual propagator. As explained, this predicate
is automatically called by the constraint solver. The first argument
is the user-defined representation of the constraint as used in
clpfd:make_propagator/2, and the second argument is a mutable state
that can be used to prevent further invocations of the propagator when
the constraint has become entailed, by using clpfd:kill/1. An example
of using the new constraint:

==
?- oneground(X, Y, Z), Y = 5.
Y = 5,
Z = 1,
X in inf..sup.
==

## Applications   {#clpfd-applications}

CLP(FD) applications that we find particularly impressive and worth
studying include:

  * Michael Hendricks uses CLP(FD) constraints for flexible reasoning
    about _dates_ and _times_ in the
    [`julian`](http://www.swi-prolog.org/pack/list?p=julian) package.
  * Julien Cumin uses CLP(FD) constraints for integer arithmetic in
    [=Brachylog=](https://github.com/JCumin/Brachylog).

## Acknowledgments {#clpfd-acknowledgments}

This library gives you a glimpse of what [**SICStus
Prolog**](https://sicstus.sics.se/) can do. The API is intentionally
mostly compatible with that of SICStus Prolog, so that you can easily
switch to a much more feature-rich and much faster CLP(FD) system when
you need it. I thank [Mats Carlsson](https://www.sics.se/~matsc/), the
designer and main implementor of SICStus Prolog, for his elegant
example. I first encountered his system as part of the excellent
[**GUPU**](http://www.complang.tuwien.ac.at/ulrich/gupu/) teaching
environment by [Ulrich
Neumerkel](http://www.complang.tuwien.ac.at/ulrich/). Ulrich was also
the first and most determined tester of the present system, filing
hundreds of comments and suggestions for improvement. [Tom
Schrijvers](https://people.cs.kuleuven.be/~tom.schrijvers/) has
contributed several constraint libraries to SWI-Prolog, and I learned
a lot from his coding style and implementation examples. [Bart
Demoen](https://people.cs.kuleuven.be/~bart.demoen/) was a driving
force behind the implementation of attributed variables in SWI-Prolog,
and this library could not even have started without his prior work
and contributions. Thank you all!

## CLP(FD) predicate index			{#clpfd-predicate-index}

In the following, each CLP(FD) predicate is described in more detail.

We recommend the following link to refer to this manual:

http://eu.swi-prolog.org/man/clpfd.html

@author [Markus Triska](https://www.metalevel.at)
*/

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   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.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

goal_expansion(A cis B, Expansion) :-
        phrase(cis_goals(B, A), Goals),
        list_goal(Goals, Expansion).
goal_expansion(A cis_lt B, B cis_gt A).
goal_expansion(A cis_leq B, B cis_geq A).
goal_expansion(A cis_geq B, cis_leq_numeric(B, N)) :- nonvar(A), A = n(N).
goal_expansion(A cis_geq B, cis_geq_numeric(A, N)) :- nonvar(B), B = n(N).
goal_expansion(A cis_gt B, cis_lt_numeric(B, N))   :- nonvar(A), A = n(N).
goal_expansion(A cis_gt B, cis_gt_numeric(A, N))   :- nonvar(B), B = n(N).

% cis_gt only works for terms of depth 0 on both sides
cis_gt(sup, B0) :- B0 \== sup.
cis_gt(n(N), B) :- cis_lt_numeric(B, N).

cis_lt_numeric(inf, _).
cis_lt_numeric(n(B), A) :- B < A.

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

cis_geq(inf, inf).
cis_geq(sup, _).
cis_geq(n(N), B) :- cis_leq_numeric(B, N).

cis_leq_numeric(inf, _).
cis_leq_numeric(n(B), A) :- B =< A.

cis_geq_numeric(sup, _).
cis_geq_numeric(n(B), A) :- 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
        ;   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.

cis_exp(inf, n(Y), R) :-
        (   even(Y) -> R = sup
        ;   R = inf
        ).
cis_exp(sup, _, sup).
cis_exp(n(N), Y, R) :- cis_exp_(Y, N, R).

cis_exp_(n(Y), N, n(R)) :- R is N^Y.
cis_exp_(sup, _, sup).
cis_exp_(inf, _, inf).

cis_goals(V, V)          --> { var(V) }, !.
cis_goals(n(N), n(N))    --> [].
cis_goals(inf, inf)      --> [].
cis_goals(sup, sup)      --> [].
cis_goals(sign(A0), R)   --> cis_goals(A0, A), [cis_sign(A, R)].
cis_goals(abs(A0), R)    --> cis_goals(A0, A), [cis_abs(A, R)].
cis_goals(-A0, R)        --> cis_goals(A0, A), [cis_uminus(A, R)].
cis_goals(A0+B0, R)      -->
        cis_goals(A0, A),
        cis_goals(B0, B),
        [cis_plus(A, B, R)].
cis_goals(A0-B0, R)      -->
        cis_goals(A0, A),
        cis_goals(B0, B),
        [cis_minus(A, B, R)].
cis_goals(min(A0,B0), R) -->
        cis_goals(A0, A),
        cis_goals(B0, B),
        [cis_min(A, B, R)].
cis_goals(max(A0,B0), R) -->
        cis_goals(A0, A),
        cis_goals(B0, B),
        [cis_max(A, B, R)].
cis_goals(A0*B0, R)      -->
        cis_goals(A0, A),
        cis_goals(B0, B),
        [cis_times(A, B, R)].
cis_goals(div(A0,B0), R) -->
        cis_goals(A0, A),
        cis_goals(B0, B),
        [cis_div(A, B, R)].
cis_goals(A0//B0, R)     -->
        cis_goals(A0, A),
        cis_goals(B0, B),
        [cis_slash(A, B, R)].
cis_goals(A0^B0, R)      -->
        cis_goals(A0, A),
        cis_goals(B0, B),
        [cis_exp(A, B, R)].

list_goal([], true).
list_goal([G|Gs], Goal) :- foldl(list_goal_, Gs, G, Goal).

list_goal_(G, G0, (G0,G)).

cis_sign(sup, n(1)).
cis_sign(inf, n(-1)).
cis_sign(n(N), n(S)) :- S is sign(N).

cis_slash(sup, Y, Z)  :- ( Y cis_geq n(0) -> Z = sup ; Z = inf ).
cis_slash(inf, Y, Z)  :- ( Y cis_geq n(0) -> Z = inf ; Z = sup ).
cis_slash(n(X), Y, Z) :- cis_slash_(Y, X, Z).

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

cis_div(sup, Y, Z)  :- ( Y cis_geq n(0) -> Z = sup ; Z = inf ).
cis_div(inf, Y, Z)  :- ( Y cis_geq n(0) -> 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 div Y, Z = n(Z0)
        ).


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   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 cis NL + NR.

domain_direction_element(from_to(n(From), n(To)), Dir, E) :-
        (   Dir == up -> between(From, To, E)
        ;   between(From, To, E0),
            E is To - (E0 - From)
        ).
domain_direction_element(split(_, D1, D2), Dir, E) :-
        (   Dir == up ->
            (   domain_direction_element(D1, Dir, E)
            ;   domain_direction_element(D2, Dir, E)
            )
        ;   (   domain_direction_element(D2, Dir, E)
            ;   domain_direction_element(D1, Dir, E)
            )
        ).

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

domain_remove_upper(sup, L0, X, D) :-
        (   L0 =:= X -> L1 is X + 1, D = from_to(n(L1),sup)
        ;   L0 > X -> D = from_to(n(L0),sup)
        ;   L1 is X + 1, U1 is X - 1,
            D = split(X, from_to(n(L0),n(U1)), from_to(n(L1),sup))
        ).
domain_remove_upper(n(U0), L0, X, D) :-
        (   L0 =:= U0, X =:= L0 -> D = empty
        ;   L0 =:= X -> L1 is X + 1, D = from_to(n(L1), n(U0))
        ;   U0 =:= X -> U1 is X - 1, D = from_to(n(L0), n(U1))
        ;   between(L0, U0, X) ->
            U1 is X - 1, L1 is X + 1,
            D = split(X, from_to(n(L0), n(U1)), from_to(n(L1), n(U0)))
        ;   D = from_to(n(L0),n(U0))
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Remove all elements greater than / less than a 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 cis 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 cis 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 cis To + n(1),
                    D = from_to(From1, To0)
                )
            ;   To1 cis From - n(1),
                (   To cis_lt To0 ->
                    From = n(S),
                    From2 cis 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)
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Complement of a domain
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_complement(D, C) :-
        default_domain(Default),
        domain_subtract(Default, D, C).

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

domain_intervals(D, Is) :- phrase(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 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 -> Dom = Right1
        ;   Right1 == empty -> Dom = Left1
        ;   Dom = split(S, Left1, Right1)
        ).

narrow(empty, _, _, empty).
narrow(from_to(L0,U0), From0, To0, Dom) :-
        From1 cis max(From0,L0), To1 cis 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 the domain by an offset.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_shift(empty, _, empty).
domain_shift(from_to(From0,To0), O, from_to(From,To)) :-
        From cis From0 + n(O), To cis 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),
            domain_expand_(D1, M1, D)
        ;   M =:= 1 -> D = D0
        ;   domain_expand_(D0, M, D)
        ).

domain_expand_(empty, _, empty).
domain_expand_(from_to(From0, To0), M, from_to(From,To)) :-
        From cis From0*n(M),
        To cis To0*n(M).
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).

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

domain_expand_more(D0, M, Op, 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, Op, D).
        domain_expand_more_(D0, M, Op, D).
        %format("yield: ~w\n", [D]).

domain_expand_more_(empty, _, _, empty).
domain_expand_more_(from_to(From0, To0), M, Op, D) :-
        M1 is abs(M),
        (   Op = // ->
            (   From0 cis_leq n(0) -> From1 cis (From0-n(1))*n(M1) + n(1)
            ;   From1 cis From0*n(M1)
            ),
            (   To0 cis_lt n(0) -> To1 cis To0*n(M1)
            ;   To1 cis (To0+n(1))*n(M1) - n(1)
            )
        ;   Op = div ->
            From1 cis From0*n(M1),
            To1 cis (To0+n(1))*n(M1)-sign(n(M))
        ),
        (   M < 0 -> domain_negate(from_to(From1,To1), D)
        ;   D = from_to(From1,To1)
        ).
domain_expand_more_(split(S0, Left0, Right0), M, Op, split(S, Left, Right)) :-
        S is M*S0,
        domain_expand_more_(Left0, M, Op, Left),
        domain_expand_more_(Right0, M, Op, Right).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   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)) :-
        (   From0 cis_geq n(0) ->
            From cis (From0 + n(M) - n(1)) // n(M)
        ;   From cis From0 // n(M)
        ),
        (   To0 cis_geq n(0) ->
            To cis To0 // n(M)
        ;   To cis (To0 - n(M) + n(1)) // n(M)
        ).
domain_contract_(split(_,Left0,Right0), M, D) :-
        %  Scaled down domains do not necessarily retain any holes of
        %  the original domain.
        domain_contract_(Left0, M, Left),
        domain_contract_(Right0, M, Right),
        domains_union(Left, Right, D).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Similar to domain_contract, tailored for division, i.e.,
   {21,23} contracted by 4 is 5. It contracts "less".
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

domain_contract_less(D0, M, Op, D) :-
        (   M < 0 -> domain_negate(D0, D1), M1 is abs(M)
        ;   D1 = D0, M1 = M
        ),
        domain_contract_less_(D1, M1, Op, D).

domain_contract_less_(empty, _, _, empty).
domain_contract_less_(from_to(From0, To0), M, Op, from_to(From,To)) :-
        (   Op = div -> From cis From0 div n(M),
            To cis To0 div n(M)
        ;   Op = // -> From cis From0 // n(M),
            To cis To0 // n(M)
        ).

domain_contract_less_(split(_,Left0,Right0), M, Op, D) :-
        %  Scaled down domains do not necessarily retain any holes of
        %  the original domain.
        domain_contract_less_(Left0, M, Op, Left),
        domain_contract_less_(Right0, M, Op, Right),
        domains_union(Left, Right, 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(From, To)) :-
        From cis -To0, To cis -From0.
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).

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


%% ?Var in +Domain
%
%  Var is an element of Domain. Domain is one of:
%
%         * Integer
%           Singleton set consisting only of _Integer_.
%         * Lower..Upper
%           All integers _I_ such that _Lower_ =< _I_ =< _Upper_.
%           _Lower_ must be an integer or the atom *inf*, which
%           denotes negative infinity. _Upper_ must be an integer or
%           the atom *sup*, which denotes positive infinity.
%         * Domain1 \/ Domain2
%           The union of Domain1 and Domain2.

Var in Dom :- clpfd_in(Var, Dom).

clpfd_in(V, D) :-
        fd_variable(V),
        drep_to_domain(D, Dom),
        domain(V, Dom).

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

%% +Vars ins +Domain
%
%  The variables in the list Vars are elements of Domain. See in/2 for
%  the syntax of Domain.

Vs ins D :-
        fd_must_be_list(Vs),
        maplist(fd_variable, Vs),
        drep_to_domain(D, Dom),
        domains(Vs, Dom).

fd_must_be_list(Ls) :-
        (   fd_var(Ls) -> type_error(list, Ls)
        ;   must_be(list, Ls)
        ).

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

indomain(Var) :- label([Var]).

order_dom_next(up, Dom, Next)   :- domain_infimum(Dom, n(Next)).
order_dom_next(down, Dom, Next) :- domain_supremum(Dom, n(Next)).
order_dom_next(random_value(_), Dom, Next) :-
        phrase(domain_to_intervals(Dom), Is),
        length(Is, L),
        R is random(L),
        nth0(R, Is, From-To),
        random_between(From, To, Next).

domain_to_intervals(from_to(n(From),n(To))) --> [From-To].
domain_to_intervals(split(_, Left, Right)) -->
        domain_to_intervals(Left),
        domain_to_intervals(Right).

%% label(+Vars)
%
% Equivalent to labeling([], Vars). See labeling/2.

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

%% labeling(+Options, +Vars)
%
% Assign a value to each variable in 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. Several categories of options exist:
%
% The variable selection strategy lets you specify which variable of
% Vars is 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.
%
%   * ffc
%   Of the variables with smallest domains, the leftmost one
%   participating in most constraints is labeled next.
%
%   * 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 value order is one of:
%
%   * up
%   Try the elements of the chosen variable's domain in ascending order.
%   This is the default.
%
%   * down
%   Try the domain elements in descending order.
%
% The branching strategy is one of:
%
%   * step
%   For each variable X, a choice is made between X = V and X #\= V,
%   where V is determined by the value ordering options. This is the
%   default.
%
%   * enum
%   For each variable X, a choice is made between X = V_1, X = V_2
%   etc., for all values V_i of the domain of X. The order is
%   determined by the value ordering options.
%
%   * bisect
%   For each variable X, a choice is made between X #=< M and X #> M,
%   where M is the midpoint of the domain of X.
%
% At most one option of each category can be specified, and an option
% must not occur repeatedly.
%
% The order of solutions can be influenced with:
%
%   * min(Expr)
%   * max(Expr)
%
% This generates solutions in ascending/descending order with respect
% to the evaluation of the arithmetic expression Expr. Labeling Vars
% must make Expr ground. If several such options are specified, they
% are interpreted from left to right, e.g.:
%
% ==
% ?- [X,Y] ins 10..20, labeling([max(X),min(Y)],[X,Y]).
% ==
%
% This generates solutions in descending order of X, and for each
% binding of X, solutions are generated in ascending order of Y. To
% obtain the incomplete behaviour that other systems exhibit with
% "maximize(Expr)" and "minimize(Expr)", use once/1, e.g.:
%
% ==
% once(labeling([max(Expr)], Vars))
% ==
%
% Labeling is always complete, always terminates, and yields no
% redundant solutions. See [core relations and
% search](<#clpfd-search>) for usage advice.

labeling(Options, Vars) :-
        must_be(list, Options),
        fd_must_be_list(Vars),
        maplist(must_be_finite_fdvar, Vars),
        label(Options, Options, default(leftmost), default(up), default(step), [], upto_ground, Vars).

finite_domain(Dom) :-
        domain_infimum(Dom, n(_)),
        domain_supremum(Dom, n(_)).

must_be_finite_fdvar(Var) :-
        (   fd_get(Var, Dom, _) ->
            (   finite_domain(Dom) -> true
            ;   instantiation_error(Var)
            )
        ;   integer(Var) -> true
        ;   must_be(integer, Var)
        ).


label([O|Os], Options, Selection, Order, Choice, Optim, Consistency, Vars) :-
        (   var(O)-> instantiation_error(O)
        ;   override(selection, Selection, O, Options, S1) ->
            label(Os, Options, S1, Order, Choice, Optim, Consistency, Vars)
        ;   override(order, Order, O, Options, O1) ->
            label(Os, Options, Selection, O1, Choice, Optim, Consistency, Vars)
        ;   override(choice, Choice, O, Options, C1) ->
            label(Os, Options, Selection, Order, C1, Optim, Consistency, Vars)
        ;   optimisation(O) ->
            label(Os, Options, Selection, Order, Choice, [O|Optim], Consistency, Vars)
        ;   consistency(O, O1) ->
            label(Os, Options, Selection, Order, Choice, Optim, O1, Vars)
        ;   domain_error(labeling_option, O)
        ).
label([], _, Selection, Order, Choice, Optim0, Consistency, Vars) :-
        maplist(arg(1), [Selection,Order,Choice], [S,O,C]),
        (   Optim0 == [] ->
            label(Vars, S, O, C, Consistency)
        ;   reverse(Optim0, Optim),
            exprs_singlevars(Optim, SVs),
            optimise(Vars, [S,O,C], SVs)
        ).

% Introduce new variables for each min/max expression to avoid
% reparsing expressions during optimisation.

exprs_singlevars([], []).
exprs_singlevars([E|Es], [SV|SVs]) :-
        E =.. [F,Expr],
        ?(Single) #= Expr,
        SV =.. [F,Single],
        exprs_singlevars(Es, SVs).

all_dead(fd_props(Bs,Gs,Os)) :-
        all_dead_(Bs),
        all_dead_(Gs),
        all_dead_(Os).

all_dead_([]).
all_dead_([propagator(_, S)|Ps]) :- S == dead, all_dead_(Ps).

label([], _, _, _, Consistency) :- !,
        (   Consistency = upto_in(I0,I) -> I0 = I
        ;   true
        ).
label(Vars, Selection, Order, Choice, Consistency) :-
        (   Vars = [V|Vs], nonvar(V) -> label(Vs, Selection, Order, Choice, Consistency)
        ;   select_var(Selection, Vars, Var, RVars),
            (   var(Var) ->
                (   Consistency = upto_in(I0,I), fd_get(Var, _, Ps), all_dead(Ps) ->
                    fd_size(Var, Size),
                    I1 is I0*Size,
                    label(RVars, Selection, Order, Choice, upto_in(I1,I))
                ;   Consistency = upto_in, fd_get(Var, _, Ps), all_dead(Ps) ->
                    label(RVars, Selection, Order, Choice, Consistency)
                ;   choice_order_variable(Choice, Order, Var, RVars, Vars, Selection, Consistency)
                )
            ;   label(RVars, Selection, Order, Choice, Consistency)
            )
        ).

choice_order_variable(step, Order, Var, Vars, Vars0, Selection, Consistency) :-
        fd_get(Var, Dom, _),
        order_dom_next(Order, Dom, Next),
        (   Var = Next,
            label(Vars, Selection, Order, step, Consistency)
        ;   neq_num(Var, Next),
            do_queue,
            label(Vars0, Selection, Order, step, Consistency)
        ).
choice_order_variable(enum, Order, Var, Vars, _, Selection, Consistency) :-
        fd_get(Var, Dom0, _),
        domain_direction_element(Dom0, Order, Var),
        label(Vars, Selection, Order, enum, Consistency).
choice_order_variable(bisect, Order, Var, _, Vars0, Selection, Consistency) :-
        fd_get(Var, Dom, _),
        domain_infimum(Dom, n(I)),
        domain_supremum(Dom, n(S)),
        Mid0 is (I + S) // 2,
        (   Mid0 =:= S -> Mid is Mid0 - 1 ; Mid = Mid0 ),
        (   Order == up -> ( Var #=< Mid ; Var #> Mid )
        ;   Order == down -> ( Var #> Mid ; Var #=< Mid )
        ;   domain_error(bisect_up_or_down, Order)
        ),
        label(Vars0, Selection, Order, bisect, Consistency).

override(What, Prev, Value, Options, Result) :-
        call(What, Value),
        override_(Prev, Value, Options, Result).

override_(default(_), Value, _, user(Value)).
override_(user(Prev), Value, Options, _) :-
        (   Value == Prev ->
            domain_error(nonrepeating_labeling_options, Options)
        ;   domain_error(consistent_labeling_options, Options)
        ).

selection(ff).
selection(ffc).
selection(min).
selection(max).
selection(leftmost).
selection(random_variable(Seed)) :-
        must_be(integer, Seed),
        set_random(seed(Seed)).

choice(step).
choice(enum).
choice(bisect).

order(up).
order(down).
% TODO: random_variable and random_value currently both set the seed,
% so exchanging the options can yield different results.
order(random_value(Seed)) :-
        must_be(integer, Seed),
        set_random(seed(Seed)).

consistency(upto_in(I), upto_in(1, I)).
consistency(upto_in, upto_in).
consistency(upto_ground, upto_ground).

optimisation(min(_)).
optimisation(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) :-
        fd_size_(V, n(S)),
        find_ff(Vs, V, S, Var),
        delete_eq([V|Vs], Var, RVars).
select_var(ffc, [V|Vs], Var, RVars) :-
        find_ffc(Vs, V, Var),
        delete_eq([V|Vs], Var, RVars).
select_var(random_variable(_), Vars0, Var, Vars) :-
        length(Vars0, L),
        I is random(L),
        nth0(I, 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, S0, FF) :-
        (   nonvar(V) -> find_ff(Vs, CM, S0, FF)
        ;   (   fd_size_(V, n(S1)), S1 < S0 ->
                find_ff(Vs, V, S1, FF)
            ;   find_ff(Vs, CM, S0, FF)
            )
        ).

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


ffc_lt(X, Y) :-
        (   fd_get(X, XD, XPs) ->
            domain_num_elements(XD, n(NXD))
        ;   NXD = 1, XPs = []
        ),
        (   fd_get(Y, YD, YPs) ->
            domain_num_elements(YD, n(NYD))
        ;   NYD = 1, YPs = []
        ),
        (   NXD < NYD -> true
        ;   NXD =:= NYD,
            props_number(XPs, NXPs),
            props_number(YPs, NYPs),
            NXPs > NYPs
        ).

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

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

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

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

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   contracting/1 -- subject to change

   This can remove additional domain elements from the boundaries.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

contracting(Vs) :-
        must_be(list, Vs),
        maplist(must_be_finite_fdvar, Vs),
        contracting(Vs, false, Vs).

contracting([], Repeat, Vars) :-
        (   Repeat -> contracting(Vars, false, Vars)
        ;   true
        ).
contracting([V|Vs], Repeat, Vars) :-
        fd_inf(V, Min),
        (   \+ \+ (V = Min) ->
            fd_sup(V, Max),
            (   \+ \+ (V = Max) ->
                contracting(Vs, Repeat, Vars)
            ;   V #\= Max,
                contracting(Vs, true, Vars)
            )
        ;   V #\= Min,
            contracting(Vs, true, Vars)
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   fds_sespsize(Vs, S).

   S is an upper bound on the search space size with respect to finite
   domain variables Vs.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

fds_sespsize(Vs, S) :-
        must_be(list, Vs),
        maplist(fd_variable, Vs),
        fds_sespsize(Vs, n(1), S1),
        bound_portray(S1, S).

fd_size_(V, S) :-
        (   fd_get(V, D, _) ->
            domain_num_elements(D, S)
        ;   S = n(1)
        ).

fds_sespsize([], S, S).
fds_sespsize([V|Vs], S0, S) :-
        fd_size_(V, S1),
        S2 cis S0*S1,
        fds_sespsize(Vs, S2, S).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Optimisation uses destructive assignment to save the computed
   extremum over backtracking. Failure is used to get rid of copies of
   attributed variables that are created in intermediate steps. At
   least that's the intention - it currently doesn't work in SWI:

   %?- X in 0..3, call_residue_vars(labeling([min(X)], [X]), Vs).
   %@ X = 0,
   %@ Vs = [_G6174, _G6177],
   %@ _G6174 in 0..3

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

optimise(Vars, Options, Whats) :-
        Whats = [What|WhatsRest],
        Extremum = extremum(none),
        (   catch(store_extremum(Vars, Options, What, Extremum),
                  time_limit_exceeded,
                  false)
        ;   Extremum = extremum(n(Val)),
            arg(1, What, Expr),
            append(WhatsRest, Options, Options1),
            (   Expr #= Val,
                labeling(Options1, Vars)
            ;   Expr #\= Val,
                optimise(Vars, Options, Whats)
            )
        ).

store_extremum(Vars, Options, What, Extremum) :-
        catch((labeling(Options, Vars), throw(w(What))), w(What1), true),
        functor(What, Direction, _),
        maplist(arg(1), [What,What1], [Expr,Expr1]),
        optimise(Direction, Options, Vars, Expr1, Expr, Extremum).

optimise(Direction, Options, Vars, Expr0, Expr, Extremum) :-
        must_be(ground, Expr0),
        nb_setarg(1, Extremum, n(Expr0)),
        catch((tighten(Direction, Expr, Expr0),
               labeling(Options, Vars),
               throw(v(Expr))), v(Expr1), true),
        optimise(Direction, Options, Vars, Expr1, Expr, Extremum).

tighten(min, E, V) :- E #< V.
tighten(max, E, V) :- E #> V.

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

%% all_different(+Vars)
%
% Like all_distinct/1, but with weaker propagation. Consider using
% all_distinct/1 instead, since all_distinct/1 is typically acceptably
% efficient and propagates much more strongly.

all_different(Ls) :-
        fd_must_be_list(Ls),
        maplist(fd_variable, Ls),
        Orig = original_goal(_, all_different(Ls)),
        all_different(Ls, [], Orig),
        do_queue.

all_different([], _, _).
all_different([X|Right], Left, Orig) :-
        (   var(X) ->
            make_propagator(pdifferent(Left,Right,X,Orig), Prop),
            init_propagator(X, Prop),
            trigger_prop(Prop)
        ;   exclude_fire(Left, Right, X)
        ),
        all_different(Right, [X|Left], Orig).

%% all_distinct(+Vars).
%
%  True iff Vars are pairwise distinct. For example, all_distinct/1
%  can detect that not all variables can assume distinct values given
%  the following domains:
%
%  ==
%  ?- maplist(in, Vs,
%             [1\/3..4, 1..2\/4, 1..2\/4, 1..3, 1..3, 1..6]),
%     all_distinct(Vs).
%  false.
%  ==

all_distinct(Ls) :-
        fd_must_be_list(Ls),
        maplist(fd_variable, Ls),
        make_propagator(pdistinct(Ls), Prop),
        distinct_attach(Ls, Prop, []),
        trigger_once(Prop).

%% sum(+Vars, +Rel, ?Expr)
%
% The sum of elements of the list Vars is in relation Rel to Expr.
% Rel is one of #=, #\=, #<, #>, #=< or #>=. For example:
%
% ==
% ?- [A,B,C] ins 0..sup, sum([A,B,C], #=, 100).
% A in 0..100,
% A+B+C#=100,
% B in 0..100,
% C in 0..100.
% ==

sum(Vs, Op, Value) :-
        must_be(list, Vs),
        same_length(Vs, Ones),
        maplist(=(1), Ones),
        scalar_product(Ones, Vs, Op, Value).

%% scalar_product(+Cs, +Vs, +Rel, ?Expr)
%
% True iff the scalar product of Cs and Vs is in relation Rel to Expr.
% Cs is a list of integers, Vs is a list of variables and integers.
% Rel is #=, #\=, #<, #>, #=< or #>=.

scalar_product(Cs, Vs, Op, Value) :-
        must_be(list(integer), Cs),
        must_be(list, Vs),
        maplist(fd_variable, Vs),
        (   Op = (#=), single_value(Value, Right), ground(Vs) ->
            foldl(coeff_int_linsum, Cs, Vs, 0, Right)
        ;   must_be(callable, Op),
            (   memberchk(Op, [#=,#\=,#<,#>,#=<,#>=]) -> true
            ;   domain_error(scalar_product_relation, Op)
            ),
            must_be(acyclic, Value),
            foldl(coeff_var_plusterm, Cs, Vs, 0, Left),
            (   left_right_linsum_const(Left, Value, Cs1, Vs1, Const) ->
                scalar_product_(Op, Cs1, Vs1, Const)
            ;   sum(Cs, Vs, 0, Op, Value)
            )
        ).

single_value(V, V)    :- var(V), !, non_monotonic(V).
single_value(V, V)    :- integer(V).
single_value(?(V), V) :- fd_variable(V).

coeff_var_plusterm(C, V, T0, T0+(C* ?(V))).

coeff_int_linsum(C, I, S0, S) :- S is S0 + C*I.

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

multiples([], [], _).
multiples([C|Cs], [V|Vs], Left) :-
        (   (   Cs = [N|_] ; Left = [N|_] ) ->
            (   N =\= 1, gcd(C,N) =:= 1 ->
                gcd(Cs, N, GCD0),
                gcd(Left, GCD0, GCD),
                (   GCD > 1 -> ?(V) #= GCD * ?(_)
                ;   true
                )
            ;   true
            )
        ;   true
        ),
        multiples(Cs, Vs, [C|Left]).

abs(N, A) :- A is abs(N).

divide(D, N, R) :- R is N // D.

scalar_product_(#=, Cs0, Vs, S0) :-
        (   Cs0 = [C|Rest] ->
            gcd(Rest, C, GCD),
            S0 mod GCD =:= 0,
            maplist(divide(GCD), [S0|Cs0], [S|Cs])
        ;   S0 =:= 0, S = S0, Cs = Cs0
        ),
        (   S0 =:= 0 ->
            maplist(abs, Cs, As),
            multiples(As, Vs, [])
        ;   true
        ),
        propagator_init_trigger(Vs, scalar_product_eq(Cs, Vs, S)).
scalar_product_(#\=, Cs, Vs, C) :-
        propagator_init_trigger(Vs, scalar_product_neq(Cs, Vs, C)).
scalar_product_(#=<, Cs, Vs, C) :-
        propagator_init_trigger(Vs, scalar_product_leq(Cs, Vs, C)).
scalar_product_(#<, Cs, Vs, C) :-
        C1 is C - 1,
        scalar_product_(#=<, Cs, Vs, C1).
scalar_product_(#>, Cs, Vs, C) :-
        C1 is C + 1,
        scalar_product_(#>=, Cs, Vs, C1).
scalar_product_(#>=, Cs, Vs, C) :-
        maplist(negative, Cs, Cs1),
        C1 is -C,
        scalar_product_(#=<, Cs1, Vs, C1).

negative(X0, X) :- X is -X0.

coeffs_variables_const([], [], [], [], I, I).
coeffs_variables_const([C|Cs], [V|Vs], Cs1, Vs1, I0, I) :-
        (   var(V) ->
            Cs1 = [C|CRest], Vs1 = [V|VRest], I1 = I0
        ;   I1 is I0 + C*V,
            Cs1 = CRest, Vs1 = VRest
        ),
        coeffs_variables_const(Cs, Vs, CRest, VRest, I1, I).

sum_finite_domains([], [], [], [], Inf, Sup, Inf, Sup).
sum_finite_domains([C|Cs], [V|Vs], Infs, Sups, Inf0, Sup0, Inf, Sup) :-
        fd_get(V, _, Inf1, Sup1, _),
        (   Inf1 = n(NInf) ->
            (   C < 0 ->
                Sup2 is Sup0 + C*NInf
            ;   Inf2 is Inf0 + C*NInf
            ),
            Sups = Sups1,
            Infs = Infs1
        ;   (   C < 0 ->
                Sup2 = Sup0,
                Sups = [C*V|Sups1],
                Infs = Infs1
            ;   Inf2 = Inf0,
                Infs = [C*V|Infs1],
                Sups = Sups1
            )
        ),
        (   Sup1 = n(NSup) ->
            (   C < 0 ->
                Inf2 is Inf0 + C*NSup
            ;   Sup2 is Sup0 + C*NSup
            ),
            Sups1 = Sups2,
            Infs1 = Infs2
        ;   (   C < 0 ->
                Inf2 = Inf0,
                Infs1 = [C*V|Infs2],
                Sups1 = Sups2
            ;   Sup2 = Sup0,
                Sups1 = [C*V|Sups2],
                Infs1 = Infs2
            )
        ),
        sum_finite_domains(Cs, Vs, Infs2, Sups2, Inf2, Sup2, Inf, Sup).

remove_dist_upper_lower([], _, _, _).
remove_dist_upper_lower([C|Cs], [V|Vs], D1, D2) :-
        (   fd_get(V, VD, VPs) ->
            (   C < 0 ->
                domain_supremum(VD, n(Sup)),
                L is Sup + D1//C,
                domain_remove_smaller_than(VD, L, VD1),
                domain_infimum(VD1, n(Inf)),
                G is Inf - D2//C,
                domain_remove_greater_than(VD1, G, VD2)
            ;   domain_infimum(VD, n(Inf)),
                G is Inf + D1//C,
                domain_remove_greater_than(VD, G, VD1),
                domain_supremum(VD1, n(Sup)),
                L is Sup - D2//C,
                domain_remove_smaller_than(VD1, L, VD2)
            ),
            fd_put(V, VD2, VPs)
        ;   true
        ),
        remove_dist_upper_lower(Cs, Vs, D1, D2).


remove_dist_upper_leq([], _, _).
remove_dist_upper_leq([C|Cs], [V|Vs], D1) :-
        (   fd_get(V, VD, VPs) ->
            (   C < 0 ->
                domain_supremum(VD, n(Sup)),
                L is Sup + D1//C,
                domain_remove_smaller_than(VD, L, VD1)
            ;   domain_infimum(VD, n(Inf)),
                G is Inf + D1//C,
                domain_remove_greater_than(VD, G, VD1)
            ),
            fd_put(V, VD1, VPs)
        ;   true
        ),
        remove_dist_upper_leq(Cs, Vs, D1).


remove_dist_upper([], _).
remove_dist_upper([C*V|CVs], D) :-
        (   fd_get(V, VD, VPs) ->
            (   C < 0 ->
                (   domain_supremum(VD, n(Sup)) ->
                    L is Sup + D//C,
                    domain_remove_smaller_than(VD, L, VD1)
                ;   VD1 = VD
                )
            ;   (   domain_infimum(VD, n(Inf)) ->
                    G is Inf + D//C,
                    domain_remove_greater_than(VD, G, VD1)
                ;   VD1 = VD
                )
            ),
            fd_put(V, VD1, VPs)
        ;   true
        ),
        remove_dist_upper(CVs, D).

remove_dist_lower([], _).
remove_dist_lower([C*V|CVs], D) :-
        (   fd_get(V, VD, VPs) ->
            (   C < 0 ->
                (   domain_infimum(VD, n(Inf)) ->
                    G is Inf - D//C,
                    domain_remove_greater_than(VD, G, VD1)
                ;   VD1 = VD
                )
            ;   (   domain_supremum(VD, n(Sup)) ->
                    L is Sup - D//C,
                    domain_remove_smaller_than(VD, L, VD1)
                ;   VD1 = VD
                )
            ),
            fd_put(V, VD1, VPs)
        ;   true
        ),
        remove_dist_lower(CVs, D).

remove_upper([], _).
remove_upper([C*X|CXs], Max) :-
        (   fd_get(X, XD, XPs) ->
            D is Max//C,
            (   C < 0 ->
                domain_remove_smaller_than(XD, D, XD1)
            ;   domain_remove_greater_than(XD, D, XD1)
            ),
            fd_put(X, XD1, XPs)
        ;   true
        ),
        remove_upper(CXs, Max).

remove_lower([], _).
remove_lower([C*X|CXs], Min) :-
        (   fd_get(X, XD, XPs) ->
            D is -Min//C,
            (   C < 0 ->
                domain_remove_greater_than(XD, D, XD1)
            ;   domain_remove_smaller_than(XD, D, XD1)
            ),
            fd_put(X, XD1, XPs)
        ;   true
        ),
        remove_lower(CXs, Min).

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

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Constraint propagation proceeds as follows: Each CLP(FD) variable
   has an attribute that stores its associated domain and constraints.
   Constraints are triggered when the event they are registered for
   occurs (for example: variable is instantiated, bounds change etc.).
   do_queue/0 works off all triggered constraints, possibly triggering
   new ones, until fixpoint.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

% FIFO queue

make_queue :- nb_setval('$clpfd_queue', fast_slow([], [])).

push_queue(E, Which) :-
        nb_getval('$clpfd_queue', Qs),
        arg(Which, Qs, Q),
        (   Q == [] ->
            setarg(Which, Qs, [E|T]-T)
        ;   Q = H-[E|T],
            setarg(Which, Qs, H-T)
        ).

pop_queue(E) :-
        nb_getval('$clpfd_queue', Qs),
        (   pop_queue(E, Qs, 1) ->  true
        ;   pop_queue(E, Qs, 2)
        ).

pop_queue(E, Qs, Which) :-
        arg(Which, Qs, [E|NH]-T),
        (   var(NH) ->
            setarg(Which, Qs, [])
        ;   setarg(Which, Qs, NH-T)
        ).

fetch_propagator(Prop) :-
        pop_queue(P),
        (   propagator_state(P, S), S == dead -> fetch_propagator(Prop)
        ;   Prop = P
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   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.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

constrain_to_integer(Var) :-
        (   integer(Var) -> true
        ;   fd_get(Var, D, Ps),
            fd_put(Var, D, Ps)
        ).

power_var_num(P, X, N) :-
        (   var(P) -> X = P, N = 1
        ;   P = Left*Right,
            power_var_num(Left, XL, L),
            power_var_num(Right, XR, R),
            XL == XR,
            X = XL,
            N is L + R
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Given expression E, we obtain the finite domain variable R by
   interpreting a simple committed-choice language that is a list of
   conditions and bodies. In conditions, g(Goal) means literally Goal,
   and m(Match) means that E can be decomposed as stated. The
   variables are to be understood as the result of parsing the
   subexpressions recursively. In the body, g(Goal) means again Goal,
   and p(Propagator) means to attach and trigger once a propagator.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

:- op(800, xfx, =>).

parse_clpfd(E, R,
            [g(cyclic_term(E)) => [g(domain_error(clpfd_expression, E))],
             g(var(E))         => [g(non_monotonic(E)),
                                   g(constrain_to_integer(E)), g(E = R)],
             g(integer(E))     => [g(R = E)],
             ?(E)              => [g(must_be_fd_integer(E)), g(R = E)],
             #(E)              => [g(must_be_fd_integer(E)), g(R = E)],
             m(A+B)            => [p(pplus(A, B, R))],
             % power_var_num/3 must occur before */2 to be useful
             g(power_var_num(E, V, N)) => [p(pexp(V, N, R))],
             m(A*B)            => [p(ptimes(A, B, R))],
             m(A-B)            => [p(pplus(R,B,A))],
             m(-A)             => [p(ptimes(-1,A,R))],
             m(max(A,B))       => [g(A #=< ?(R)), g(B #=< R), p(pmax(A, B, R))],
             m(min(A,B))       => [g(A #>= ?(R)), g(B #>= R), p(pmin(A, B, R))],
             m(A mod B)        => [g(B #\= 0), p(pmod(A, B, R))],
             m(A rem B)        => [g(B #\= 0), p(prem(A, B, R))],
             m(abs(A))         => [g(?(R) #>= 0), p(pabs(A, R))],
%             m(A/B)            => [g(B #\= 0), p(ptzdiv(A, B, R))],
             m(A//B)           => [g(B #\= 0), p(ptzdiv(A, B, R))],
             m(A div B)        => [g(B #\= 0), p(pdiv(A, B, R))],
             m(A rdiv B)       => [g(B #\= 0), p(prdiv(A, B, R))],
             m(A^B)            => [p(pexp(A, B, R))],
             % bitwise operations
             m(\A)             => [p(pfunction(\, A, R))],
             m(msb(A))         => [p(pfunction(msb, A, R))],
             m(lsb(A))         => [p(pfunction(lsb, A, R))],
             m(popcount(A))    => [p(pfunction(popcount, A, R))],
             m(A<<B)           => [p(pshift(A, B, R, 1))],
             m(A>>B)           => [p(pshift(A, B, R, -1))],
             m(A/\B)           => [p(pfunction(/\, A, B, R))],
             m(A\/B)           => [p(pfunction(\/, A, B, R))],
             m(A xor B)        => [p(pfunction(xor, A, B, R))],
             g(true)           => [g(domain_error(clpfd_expression, E))]
            ]).

non_monotonic(X) :-
        (   \+ fd_var(X), current_prolog_flag(clpfd_monotonic, true) ->
            instantiation_error(X)
        ;   true
        ).

% Here, we compile the committed choice language to a single
% predicate, parse_clpfd/2.

make_parse_clpfd(Clauses) :-
        parse_clpfd_clauses(Clauses0),
        maplist(goals_goal, Clauses0, Clauses).

goals_goal((Head :- Goals), (Head :- Body)) :-
        list_goal(Goals, Body).

parse_clpfd_clauses(Clauses) :-
        parse_clpfd(E, R, Matchers),
        maplist(parse_matcher(E, R), Matchers, Clauses).

parse_matcher(E, R, Matcher, Clause) :-
        Matcher = (Condition0 => Goals0),
        phrase((parse_condition(Condition0, E, Head),
                parse_goals(Goals0)), Goals),
        Clause = (parse_clpfd(Head, R) :- Goals).

parse_condition(g(Goal), E, E)       --> [Goal, !].
parse_condition(?(E), _, ?(E))       --> [!].
parse_condition(#(E), _, #(E))       --> [!].
parse_condition(m(Match), _, Match0) -->
        [!],
        { copy_term(Match, Match0),
          term_variables(Match0, Vs0),
          term_variables(Match, Vs)
        },
        parse_match_variables(Vs0, Vs).

parse_match_variables([], []) --> [].
parse_match_variables([V0|Vs0], [V|Vs]) -->
        [parse_clpfd(V0, V)],
        parse_match_variables(Vs0, Vs).

parse_goals([]) --> [].
parse_goals([G|Gs]) --> parse_goal(G), parse_goals(Gs).

parse_goal(g(Goal)) --> [Goal].
parse_goal(p(Prop)) -->
        [make_propagator(Prop, P)],
        { term_variables(Prop, Vs) },
        parse_init(Vs, P),
        [trigger_once(P)].

parse_init([], _)     --> [].
parse_init([V|Vs], P) --> [init_propagator(V, P)], parse_init(Vs, P).

%?- set_prolog_flag(answer_write_options, [portray(true)]),
%   clpfd:parse_clpfd_clauses(Clauses), maplist(portray_clause, Clauses).


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

trigger_once(Prop) :- trigger_prop(Prop), do_queue.

neq(A, B) :- propagator_init_trigger(pneq(A, B)).

propagator_init_trigger(P) -->
        { term_variables(P, Vs) },
        propagator_init_trigger(Vs, P).

propagator_init_trigger(Vs, P) -->
        [p(Prop)],
        { make_propagator(P, Prop),
          maplist(prop_init(Prop), Vs),
          trigger_once(Prop) }.

propagator_init_trigger(P) :-
        phrase(propagator_init_trigger(P), _).

propagator_init_trigger(Vs, P) :-
        phrase(propagator_init_trigger(Vs, P), _).

prop_init(Prop, V) :- init_propagator(V, Prop).

geq(A, B) :-
        (   fd_get(A, AD, APs) ->
            domain_infimum(AD, AI),
            (   fd_get(B, BD, _) ->
                domain_supremum(BD, BS),
                (   AI cis_geq BS -> true
                ;   propagator_init_trigger(pgeq(A,B))
                )
            ;   (   AI cis_geq n(B) -> true
                ;   domain_remove_smaller_than(AD, B, AD1),
                    fd_put(A, AD1, APs),
                    do_queue
                )
            )
        ;   fd_get(B, BD, BPs) ->
            domain_remove_greater_than(BD, A, BD1),
            fd_put(B, BD1, BPs),
            do_queue
        ;   A >= B
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Naive parsing of inequalities and disequalities can result in a lot
   of unnecessary work if expressions of non-trivial depth are
   involved: Auxiliary variables are introduced for sub-expressions,
   and propagation proceeds on them as if they were involved in a
   tighter constraint (like equality), whereas eventually only very
   little of the propagated information is actually used. For example,
   only extremal values are of interest in inequalities. Introducing
   auxiliary variables should be avoided when possible, and
   specialised propagators should be used for common constraints.

   We again use a simple committed-choice language for matching
   special cases of constraints. m_c(M,C) means that M matches and C
   holds. d(X, Y) means decomposition, i.e., it is short for
   g(parse_clpfd(X, Y)). r(X, Y) means to rematch with X and Y.

   Two things are important: First, although the actual constraint
   functors (#\=2, #=/2 etc.) are used in the description, they must
   expand to the respective auxiliary predicates (match_expand/2)
   because the actual constraints are subject to goal expansion.
   Second, when specialised constraints (like scalar product) post
   simpler constraints on their own, these simpler versions must be
   handled separately and must occur before.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

match_expand(#>=, clpfd_geq_).
match_expand(#=, clpfd_equal_).
match_expand(#\=, clpfd_neq).

symmetric(#=).
symmetric(#\=).

matches([
         m_c(any(X) #>= any(Y), left_right_linsum_const(X, Y, Cs, Vs, Const)) =>
            [g((   Cs = [1], Vs = [A] -> geq(A, Const)
               ;   Cs = [-1], Vs = [A] -> Const1 is -Const, geq(Const1, A)
               ;   Cs = [1,1], Vs = [A,B] -> ?(A) + ?(B) #= ?(S), geq(S, Const)
               ;   Cs = [1,-1], Vs = [A,B] ->
                   (   Const =:= 0 -> geq(A, B)
                   ;   C1 is -Const,
                       propagator_init_trigger(x_leq_y_plus_c(B, A, C1))
                   )
               ;   Cs = [-1,1], Vs = [A,B] ->
                   (   Const =:= 0 -> geq(B, A)
                   ;   C1 is -Const,
                       propagator_init_trigger(x_leq_y_plus_c(A, B, C1))
                   )
               ;   Cs = [-1,-1], Vs = [A,B] ->
                   ?(A) + ?(B) #= ?(S), Const1 is -Const, geq(Const1, S)
               ;   scalar_product_(#>=, Cs, Vs, Const)
               ))],
         m(any(X) - any(Y) #>= integer(C))     => [d(X, X1), d(Y, Y1), g(C1 is -C), p(x_leq_y_plus_c(Y1, X1, C1))],
         m(integer(X) #>= any(Z) + integer(A)) => [g(C is X - A), r(C, Z)],
         m(abs(any(X)-any(Y)) #>= integer(I))  => [d(X, X1), d(Y, Y1), p(absdiff_geq(X1, Y1, I))],
         m(abs(any(X)) #>= integer(I))         => [d(X, RX), g((I>0 -> I1 is -I, RX in inf..I1 \/ I..sup; true))],
         m(integer(I) #>= abs(any(X)))         => [d(X, RX), g(I>=0), g(I1 is -I), g(RX in I1..I)],
         m(any(X) #>= any(Y))                  => [d(X, RX), d(Y, RY), g(geq(RX, RY))],

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

         m(var(X) #= var(Y))        => [g(constrain_to_integer(X)), g(X=Y)],
         m(var(X) #= var(Y)+var(Z)) => [p(pplus(Y,Z,X))],
         m(var(X) #= var(Y)-var(Z)) => [p(pplus(X,Z,Y))],
         m(var(X) #= var(Y)*var(Z)) => [p(ptimes(Y,Z,X))],
         m(var(X) #= -var(Z))       => [p(ptimes(-1, Z, X))],
         m_c(any(X) #= any(Y), left_right_linsum_const(X, Y, Cs, Vs, S)) =>
            [g(scalar_product_(#=, Cs, Vs, S))],
         m(var(X) #= any(Y))       => [d(Y,X)],
         m(any(X) #= any(Y))       => [d(X, RX), d(Y, RX)],

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

         m(var(X) #\= integer(Y))             => [g(neq_num(X, Y))],
         m(var(X) #\= var(Y))                 => [g(neq(X,Y))],
         m(var(X) #\= var(Y) + var(Z))        => [p(x_neq_y_plus_z(X, Y, Z))],
         m(var(X) #\= var(Y) - var(Z))        => [p(x_neq_y_plus_z(Y, X, Z))],
         m(var(X) #\= var(Y)*var(Z))          => [p(ptimes(Y,Z,P)), g(neq(X,P))],
         m(integer(X) #\= abs(any(Y)-any(Z))) => [d(Y, Y1), d(Z, Z1), p(absdiff_neq(Y1, Z1, X))],
         m_c(any(X) #\= any(Y), left_right_linsum_const(X, Y, Cs, Vs, S)) =>
            [g(scalar_product_(#\=, Cs, Vs, S))],
         m(any(X) #\= any(Y) + any(Z))        => [d(X, X1), d(Y, Y1), d(Z, Z1), p(x_neq_y_plus_z(X1, Y1, Z1))],
         m(any(X) #\= any(Y) - any(Z))        => [d(X, X1), d(Y, Y1), d(Z, Z1), p(x_neq_y_plus_z(Y1, X1, Z1))],
         m(any(X) #\= any(Y)) => [d(X, RX), d(Y, RY), g(neq(RX, RY))]
        ]).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   We again compile the committed-choice matching language to the
   intended auxiliary predicates. We now must take care not to
   unintentionally unify a variable with a complex term.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

make_matches(Clauses) :-
        matches(Ms),
        findall(F, (member(M=>_, Ms), arg(1, M, M1), functor(M1, F, _)), Fs0),
        sort(Fs0, Fs),
        maplist(prevent_cyclic_argument, Fs, PrevCyclicClauses),
        phrase(matchers(Ms), Clauses0),
        maplist(goals_goal, Clauses0, MatcherClauses),
        append(PrevCyclicClauses, MatcherClauses, Clauses1),
        sort_by_predicate(Clauses1, Clauses).

sort_by_predicate(Clauses, ByPred) :-
        map_list_to_pairs(predname, Clauses, Keyed),
        keysort(Keyed, KeyedByPred),
        pairs_values(KeyedByPred, ByPred).

predname((H:-_), Key)   :- !, predname(H, Key).
predname(M:H, M:Key)    :- !, predname(H, Key).
predname(H, Name/Arity) :- !, functor(H, Name, Arity).

prevent_cyclic_argument(F0, Clause) :-
        match_expand(F0, F),
        Head =.. [F,X,Y],
        Clause = (Head :- (   cyclic_term(X) ->
                              domain_error(clpfd_expression, X)
                          ;   cyclic_term(Y) ->
                              domain_error(clpfd_expression, Y)
                          ;   false
                          )).

matchers([]) --> [].
matchers([Condition => Goals|Ms]) -->
        matcher(Condition, Goals),
        matchers(Ms).

matcher(m(M), Gs) --> matcher(m_c(M,true), Gs).
matcher(m_c(Matcher,Cond), Gs) -->
        [(Head :- Goals0)],
        { Matcher =.. [F,A,B],
          match_expand(F, Expand),
          Head =.. [Expand,X,Y],
          phrase((match(A, X), match(B, Y)), Goals0, [Cond,!|Goals1]),
          phrase(match_goals(Gs, Expand), Goals1) },
        (   { symmetric(F), \+ (subsumes_term(A, B), subsumes_term(B, A)) } ->
            { Head1 =.. [Expand,Y,X] },
            [(Head1 :- Goals0)]
        ;   []
        ).

match(any(A), T)     --> [A = T].
match(var(V), T)     --> [( nonvar(T), ( T = ?(Var) ; T = #(Var) ) ->
                            must_be_fd_integer(Var), V = Var
                          ; v_or_i(T), V = T
                          )].
match(integer(I), T) --> [integer(T), I = T].
match(-X, T)         --> [nonvar(T), T = -A], match(X, A).
match(abs(X), T)     --> [nonvar(T), T = abs(A)], match(X, A).
match(Binary, T)     -->
        { Binary =.. [Op,X,Y], Term =.. [Op,A,B] },
        [nonvar(T), T = Term],
        match(X, A), match(Y, B).

match_goals([], _)     --> [].
match_goals([G|Gs], F) --> match_goal(G, F), match_goals(Gs, F).

match_goal(r(X,Y), F)  --> { G =.. [F,X,Y] }, [G].
match_goal(d(X,Y), _)  --> [parse_clpfd(X, Y)].
match_goal(g(Goal), _) --> [Goal].
match_goal(p(Prop), _) -->
        [make_propagator(Prop, P)],
        { term_variables(Prop, Vs) },
        parse_init(Vs, P),
        [trigger_once(P)].


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



%% ?X #>= ?Y
%
% Same as Y #=< X. When reasoning over integers, replace `(>=)/2` by
% #>=/2 to obtain more general relations. See [declarative integer
% arithmetic](<#clpfd-integer-arith>).

X #>= Y :- clpfd_geq(X, Y).

clpfd_geq(X, Y) :- clpfd_geq_(X, Y), reinforce(X), reinforce(Y).

%% ?X #=< ?Y
%
% The arithmetic expression X is less than or equal to Y. When
% reasoning over integers, replace `(=<)/2` by #=</2 to obtain more
% general relations. See [declarative integer
% arithmetic](<#clpfd-integer-arith>).

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

%% ?X #= ?Y
%
% The arithmetic expression X equals Y. This is the most important
% [arithmetic constraint](<#clpfd-arith-constraints>), subsuming and
% replacing both `(is)/2` _and_ `(=:=)/2` over integers. See
% [declarative integer arithmetic](<#clpfd-integer-arith>).

X #= Y :- clpfd_equal(X, Y).

clpfd_equal(X, Y) :- clpfd_equal_(X, Y), reinforce(X).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Conditions under which an equality can be compiled to built-in
   arithmetic. Their order is significant. (/)/2 becomes (//)/2.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

expr_conds(E, E)                 --> [integer(E)],
        { var(E), !, \+ current_prolog_flag(clpfd_monotonic, true) }.
expr_conds(E, E)                 --> { integer(E) }.
expr_conds(?(E), E)              --> [integer(E)].
expr_conds(#(E), E)              --> [integer(E)].
expr_conds(-E0, -E)              --> expr_conds(E0, E).
expr_conds(abs(E0), abs(E))      --> expr_conds(E0, E).
expr_conds(A0+B0, A+B)           --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0*B0, A*B)           --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0-B0, A-B)           --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0//B0, A//B)         -->
        expr_conds(A0, A), expr_conds(B0, B),
        [B =\= 0].
%expr_conds(A0/B0, AB)            --> expr_conds(A0//B0, AB).
expr_conds(min(A0,B0), min(A,B)) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(max(A0,B0), max(A,B)) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0 mod B0, A mod B)   -->
        expr_conds(A0, A), expr_conds(B0, B),
        [B =\= 0].
expr_conds(A0^B0, A^B)           -->
        expr_conds(A0, A), expr_conds(B0, B),
        [(B >= 0 ; A =:= -1)].
% Bitwise operations, added to make CLP(FD) usable in more cases
expr_conds(\ A0, \ A) --> expr_conds(A0, A).
expr_conds(A0<<B0, A<<B) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0>>B0, A>>B) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0/\B0, A/\B) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0\/B0, A\/B) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(A0 xor B0, A xor B) --> expr_conds(A0, A), expr_conds(B0, B).
expr_conds(lsb(A0), lsb(A)) --> expr_conds(A0, A).
expr_conds(msb(A0), msb(A)) --> expr_conds(A0, A).
expr_conds(popcount(A0), popcount(A)) --> expr_conds(A0, A).

:- multifile
        system:goal_expansion/2.
:- dynamic
        system:goal_expansion/2.

system:goal_expansion(Goal, Expansion) :-
        \+ current_prolog_flag(optimise_clpfd, false),
        clpfd_expandable(Goal),
        prolog_load_context(module, M),
	(   M == clpfd
	->  true
	;   predicate_property(M:Goal, imported_from(clpfd))
	),
        clpfd_expansion(Goal, Expansion).

clpfd_expandable(_ in _).
clpfd_expandable(_ #= _).
clpfd_expandable(_ #>= _).
clpfd_expandable(_ #=< _).
clpfd_expandable(_ #> _).
clpfd_expandable(_ #< _).

clpfd_expansion(Var in Dom, In) :-
        (   ground(Dom), Dom = L..U, integer(L), integer(U) ->
            expansion_simpler(
                (   integer(Var) ->
                    between(L, U, Var)
                ;   clpfd:clpfd_in(Var, Dom)
                ), In)
        ;   In = clpfd:clpfd_in(Var, Dom)
        ).
clpfd_expansion(X0 #= Y0, Equal) :-
        phrase(expr_conds(X0, X), CsX),
        phrase(expr_conds(Y0, Y), CsY),
        list_goal(CsX, CondX),
        list_goal(CsY, CondY),
        expansion_simpler(
                (   CondX ->
                    (   var(Y) -> Y is X
                    ;   CondY -> X =:= Y
                    ;   T is X, clpfd:clpfd_equal(T, Y0)
                    )
                ;   CondY ->
                    (   var(X) -> X is Y
                    ;   T is Y, clpfd:clpfd_equal(X0, T)
                    )
                ;   clpfd:clpfd_equal(X0, Y0)
                ), Equal).
clpfd_expansion(X0 #>= Y0, Geq) :-
        phrase(expr_conds(X0, X), CsX),
        phrase(expr_conds(Y0, Y), CsY),
        list_goal(CsX, CondX),
        list_goal(CsY, CondY),
        expansion_simpler(
              (   CondX ->
                  (   CondY -> X >= Y
                  ;   T is X, clpfd:clpfd_geq(T, Y0)
                  )
              ;   CondY -> T is Y, clpfd:clpfd_geq(X0, T)
              ;   clpfd:clpfd_geq(X0, Y0)
              ), Geq).
clpfd_expansion(X #=< Y,  Leq) :- clpfd_expansion(Y #>= X, Leq).
clpfd_expansion(X #> Y, Gt)    :- clpfd_expansion(X #>= Y+1, Gt).
clpfd_expansion(X #< Y, Lt)    :- clpfd_expansion(Y #> X, Lt).

expansion_simpler((True->Then0;_), Then) :-
        is_true(True), !,
        expansion_simpler(Then0, Then).
expansion_simpler((False->_;Else0), Else) :-
        is_false(False), !,
        expansion_simpler(Else0, Else).
expansion_simpler((If->Then0;Else0), (If->Then;Else)) :- !,
        expansion_simpler(Then0, Then),
        expansion_simpler(Else0, Else).
expansion_simpler((A0,B0), (A,B)) :-
        expansion_simpler(A0, A),
        expansion_simpler(B0, B).
expansion_simpler(Var is Expr0, Goal) :-
        ground(Expr0), !,
        phrase(expr_conds(Expr0, Expr), Gs),
        (   maplist(call, Gs) -> Value is Expr, Goal = (Var = Value)
        ;   Goal = false
        ).
expansion_simpler(Var =:= Expr0, Goal) :-
        ground(Expr0), !,
        phrase(expr_conds(Expr0, Expr), Gs),
        (   maplist(call, Gs) -> Value is Expr, Goal = (Var =:= Value)
        ;   Goal = false
        ).
expansion_simpler(Var is Expr, Var = Expr) :- var(Expr), !.
expansion_simpler(Var is Expr, Goal) :- !,
        (   var(Var), nonvar(Expr),
            Expr = E mod M, nonvar(E), E = A^B ->
            Goal = ( ( integer(A), integer(B), integer(M),
                       A >= 0, B >= 0, M > 0 ->
                       Var is powm(A, B, M)
                     ; Var is Expr
                     ) )
        ;   Goal = ( Var is Expr )
        ).
expansion_simpler(between(L,U,V), Goal) :- maplist(integer, [L,U,V]), !,
        (   between(L,U,V) -> Goal = true
        ;   Goal = false
        ).
expansion_simpler(Goal, Goal).

is_true(true).
is_true(integer(I))  :- integer(I).
:- if(current_predicate(var_property/2)).
is_true(var(X))      :- var(X), var_property(X, fresh(true)).
is_false(integer(X)) :- var(X), var_property(X, fresh(true)).
is_false((A,B))      :- is_false(A) ; is_false(B).
:- endif.
is_false(var(X)) :- nonvar(X).


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

linsum(X, S, S)    --> { var(X), !, non_monotonic(X) }, [vn(X,1)].
linsum(I, S0, S)   --> { integer(I), S is S0 + I }.
linsum(?(X), S, S) --> { must_be_fd_integer(X) }, [vn(X,1)].
linsum(#(X), S, S) --> { must_be_fd_integer(X) }, [vn(X,1)].
linsum(-A, S0, S)  --> mulsum(A, -1, S0, S).
linsum(N*A, S0, S) --> { integer(N) }, !, mulsum(A, N, S0, S).
linsum(A*N, S0, S) --> { integer(N) }, !, mulsum(A, N, S0, S).
linsum(A+B, S0, S) --> linsum(A, S0, S1), linsum(B, S1, S).
linsum(A-B, S0, S) --> linsum(A, S0, S1), mulsum(B, -1, S1, S).

mulsum(A, M, S0, S) -->
        { phrase(linsum(A, 0, CA), As), S is S0 + M*CA },
        lin_mul(As, M).

lin_mul([], _)             --> [].
lin_mul([vn(X,N0)|VNs], M) --> { N is N0*M }, [vn(X,N)], lin_mul(VNs, M).

v_or_i(V) :- var(V), !, non_monotonic(V).
v_or_i(I) :- integer(I).

must_be_fd_integer(X) :-
        (   var(X) -> constrain_to_integer(X)
        ;   must_be(integer, X)
        ).

left_right_linsum_const(Left, Right, Cs, Vs, Const) :-
        phrase(linsum(Left, 0, CL), Lefts0, Rights),
        phrase(linsum(Right, 0, CR), Rights0),
        maplist(linterm_negate, Rights0, Rights),
        msort(Lefts0, Lefts),
        Lefts = [vn(First,N)|LeftsRest],
        vns_coeffs_variables(LeftsRest, N, First, Cs0, Vs0),
        filter_linsum(Cs0, Vs0, Cs, Vs),
        Const is CR - CL.
        %format("linear sum: ~w ~w ~w\n", [Cs,Vs,Const]).

linterm_negate(vn(V,N0), vn(V,N)) :- N is -N0.

vns_coeffs_variables([], N, V, [N], [V]).
vns_coeffs_variables([vn(V,N)|VNs], N0, V0, Ns, Vs) :-
        (   V == V0 ->
            N1 is N0 + N,
            vns_coeffs_variables(VNs, N1, V0, Ns, Vs)
        ;   Ns = [N0|NRest],
            Vs = [V0|VRest],
            vns_coeffs_variables(VNs, N, V, NRest, VRest)
        ).

filter_linsum([], [], [], []).
filter_linsum([C0|Cs0], [V0|Vs0], Cs, Vs) :-
        (   C0 =:= 0 ->
            constrain_to_integer(V0),
            filter_linsum(Cs0, Vs0, Cs, Vs)
        ;   Cs = [C0|Cs1], Vs = [V0|Vs1],
            filter_linsum(Cs0, Vs0, Cs1, Vs1)
        ).

gcd([], G, G).
gcd([N|Ns], G0, G) :-
        G1 is gcd(N, G0),
        gcd(Ns, G1, G).

even(N) :- N mod 2 =:= 0.

odd(N) :- \+ even(N).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   k-th root of N, if N is a k-th power.

   TODO: Replace this when the GMP function becomes available.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

integer_kth_root(N, K, R) :-
        (   even(K) ->
            N >= 0
        ;   true
        ),
        (   N < 0 ->
            odd(K),
            integer_kroot(N, 0, N, K, R)
        ;   integer_kroot(0, N, N, K, R)
        ).

integer_kroot(L, U, N, K, R) :-
        (   L =:= U -> N =:= L^K, R = L
        ;   L + 1 =:= U ->
            (   L^K =:= N -> R = L
            ;   U^K =:= N -> R = U
            ;   false
            )
        ;   Mid is (L + U)//2,
            (   Mid^K > N ->
                integer_kroot(L, Mid, N, K, R)
            ;   integer_kroot(Mid, U, N, K, R)
            )
        ).

integer_log_b(N, B, Log0, Log) :-
        T is B^Log0,
        (   T =:= N -> Log = Log0
        ;   T < N,
            Log1 is Log0 + 1,
            integer_log_b(N, B, Log1, Log)
        ).

floor_integer_log_b(N, B, Log0, Log) :-
        T is B^Log0,
        (   T > N -> Log is Log0 - 1
        ;   T =:= N -> Log = Log0
        ;   T < N,
            Log1 is Log0 + 1,
            floor_integer_log_b(N, B, Log1, Log)
        ).


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Largest R such that R^K =< N.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

:- if(current_predicate(nth_integer_root_and_remainder/4)).

% This currently only works for K >= 1, which is all that is needed for now.
integer_kth_root_leq(N, K, R) :-
        nth_integer_root_and_remainder(K, N, R, _).

:- else.

integer_kth_root_leq(N, K, R) :-
        (   even(K) ->
            N >= 0
        ;   true
        ),
        (   N < 0 ->
            odd(K),
            integer_kroot_leq(N, 0, N, K, R)
        ;   integer_kroot_leq(0, N, N, K, R)
        ).

integer_kroot_leq(L, U, N, K, R) :-
        (   L =:= U -> R = L
        ;   L + 1 =:= U ->
            (   U^K =< N -> R = U
            ;   R = L
            )
        ;   Mid is (L + U)//2,
            (   Mid^K > N ->
                integer_kroot_leq(L, Mid, N, K, R)
            ;   integer_kroot_leq(Mid, U, N, K, R)
            )
        ).

:- endif.

%% ?X #\= ?Y
%
% The arithmetic expressions X and Y evaluate to distinct integers.
% When reasoning over integers, replace `(=\=)/2` by #\=/2 to obtain
% more general relations. See [declarative integer
% arithmetic](<#clpfd-integer-arith>).

X #\= Y :- clpfd_neq(X, Y), do_queue.

% X #\= Y + Z

x_neq_y_plus_z(X, Y, Z) :-
        propagator_init_trigger(x_neq_y_plus_z(X,Y,Z)).

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

neq_num(X, N) :-
        (   fd_get(X, XD, XPs) ->
            domain_remove(XD, N, XD1),
            fd_put(X, XD1, XPs)
        ;   X =\= N
        ).

%% ?X #> ?Y
%
% Same as Y #< X. When reasoning over integers, replace `(>)/2` by
% #>/2 to obtain more general relations See [declarative integer
% arithmetic](<#clpfd-integer-arith>).

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

%% #<(?X, ?Y)
%
% The arithmetic expression X is less than Y. When reasoning over
% integers, replace `(<)/2` by #</2 to obtain more general relations. See
% [declarative integer arithmetic](<#clpfd-integer-arith>).
%
% In addition to its regular use in tasks that require it, this
% constraint can also be useful to eliminate uninteresting symmetries
% from a problem. For example, all possible matches between pairs
% built from four players in total:
%
% ==
% ?- Vs = [A,B,C,D], Vs ins 1..4,
%         all_different(Vs),
%         A #< B, C #< D, A #< C,
%    findall(pair(A,B)-pair(C,D), label(Vs), Ms).
% Ms = [ pair(1, 2)-pair(3, 4),
%        pair(1, 3)-pair(2, 4),
%        pair(1, 4)-pair(2, 3)].
% ==

X #< Y  :- Y #> X.

%% #\ (+Q)
%
% Q does _not_ hold. See [reification](<#clpfd-reification>).
%
% For example, to obtain the complement of a domain:
%
% ==
% ?- #\ X in -3..0\/10..80.
% X in inf.. -4\/1..9\/81..sup.
% ==

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

%% ?P #<==> ?Q
%
% P and Q are equivalent. See [reification](<#clpfd-reification>).
%
% For example:
%
% ==
% ?- X #= 4 #<==> B, X #\= 4.
% B = 0,
% X in inf..3\/5..sup.
% ==
% The following example uses reified constraints to relate a list of
% finite domain variables to the number of occurrences of a given value:
%
% ==
% vs_n_num(Vs, N, Num) :-
%         maplist(eq_b(N), Vs, Bs),
%         sum(Bs, #=, Num).
%
% eq_b(X, Y, B) :- X #= Y #<==> B.
% ==
%
% Sample queries and their results:
%
% ==
% ?- Vs = [X,Y,Z], Vs ins 0..1, vs_n_num(Vs, 4, Num).
% Vs = [X, Y, Z],
% Num = 0,
% X in 0..1,
% Y in 0..1,
% Z in 0..1.
%
% ?- vs_n_num([X,Y,Z], 2, 3).
% X = 2,
% Y = 2,
% Z = 2.
% ==

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

%% ?P #==> ?Q
%
% P implies Q. See [reification](<#clpfd-reification>).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Implication is special in that created auxiliary constraints can be
   retracted when the implication becomes entailed, for example:

   %?- X + 1 #= Y #==> Z, Z #= 1.
   %@ Z = 1,
   %@ X in inf..sup,
   %@ Y in inf..sup.

   We cannot use propagator_init_trigger/1 here because the states of
   auxiliary propagators are themselves part of the propagator.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

L #==> R   :-
        reify(L, LB, LPs),
        reify(R, RB, RPs),
        append(LPs, RPs, Ps),
        propagator_init_trigger([LB,RB], pimpl(LB,RB,Ps)).

%% ?P #<== ?Q
%
% Q implies P. See [reification](<#clpfd-reification>).

L #<== R   :- R #==> L.

%% ?P #/\ ?Q
%
% P and Q hold. See [reification](<#clpfd-reification>).

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

conjunctive_neqs_var_drep(Eqs, Var, Drep) :-
        conjunctive_neqs_var(Eqs, Var),
        phrase(conjunctive_neqs_vals(Eqs), Vals),
        list_to_domain(Vals, Dom),
        domain_complement(Dom, C),
        domain_to_drep(C, Drep).

conjunctive_neqs_var(V, _) :- var(V), !, false.
conjunctive_neqs_var(L #\= R, Var) :-
        (   var(L), integer(R) -> Var = L
        ;   integer(L), var(R) -> Var = R
        ;   false
        ).
conjunctive_neqs_var(A #/\ B, VA) :-
        conjunctive_neqs_var(A, VA),
        conjunctive_neqs_var(B, VB),
        VA == VB.

conjunctive_neqs_vals(L #\= R) --> ( { integer(L) } -> [L] ; [R] ).
conjunctive_neqs_vals(A #/\ B) -->
        conjunctive_neqs_vals(A),
        conjunctive_neqs_vals(B).

%% ?P #\/ ?Q
%
% P or Q holds. See [reification](<#clpfd-reification>).
%
% For example, the sum of natural numbers below 1000 that are
% multiples of 3 or 5:
%
% ==
% ?- findall(N, (N mod 3 #= 0 #\/ N mod 5 #= 0, N in 0..999,
%                indomain(N)),
%            Ns),
%    sum(Ns, #=, Sum).
% Ns = [0, 3, 5, 6, 9, 10, 12, 15, 18|...],
% Sum = 233168.
% ==

L #\/ R :-
        (   disjunctive_eqs_var_drep(L #\/ R, Var, Drep) -> Var in Drep
        ;   reify(L, X, Ps1),
            reify(R, Y, Ps2),
            propagator_init_trigger([X,Y], reified_or(X,Ps1,Y,Ps2,1))
        ).

disjunctive_eqs_var_drep(Eqs, Var, Drep) :-
        disjunctive_eqs_var(Eqs, Var),
        phrase(disjunctive_eqs_vals(Eqs), Vals),
        list_to_drep(Vals, Drep).

disjunctive_eqs_var(V, _) :- var(V), !, false.
disjunctive_eqs_var(V in I, V) :- var(V), integer(I).
disjunctive_eqs_var(L #= R, Var) :-
        (   var(L), integer(R) -> Var = L
        ;   integer(L), var(R) -> Var = R
        ;   false
        ).
disjunctive_eqs_var(A #\/ B, VA) :-
        disjunctive_eqs_var(A, VA),
        disjunctive_eqs_var(B, VB),
        VA == VB.

disjunctive_eqs_vals(L #= R)  --> ( { integer(L) } -> [L] ; [R] ).
disjunctive_eqs_vals(_ in I)  --> [I].
disjunctive_eqs_vals(A #\/ B) -->
        disjunctive_eqs_vals(A),
        disjunctive_eqs_vals(B).

%% ?P #\ ?Q
%
% Either P holds or Q holds, but not both. See
% [reification](<#clpfd-reification>).

L #\ R :- (L #\/ R) #/\ #\ (L #/\ R).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   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.
   Therefore, the committed-choice language is extended by an element
   d(D) that states D is 1 iff all subexpressions are defined. a(V)
   means that V is an auxiliary variable that was introduced while
   parsing a compound expression. a(X,V) means V is auxiliary unless
   it is ==/2 X, and a(X,Y,V) means V is auxiliary unless it is ==/2 X
   or Y. l(L) means the literal L occurs in the described list.

   When a constraint becomes entailed or subexpressions become
   undefined, created auxiliary constraints are killed, and the
   "clpfd" attribute is removed from auxiliary variables.

   For (/)/2, mod/2 and rem/2, we create a skeleton propagator and
   remember it as an auxiliary constraint. The pskeleton propagator
   can use the skeleton when the constraint is defined.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

parse_reified(E, R, D,
              [g(cyclic_term(E)) => [g(domain_error(clpfd_expression, E))],
               g(var(E))     => [g(non_monotonic(E)),
                                 g(constrain_to_integer(E)), g(R = E), g(D=1)],
               g(integer(E)) => [g(R=E), g(D=1)],
               ?(E)          => [g(must_be_fd_integer(E)), g(R=E), g(D=1)],
               #(E)          => [g(must_be_fd_integer(E)), g(R=E), g(D=1)],
               m(A+B)        => [d(D), p(pplus(A,B,R)), a(A,B,R)],
               m(A*B)        => [d(D), p(ptimes(A,B,R)), a(A,B,R)],
               m(A-B)        => [d(D), p(pplus(R,B,A)), a(A,B,R)],
               m(-A)         => [d(D), p(ptimes(-1,A,R)), a(R)],
               m(max(A,B))   => [d(D), p(pgeq(R, A)), p(pgeq(R, B)), p(pmax(A,B,R)), a(A,B,R)],
               m(min(A,B))   => [d(D), p(pgeq(A, R)), p(pgeq(B, R)), p(pmin(A,B,R)), a(A,B,R)],
               m(abs(A))     => [g(?(R)#>=0), d(D), p(pabs(A, R)), a(A,R)],
%               m(A/B)        => [skeleton(A,B,D,R,ptzdiv)],
               m(A//B)       => [skeleton(A,B,D,R,ptzdiv)],
               m(A div B)    => [skeleton(A,B,D,R,pdiv)],
               m(A rdiv B)   => [skeleton(A,B,D,R,prdiv)],
               m(A mod B)    => [skeleton(A,B,D,R,pmod)],
               m(A rem B)    => [skeleton(A,B,D,R,prem)],
               m(A^B)        => [d(D), p(pexp(A,B,R)), a(A,B,R)],
               % bitwise operations
               m(\A)         => [function(D,\,A,R)],
               m(msb(A))     => [function(D,msb,A,R)],
               m(lsb(A))     => [function(D,lsb,A,R)],
               m(popcount(A)) => [function(D,popcount,A,R)],
               m(A<<B)       => [d(D), p(pshift(A,B,R,1)), a(A,B,R)],
               m(A>>B)       => [d(D), p(pshift(A,B,R,-1)), a(A,B,R)],
               m(A/\B)       => [function(D,/\,A,B,R)],
               m(A\/B)       => [function(D,\/,A,B,R)],
               m(A xor B)    => [function(D,xor,A,B,R)],
               g(true)       => [g(domain_error(clpfd_expression, E))]]
             ).

% Again, we compile this to a predicate, parse_reified_clpfd//3. This
% time, it is a DCG that describes the list of auxiliary variables and
% propagators for the given expression, in addition to relating it to
% its reified (Boolean) finite domain variable and its Boolean
% definedness.

make_parse_reified(Clauses) :-
        parse_reified_clauses(Clauses0),
        maplist(goals_goal_dcg, Clauses0, Clauses).

goals_goal_dcg((Head --> Goals), Clause) :-
        list_goal(Goals, Body),
        expand_term((Head --> Body), Clause).

parse_reified_clauses(Clauses) :-
        parse_reified(E, R, D, Matchers),
        maplist(parse_reified(E, R, D), Matchers, Clauses).

parse_reified(E, R, D, Matcher, Clause) :-
        Matcher = (Condition0 => Goals0),
        phrase((reified_condition(Condition0, E, Head, Ds),
                reified_goals(Goals0, Ds)), Goals, [a(D)]),
        Clause = (parse_reified_clpfd(Head, R, D) --> Goals).

reified_condition(g(Goal), E, E, []) --> [{Goal}, !].
reified_condition(?(E), _, ?(E), []) --> [!].
reified_condition(#(E), _, #(E), []) --> [!].
reified_condition(m(Match), _, Match0, Ds) -->
        [!],
        { copy_term(Match, Match0),
          term_variables(Match0, Vs0),
          term_variables(Match, Vs)
        },
        reified_variables(Vs0, Vs, Ds).

reified_variables([], [], []) --> [].
reified_variables([V0|Vs0], [V|Vs], [D|Ds]) -->
        [parse_reified_clpfd(V0, V, D)],
        reified_variables(Vs0, Vs, Ds).

reified_goals([], _) --> [].
reified_goals([G|Gs], Ds) --> reified_goal(G, Ds), reified_goals(Gs, Ds).

reified_goal(d(D), Ds) -->
        (   { Ds = [X] } -> [{D=X}]
        ;   { Ds = [X,Y] } ->
            { phrase(reified_goal(p(reified_and(X,[],Y,[],D)), _), Gs),
              list_goal(Gs, Goal) },
            [( {X==1, Y==1} -> {D = 1} ; Goal )]
        ;   { domain_error(one_or_two_element_list, Ds) }
        ).
reified_goal(g(Goal), _) --> [{Goal}].
reified_goal(p(Vs, Prop), _) -->
        [{make_propagator(Prop, P)}],
        parse_init_dcg(Vs, P),
        [{trigger_once(P)}],
        [( { propagator_state(P, S), S == dead } -> [] ; [p(P)])].
reified_goal(p(Prop), Ds) -->
        { term_variables(Prop, Vs) },
        reified_goal(p(Vs,Prop), Ds).
reified_goal(function(D,Op,A,B,R), Ds) -->
        reified_goals([d(D),p(pfunction(Op,A,B,R)),a(A,B,R)], Ds).
reified_goal(function(D,Op,A,R), Ds) -->
        reified_goals([d(D),p(pfunction(Op,A,R)),a(A,R)], Ds).
reified_goal(skeleton(A,B,D,R,F), Ds) -->
        { Prop =.. [F,X,Y,Z] },
        reified_goals([d(D1),l(p(P)),g(make_propagator(Prop, P)),
                       p([A,B,D2,R], pskeleton(A,B,D2,[X,Y,Z]-P,R,F)),
                       p(reified_and(D1,[],D2,[],D)),a(D2),a(A,B,R)], Ds).
reified_goal(a(V), _)     --> [a(V)].
reified_goal(a(X,V), _)   --> [a(X,V)].
reified_goal(a(X,Y,V), _) --> [a(X,Y,V)].
reified_goal(l(L), _)     --> [[L]].

parse_init_dcg([], _)     --> [].
parse_init_dcg([V|Vs], P) --> [{init_propagator(V, P)}], parse_init_dcg(Vs, P).

%?- set_prolog_flag(answer_write_options, [portray(true)]),
%   clpfd:parse_reified_clauses(Cs), maplist(portray_clause, Cs).

reify(E, B) :- reify(E, B, _).

reify(Expr, B, Ps) :-
        (   acyclic_term(Expr), reifiable(Expr) -> phrase(reify(Expr, B), Ps)
        ;   domain_error(clpfd_reifiable_expression, Expr)
        ).

reifiable(E)      :- var(E), non_monotonic(E).
reifiable(E)      :- integer(E), E in 0..1.
reifiable(?(E))   :- must_be_fd_integer(E).
reifiable(#(E))   :- must_be_fd_integer(E).
reifiable(V in _) :- fd_variable(V).
reifiable(V in_set _) :- fd_variable(V).
reifiable(Expr)   :-
        Expr =.. [Op,Left,Right],
        (   memberchk(Op, [#>=,#>,#=<,#<,#=,#\=])
        ;   memberchk(Op, [#==>,#<==,#<==>,#/\,#\/,#\]),
            reifiable(Left),
            reifiable(Right)
        ).
reifiable(#\ E) :- reifiable(E).
reifiable(tuples_in(Tuples, Relation)) :-
        must_be(list(list), Tuples),
        maplist(maplist(fd_variable), Tuples),
        must_be(list(list(integer)), Relation).
reifiable(finite_domain(V)) :- fd_variable(V).

reify(E, B) --> { B in 0..1 }, reify_(E, B).

reify_(E, B) --> { var(E), !, E = B }.
reify_(E, B) --> { integer(E), E = B }.
reify_(?(B), B) --> [].
reify_(#(B), B) --> [].
reify_(V in Drep, B) -->
        { drep_to_domain(Drep, Dom) },
        propagator_init_trigger(reified_in(V,Dom,B)),
        a(B).
reify_(V in_set Dom, B) -->
        propagator_init_trigger(reified_in(V,Dom,B)),
        a(B).
reify_(tuples_in(Tuples, Relation), B) -->
        { maplist(relation_tuple_b_prop(Relation), Tuples, Bs, Ps),
          maplist(monotonic, Bs, Bs1),
          fold_statement(conjunction, Bs1, And),
          ?(B) #<==> And },
        propagator_init_trigger([B], tuples_not_in(Tuples, Relation, B)),
        kill_reified_tuples(Bs, Ps, Bs),
        list(Ps),
        as([B|Bs]).
reify_(finite_domain(V), B) -->
        propagator_init_trigger(reified_fd(V,B)),
        a(B).
reify_(L #>= R, B) --> arithmetic(L, R, B, reified_geq).
reify_(L #= R, B)  --> arithmetic(L, R, B, reified_eq).
reify_(L #\= R, B) --> arithmetic(L, R, B, reified_neq).
reify_(L #> R, B)  --> reify_(L #>= (R+1), B).
reify_(L #=< R, B) --> reify_(R #>= L, B).
reify_(L #< R, B)  --> reify_(R #>= (L+1), B).
reify_(L #==> R, B)  --> reify_((#\ L) #\/ R, B).
reify_(L #<== R, B)  --> reify_(R #==> L, B).
reify_(L #<==> R, B) --> reify_((L #==> R) #/\ (R #==> L), B).
reify_(L #\ R, B) --> reify_((L #\/ R) #/\ #\ (L #/\ R), B).
reify_(L #/\ R, B)   -->
        (   { conjunctive_neqs_var_drep(L #/\ R, V, D) } -> reify_(V in D, B)
        ;   boolean(L, R, B, reified_and)
        ).
reify_(L #\/ R, B) -->
        (   { disjunctive_eqs_var_drep(L #\/ R, V, D) } -> reify_(V in D, B)
        ;   boolean(L, R, B, reified_or)
        ).
reify_(#\ Q, B) -->
        reify(Q, QR),
        propagator_init_trigger(reified_not(QR,B)),
        a(B).

arithmetic(L, R, B, Functor) -->
        { phrase((parse_reified_clpfd(L, LR, LD),
                  parse_reified_clpfd(R, RR, RD)), Ps),
          Prop =.. [Functor,LD,LR,RD,RR,Ps,B] },
        list(Ps),
        propagator_init_trigger([LD,LR,RD,RR,B], Prop),
        a(B).

boolean(L, R, B, Functor) -->
        { reify(L, LR, Ps1), reify(R, RR, Ps2),
          Prop =.. [Functor,LR,Ps1,RR,Ps2,B] },
        list(Ps1), list(Ps2),
        propagator_init_trigger([LR,RR,B], Prop),
        a(LR, RR, B).

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

a(X,Y,B) -->
        (   { nonvar(X) } -> a(Y, B)
        ;   { nonvar(Y) } -> a(X, B)
        ;   [a(X,Y,B)]
        ).

a(X, B) -->
        (   { var(X) } -> [a(X, B)]
        ;   a(B)
        ).

a(B) -->
        (   { var(B) } -> [a(B)]
        ;   []
        ).

as([])     --> [].
as([B|Bs]) --> a(B), as(Bs).

kill_reified_tuples([], _, _) --> [].
kill_reified_tuples([B|Bs], Ps, All) -->
        propagator_init_trigger([B], kill_reified_tuples(B, Ps, All)),
        kill_reified_tuples(Bs, Ps, All).

relation_tuple_b_prop(Relation, Tuple, B, p(Prop)) :-
        put_attr(R, clpfd_relation, Relation),
        make_propagator(reified_tuple_in(Tuple, R, B), Prop),
        tuple_freeze_(Tuple, Prop),
        init_propagator(B, Prop).


tuples_in_conjunction(Tuples, Relation, Conj) :-
        maplist(tuple_in_disjunction(Relation), Tuples, Disjs),
        fold_statement(conjunction, Disjs, Conj).

tuple_in_disjunction(Relation, Tuple, Disj) :-
        maplist(tuple_in_conjunction(Tuple), Relation, Conjs),
        fold_statement(disjunction, Conjs, Disj).

tuple_in_conjunction(Tuple, Element, Conj) :-
        maplist(var_eq, Tuple, Element, Eqs),
        fold_statement(conjunction, Eqs, Conj).

fold_statement(Operation, List, Statement) :-
        (   List = [] -> Statement = 1
        ;   List = [First|Rest],
            foldl(Operation, Rest, First, Statement)
        ).

conjunction(E, Conj, Conj #/\ E).

disjunction(E, Disj, Disj #\/ E).

var_eq(V, N, ?(V) #= N).

% Match variables to created skeleton.

skeleton(Vs, Vs-Prop) :-
        maplist(prop_init(Prop), Vs),
        trigger_once(Prop).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   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(N)      :- integer(N).
is_drep(N..M)   :- drep_bound(N), drep_bound(M), N \== sup, M \== inf.
is_drep(D1\/D2) :- is_drep(D1), is_drep(D2).
is_drep({AI})   :- is_and_integers(AI).
is_drep(\D)     :- is_drep(D).

is_and_integers(I)     :- integer(I).
is_and_integers((A,B)) :- is_and_integers(A), is_and_integers(B).

drep_bound(I)   :- integer(I).
drep_bound(sup).
drep_bound(inf).

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_intervals(\D0) -->
        { drep_to_domain(D0, D1),
          domain_complement(D1, D),
          domain_to_drep(D, Drep) },
        drep_to_intervals(Drep).
drep_to_intervals({AI}) -->
        and_integers_(AI).

and_integers_(I)     --> { integer(I) }, [n(I)-n(I)].
and_integers_((A,B)) --> and_integers_(A), and_integers_(B).

drep_to_domain(DR, D) :-
        must_be(ground, DR),
        (   is_drep(DR) -> true
        ;   domain_error(clpfd_domain, DR)
        ),
        phrase(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 cis B0 + n(1),
        (   N cis_gt Next -> B = B0, Rest = [N-M|NMs]
        ;   B1 cis max(B0,M),
            merge_remaining(NMs, B1, B, Rest)
        ).

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

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

props_number(fd_props(Gs,Bs,Os), N) :-
        length(Gs, N1),
        length(Bs, N2),
        length(Os, N3),
        N is N1 + N2 + N3.

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

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

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   By default, propagation always terminates. Currently, this is
   ensured 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. Set the experimental
   Prolog flag 'clpfd_propagation' to 'full' to make the solver
   propagate as much as possible. This can make queries
   non-terminating, like: X #> abs(X), or: X #> Y, Y #> X, X #> 0.
   Importantly, it can also make labeling non-terminating, as in:

   ?- B #==> X #> abs(X), indomain(B).
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

fd_put(X, Dom, Ps) :-
        (   current_prolog_flag(clpfd_propagation, full) ->
            put_full(X, Dom, Ps)
        ;   put_terminating(X, Dom, Ps)
        ).

put_terminating(X, Dom, Ps) :-
        Dom \== empty,
        (   Dom = from_to(F, F) -> F = n(X)
        ;   (   get_attr(X, clpfd, Attr) ->
                Attr = clpfd_attr(Left,Right,Spread,OldDom, _OldPs),
                put_attr(X, clpfd, clpfd_attr(Left,Right,Spread,Dom,Ps)),
                (   OldDom == Dom -> true
                ;   (   Left == (.) -> Bounded = yes
                    ;   domain_infimum(Dom, Inf), domain_supremum(Dom, Sup),
                        (   Inf = n(_), Sup = n(_) ->
                            Bounded = yes
                        ;   Bounded = no
                        )
                    ),
                    (   Bounded == yes ->
                        put_attr(X, clpfd, clpfd_attr(.,.,.,Dom,Ps)),
                        trigger_props(Ps, X, OldDom, Dom)
                    ;   % 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
                        ;   NewSpread cis_lt OldSpread -> SpreadP = no
                        ;   SpreadP = yes
                        ),
                        put_attr(X, clpfd, clpfd_attr(LeftP,RightP,SpreadP,Dom,Ps)),
                        (   RightP == yes, Right = yes -> true
                        ;   LeftP == yes, Left = yes -> true
                        ;   SpreadP == yes, Spread = yes -> true
                        ;   trigger_props(Ps, X, OldDom, Dom)
                        )
                    )
                )
            ;   var(X) ->
                put_attr(X, clpfd, clpfd_attr(no,no,no,Dom, Ps))
            ;   true
            )
        ).

domain_spread(Dom, Spread) :-
        domain_smallest_finite(Dom, S),
        domain_largest_finite(Dom, L),
        Spread cis 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, full) ->
            % full propagation propagates everything in any case
            true
        ;   term_variables(X, Vs),
            maplist(reinforce_, Vs),
            do_queue
        ).

reinforce_(X) :-
        (   fd_var(X), fd_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_attr(_,_,_,OldDom, _OldPs),
                put_attr(X, clpfd, clpfd_attr(no,no,no,Dom, Ps)),
                %format("putting dom: ~w\n", [Dom]),
                (   OldDom == Dom -> true
                ;   trigger_props(Ps, X, OldDom, Dom)
                )
            ;   var(X) -> %format('\t~w in ~w .. ~w\n',[X,L,U]),
                put_attr(X, clpfd, clpfd_attr(no,no,no,Dom, Ps))
            ;   true
            )
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   A propagator is a term of the form propagator(C, State), where C
   represents a constraint, and State is a free variable that can be
   used to destructively change the state of the propagator via
   attributes. This can be used to avoid redundant invocation of the
   same propagator, or to disable the propagator.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

make_propagator(C, propagator(C, _)).

propagator_state(propagator(_,S), S).

trigger_props(fd_props(Gs,Bs,Os), X, D0, D) :-
        (   ground(X) ->
            trigger_props_(Gs),
            trigger_props_(Bs)
        ;   Bs \== [] ->
            domain_infimum(D0, I0),
            domain_infimum(D, I),
            (   I == I0 ->
                domain_supremum(D0, S0),
                domain_supremum(D, S),
                (   S == S0 -> true
                ;   trigger_props_(Bs)
                )
            ;   trigger_props_(Bs)
            )
        ;   true
        ),
        trigger_props_(Os).

trigger_props(fd_props(Gs,Bs,Os), X) :-
        trigger_props_(Os),
        trigger_props_(Bs),
        (   ground(X) ->
            trigger_props_(Gs)
        ;   true
        ).

trigger_props(fd_props(Gs,Bs,Os)) :-
        trigger_props_(Gs),
        trigger_props_(Bs),
        trigger_props_(Os).

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

trigger_prop(Propagator) :-
        propagator_state(Propagator, State),
        (   State == dead -> true
        ;   get_attr(State, clpfd_aux, queued) -> true
        ;   b_getval('$clpfd_current_propagator', C), C == State -> true
        ;   % passive
            % format("triggering: ~w\n", [Propagator]),
            put_attr(State, clpfd_aux, queued),
            (   arg(1, Propagator, C), functor(C, F, _), global_constraint(F) ->
                push_queue(Propagator, 2)
            ;   push_queue(Propagator, 1)
            )
        ).

kill(State) :- del_attr(State, clpfd_aux), State = dead.

kill(State, Ps) :-
        kill(State),
        maplist(kill_entailed, Ps).

kill_entailed(p(Prop)) :-
        propagator_state(Prop, State),
        kill(State).
kill_entailed(a(V)) :-
        del_attr(V, clpfd).
kill_entailed(a(X,B)) :-
        (   X == B -> true
        ;   del_attr(B, clpfd)
        ).
kill_entailed(a(X,Y,B)) :-
        (   X == B -> true
        ;   Y == B -> true
        ;   del_attr(B, clpfd)
        ).

no_reactivation(rel_tuple(_,_)).
no_reactivation(pdistinct(_)).
no_reactivation(pgcc(_,_,_)).
no_reactivation(pgcc_single(_,_)).
%no_reactivation(scalar_product(_,_,_,_)).

activate_propagator(propagator(P,State)) :-
        % format("running: ~w\n", [P]),
        del_attr(State, clpfd_aux),
        (   no_reactivation(P) ->
            b_setval('$clpfd_current_propagator', State),
            run_propagator(P, State),
            b_setval('$clpfd_current_propagator', [])
        ;   run_propagator(P, State)
        ).

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

portray_propagator(propagator(P,_), F) :- functor(P, F, _).

portray_queue(V, []) :- var(V), !.
portray_queue([P|Ps], [F|Fs]) :-
        portray_propagator(P, F),
        portray_queue(Ps, Fs).

do_queue :-
        % b_getval('$clpfd_queue', H-_),
        % portray_queue(H, Port),
        % format("queue: ~w\n", [Port]),
        (   b_getval('$clpfd_queue_status', enabled) ->
            (   fetch_propagator(Propagator) ->
                activate_propagator(Propagator),
                do_queue
            ;   true
            )
        ;   true
        ).

init_propagator(Var, Prop) :-
        (   fd_get(Var, Dom, Ps0) ->
            insert_propagator(Prop, Ps0, Ps),
            fd_put(Var, Dom, Ps)
        ;   true
        ).

constraint_wake(pneq, ground).
constraint_wake(x_neq_y_plus_z, ground).
constraint_wake(absdiff_neq, ground).
constraint_wake(pdifferent, ground).
constraint_wake(pexclude, ground).
constraint_wake(scalar_product_neq, ground).

constraint_wake(x_leq_y_plus_c, bounds).
constraint_wake(scalar_product_eq, bounds).
constraint_wake(scalar_product_leq, bounds).
constraint_wake(pplus, bounds).
constraint_wake(pgeq, bounds).
constraint_wake(pgcc_single, bounds).
constraint_wake(pgcc_check_single, bounds).

global_constraint(pdistinct).
global_constraint(pgcc).
global_constraint(pgcc_single).
global_constraint(pcircuit).
%global_constraint(rel_tuple).
%global_constraint(scalar_product_eq).

insert_propagator(Prop, Ps0, Ps) :-
        Ps0 = fd_props(Gs,Bs,Os),
        arg(1, Prop, Constraint),
        functor(Constraint, F, _),
        (   constraint_wake(F, ground) ->
            Ps = fd_props([Prop|Gs], Bs, Os)
        ;   constraint_wake(F, bounds) ->
            Ps = fd_props(Gs, [Prop|Bs], Os)
        ;   Ps = fd_props(Gs, Bs, [Prop|Os])
        ).

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

%% lex_chain(+Lists)
%
% Lists are lexicographically non-decreasing.

lex_chain(Lss) :-
        must_be(list(list), Lss),
        maplist(maplist(fd_variable), Lss),
        (   Lss == [] -> true
        ;   Lss = [First|Rest],
            make_propagator(presidual(lex_chain(Lss)), Prop),
            foldl(lex_chain_(Prop), Rest, First, _)
        ).

lex_chain_(Prop, Ls, Prev, Ls) :-
        maplist(prop_init(Prop), Ls),
        lex_le(Prev, 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).
%
% True iff all Tuples are elements of Relation. Each element of the
% list Tuples is a list of integers or finite domain variables.
% Relation is a list of lists of integers. Arbitrary finite relations,
% such as compatibility tables, can be modeled in this way. For
% example, if 1 is compatible with 2 and 5, and 4 is compatible with 0
% and 3:
%
% ==
% ?- tuples_in([[X,Y]], [[1,2],[1,5],[4,0],[4,3]]), X = 4.
% X = 4,
% Y in 0\/3.
% ==
%
% As another example, consider a train schedule represented as a list
% of quadruples, denoting departure and arrival places and times for
% each train. In the following program, Ps is a feasible journey of
% length 3 from A to D via trains that are part of the given schedule.
%
% ==
% trains([[1,2,0,1],
%         [2,3,4,5],
%         [2,3,0,1],
%         [3,4,5,6],
%         [3,4,2,3],
%         [3,4,8,9]]).
%
% threepath(A, D, Ps) :-
%         Ps = [[A,B,_T0,T1],[B,C,T2,T3],[C,D,T4,_T5]],
%         T2 #> T1,
%         T4 #> T3,
%         trains(Ts),
%         tuples_in(Ps, Ts).
% ==
%
% In this example, the unique solution is found without labeling:
%
% ==
% ?- threepath(1, 4, Ps).
% Ps = [[1, 2, 0, 1], [2, 3, 4, 5], [3, 4, 8, 9]].
% ==

tuples_in(Tuples, Relation) :-
        must_be(list(list), Tuples),
        maplist(maplist(fd_variable), Tuples),
        must_be(list(list(integer)), Relation),
        maplist(relation_tuple(Relation), Tuples),
        do_queue.

relation_tuple(Relation, Tuple) :-
        relation_unifiable(Relation, Tuple, Us, _, _),
        (   ground(Tuple) -> memberchk(Tuple, Relation)
        ;   tuple_domain(Tuple, Us),
            (   Tuple = [_,_|_] -> tuple_freeze(Tuple, Us)
            ;   true
            )
        ).

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

tuple_freeze(Tuple, Relation) :-
        put_attr(R, clpfd_relation, Relation),
        make_propagator(rel_tuple(R, Tuple), Prop),
        tuple_freeze_(Tuple, Prop).

tuple_freeze_([], _).
tuple_freeze_([T|Ts], Prop) :-
        (   var(T) ->
            init_propagator(T, Prop),
            trigger_prop(Prop)
        ;   true
        ),
        tuple_freeze_(Ts, 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, true, Changed)
        ).

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

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

% trivial propagator, used only to remember pending constraints
run_propagator(presidual(_), _).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(pdifferent(Left,Right,X,_), MState) :-
        run_propagator(pexclude(Left,Right,X), MState).

run_propagator(weak_distinct(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
            %)
        ).

run_propagator(pexclude(Left,Right,X), _) :-
        (   ground(X) ->
            disable_queue,
            exclude_fire(Left, Right, X),
            enable_queue
        ;   true
        ).

run_propagator(pdistinct(Ls), _MState) :-
        distinct(Ls).

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

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

run_propagator(pelement(N, Is, V), MState) :-
        (   fd_get(N, NDom, _) ->
            (   fd_get(V, VDom, VPs) ->
                integers_remaining(Is, 1, NDom, empty, VDom1),
                domains_intersection(VDom, VDom1, VDom2),
                fd_put(V, VDom2, VPs)
            ;   true
            )
        ;   kill(MState), nth1(N, Is, V)
        ).

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

run_propagator(pgcc_single(Vs, Pairs), _) :- gcc_global(Vs, Pairs).

run_propagator(pgcc_check_single(Pairs), _) :- gcc_check(Pairs).

run_propagator(pgcc_check(Pairs), _) :- gcc_check(Pairs).

run_propagator(pgcc(Vs, _, Pairs), _) :- gcc_global(Vs, Pairs).

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

run_propagator(pcircuit(Vs), _MState) :-
        distinct(Vs),
        propagate_circuit(Vs).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(pneq(A, B), MState) :-
        (   nonvar(A) ->
            (   nonvar(B) -> A =\= B, kill(MState)
            ;   fd_get(B, BD0, BExp0),
                domain_remove(BD0, A, BD1),
                kill(MState),
                fd_put(B, BD1, BExp0)
            )
        ;   nonvar(B) -> run_propagator(pneq(B, A), MState)
        ;   A \== B,
            fd_get(A, _, AI, AS, _), fd_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 -> kill(MState)
        ;   nonvar(A) ->
            (   nonvar(B) -> kill(MState), A >= B
            ;   fd_get(B, BD, BPs),
                domain_remove_greater_than(BD, A, BD1),
                kill(MState),
                fd_put(B, BD1, BPs)
            )
        ;   nonvar(B) ->
            fd_get(A, AD, APs),
            domain_remove_smaller_than(AD, B, AD1),
            kill(MState),
            fd_put(A, AD1, APs)
        ;   fd_get(A, AD, AL, AU, APs),
            fd_get(B, _, BL, BU, _),
            AU cis_geq BL,
            (   AL cis_geq BU -> kill(MState)
            ;   AU == BL -> kill(MState), A = B
            ;   NAL cis max(AL,BL),
                domains_intersection(AD, from_to(NAL,AU), NAD),
                fd_put(A, NAD, APs),
                (   fd_get(B, BD2, BL2, BU2, BPs2) ->
                    NBU cis min(BU2, AU),
                    domains_intersection(BD2, from_to(BL2,NBU), NBD),
                    fd_put(B, NBD, BPs2)
                ;   true
                )
            )
        ).

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

run_propagator(rel_tuple(R, Tuple), MState) :-
        get_attr(R, clpfd_relation, Relation),
        (   ground(Tuple) -> kill(MState), memberchk(Tuple, Relation)
        ;   relation_unifiable(Relation, Tuple, Us, false, Changed),
            Us = [_|_],
            (   Tuple = [First,Second], ( ground(First) ; ground(Second) ) ->
                kill(MState)
            ;   true
            ),
            (   Us = [Single] -> kill(MState), Single = Tuple
            ;   Changed ->
                put_attr(R, clpfd_relation, Us),
                disable_queue,
                tuple_domain(Tuple, Us),
                enable_queue
            ;   true
            )
        ).

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

run_propagator(pserialized(S_I, D_I, S_J, D_J, _), MState) :-
        (   nonvar(S_I), nonvar(S_J) ->
            kill(MState),
            (   S_I + D_I =< S_J -> true
            ;   S_J + D_J =< S_I -> true
            ;   false
            )
        ;   serialize_lower_upper(S_I, D_I, S_J, D_J, MState),
            serialize_lower_upper(S_J, D_J, S_I, D_I, MState)
        ).

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

% abs(X-Y) #\= C
run_propagator(absdiff_neq(X,Y,C), MState) :-
        (   C < 0 -> kill(MState)
        ;   nonvar(X) ->
            kill(MState),
            (   nonvar(Y) -> abs(X - Y) =\= C
            ;   V1 is X - C, neq_num(Y, V1),
                V2 is C + X, neq_num(Y, V2)
            )
        ;   nonvar(Y) -> kill(MState),
            V1 is C + Y, neq_num(X, V1),
            V2 is Y - C, neq_num(X, V2)
        ;   true
        ).

% abs(X-Y) #>= C
run_propagator(absdiff_geq(X,Y,C), MState) :-
        (   C =< 0 -> kill(MState)
        ;   nonvar(X) ->
            kill(MState),
            (   nonvar(Y) -> abs(X-Y) >= C
            ;   P1 is X - C, P2 is X + C,
                Y in inf..P1 \/ P2..sup
            )
        ;   nonvar(Y) ->
            kill(MState),
            P1 is Y - C, P2 is Y + C,
            X in inf..P1 \/ P2..sup
        ;   true
        ).

% X #\= Y + Z
run_propagator(x_neq_y_plus_z(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) ->
                (   nonvar(Z) -> kill(MState), X =\= Y + Z
                ;   kill(MState), XY is X - Y, neq_num(Z, XY)
                )
            ;   nonvar(Z) -> kill(MState), XZ is X - Z, neq_num(Y, XZ)
            ;   true
            )
        ;   nonvar(Y) ->
            (   nonvar(Z) ->
                kill(MState), YZ is Y + Z, neq_num(X, YZ)
            ;   Y =:= 0 -> kill(MState), neq(X, Z)
            ;   true
            )
        ;   Z == 0 -> kill(MState), neq(X, Y)
        ;   true
        ).

% X #=< Y + C
run_propagator(x_leq_y_plus_c(X,Y,C), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), X =< Y + C
            ;   kill(MState),
                R is X - C,
                fd_get(Y, YD, YPs),
                domain_remove_smaller_than(YD, R, YD1),
                fd_put(Y, YD1, YPs)
            )
        ;   nonvar(Y) ->
            kill(MState),
            R is Y + C,
            fd_get(X, XD, XPs),
            domain_remove_greater_than(XD, R, XD1),
            fd_put(X, XD1, XPs)
        ;   (   X == Y -> C >= 0, kill(MState)
            ;   fd_get(Y, YD, _),
                (   domain_supremum(YD, n(YSup)) ->
                    YS1 is YSup + C,
                    fd_get(X, XD, XPs),
                    domain_remove_greater_than(XD, YS1, XD1),
                    fd_put(X, XD1, XPs)
                ;   true
                ),
                (   fd_get(X, XD2, _), domain_infimum(XD2, n(XInf)) ->
                    XI1 is XInf - C,
                    (   fd_get(Y, YD1, YPs1) ->
                        domain_remove_smaller_than(YD1, XI1, YD2),
                        (   domain_infimum(YD2, n(YInf)),
                            domain_supremum(XD2, n(XSup)),
                            XSup =< YInf + C ->
                            kill(MState)
                        ;   true
                        ),
                        fd_put(Y, YD2, YPs1)
                    ;   true
                    )
                ;   true
                )
            )
        ).

run_propagator(scalar_product_neq(Cs0,Vs0,P0), MState) :-
        coeffs_variables_const(Cs0, Vs0, Cs, Vs, 0, I),
        P is P0 - I,
        (   Vs = [] -> kill(MState), P =\= 0
        ;   Vs = [V], Cs = [C] ->
            kill(MState),
            (   C =:= 1 -> neq_num(V, P)
            ;   C*V #\= P
            )
        ;   Cs == [1,-1] -> kill(MState), Vs = [A,B], x_neq_y_plus_z(A, B, P)
        ;   Cs == [-1,1] -> kill(MState), Vs = [A,B], x_neq_y_plus_z(B, A, P)
        ;   P =:= 0, Cs = [1,1,-1] ->
            kill(MState), Vs = [A,B,C], x_neq_y_plus_z(C, A, B)
        ;   P =:= 0, Cs = [1,-1,1] ->
            kill(MState), Vs = [A,B,C], x_neq_y_plus_z(B, A, C)
        ;   P =:= 0, Cs = [-1,1,1] ->
            kill(MState), Vs = [A,B,C], x_neq_y_plus_z(A, B, C)
        ;   true
        ).

run_propagator(scalar_product_leq(Cs0,Vs0,P0), MState) :-
        coeffs_variables_const(Cs0, Vs0, Cs, Vs, 0, I),
        P is P0 - I,
        (   Vs = [] -> kill(MState), P >= 0
        ;   sum_finite_domains(Cs, Vs, Infs, Sups, 0, 0, Inf, Sup),
            D1 is P - Inf,
            disable_queue,
            (   Infs == [], Sups == [] ->
                Inf =< P,
                (   Sup =< P -> kill(MState)
                ;   remove_dist_upper_leq(Cs, Vs, D1)
                )
            ;   Infs == [] -> Inf =< P, remove_dist_upper(Sups, D1)
            ;   Sups = [_], Infs = [_] ->
                remove_upper(Infs, D1)
            ;   Infs = [_] -> remove_upper(Infs, D1)
            ;   true
            ),
            enable_queue
        ).

run_propagator(scalar_product_eq(Cs0,Vs0,P0), MState) :-
        coeffs_variables_const(Cs0, Vs0, Cs, Vs, 0, I),
        P is P0 - I,
        (   Vs = [] -> kill(MState), P =:= 0
        ;   Vs = [V], Cs = [C] -> kill(MState), P mod C =:= 0, V is P // C
        ;   Cs == [1,1] -> kill(MState), Vs = [A,B], A + B #= P
        ;   Cs == [1,-1] -> kill(MState), Vs = [A,B], A #= P + B
        ;   Cs == [-1,1] -> kill(MState), Vs = [A,B], B #= P + A
        ;   Cs == [-1,-1] -> kill(MState), Vs = [A,B], P1 is -P, A + B #= P1
        ;   P =:= 0, Cs == [1,1,-1] -> kill(MState), Vs = [A,B,C], A + B #= C
        ;   P =:= 0, Cs == [1,-1,1] -> kill(MState), Vs = [A,B,C], A + C #= B
        ;   P =:= 0, Cs == [-1,1,1] -> kill(MState), Vs = [A,B,C], B + C #= A
        ;   sum_finite_domains(Cs, Vs, Infs, Sups, 0, 0, Inf, Sup),
            % nl, writeln(Infs-Sups-Inf-Sup),
            D1 is P - Inf,
            D2 is Sup - P,
            disable_queue,
            (   Infs == [], Sups == [] ->
                between(Inf, Sup, P),
                remove_dist_upper_lower(Cs, Vs, D1, D2)
            ;   Sups = [] -> P =< Sup, remove_dist_lower(Infs, D2)
            ;   Infs = [] -> Inf =< P, remove_dist_upper(Sups, D1)
            ;   Sups = [_], Infs = [_] ->
                remove_lower(Sups, D2),
                remove_upper(Infs, D1)
            ;   Infs = [_] -> remove_upper(Infs, D1)
            ;   Sups = [_] -> remove_lower(Sups, D2)
            ;   true
            ),
            enable_queue
        ).

% X + Y = Z
run_propagator(pplus(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   X =:= 0 -> kill(MState), Y = Z
            ;   Y == Z -> kill(MState), X =:= 0
            ;   nonvar(Y) -> kill(MState), Z is X + Y
            ;   nonvar(Z) -> kill(MState), Y is Z - X
            ;   fd_get(Z, ZD, ZPs),
                fd_get(Y, YD, _),
                domain_shift(YD, X, Shifted_YD),
                domains_intersection(ZD, Shifted_YD, ZD1),
                fd_put(Z, ZD1, ZPs),
                (   fd_get(Y, YD1, YPs) ->
                    O is -X,
                    domain_shift(ZD1, O, YD2),
                    domains_intersection(YD1, YD2, YD3),
                    fd_put(Y, YD3, YPs)
                ;   true
                )
            )
        ;   nonvar(Y) -> run_propagator(pplus(Y,X,Z), MState)
        ;   nonvar(Z) ->
            (   X == Y -> kill(MState), even(Z), X is Z // 2
            ;   fd_get(X, XD, _),
                fd_get(Y, YD, YPs),
                domain_negate(XD, XDN),
                domain_shift(XDN, Z, YD1),
                domains_intersection(YD, YD1, YD2),
                fd_put(Y, YD2, YPs),
                (   fd_get(X, XD1, XPs) ->
                    domain_negate(YD2, YD2N),
                    domain_shift(YD2N, Z, XD2),
                    domains_intersection(XD1, XD2, XD3),
                    fd_put(X, XD3, XPs)
                ;   true
                )
            )
        ;   (   X == Y -> kill(MState), 2*X #= Z
            ;   X == Z -> kill(MState), Y = 0
            ;   Y == Z -> kill(MState), X = 0
            ;   fd_get(X, XD, XL, XU, XPs),
                fd_get(Y, _, YL, YU, _),
                fd_get(Z, _, ZL, ZU, _),
                NXL cis max(XL, ZL-YU),
                NXU cis min(XU, ZU-YL),
                update_bounds(X, XD, XPs, XL, XU, NXL, NXU),
                (   fd_get(Y, YD2, YL2, YU2, YPs2) ->
                    NYL cis max(YL2, ZL-NXU),
                    NYU cis min(YU2, ZU-NXL),
                    update_bounds(Y, YD2, YPs2, YL2, YU2, NYL, NYU)
                ;   NYL = n(Y), NYU = n(Y)
                ),
                (   fd_get(Z, ZD2, ZL2, ZU2, ZPs2) ->
                    NZL cis max(ZL2,NXL+NYL),
                    NZU cis min(ZU2,NXU+NYU),
                    update_bounds(Z, ZD2, ZPs2, ZL2, ZU2, NZL, NZU)
                ;   true
                )
            )
        ).

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

run_propagator(ptimes(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Z is X * Y
            ;   X =:= 0 -> kill(MState), Z = 0
            ;   X =:= 1 -> kill(MState), Z = Y
            ;   nonvar(Z) -> kill(MState), 0 =:= Z mod X, Y is Z // X
            ;   (   Y == Z -> kill(MState), Y = 0
                ;   fd_get(Y, YD, _),
                    fd_get(Z, ZD, ZPs),
                    domain_expand(YD, X, Scaled_YD),
                    domains_intersection(ZD, Scaled_YD, ZD1),
                    fd_put(Z, ZD1, ZPs),
                    (   fd_get(Y, YDom2, YPs2) ->
                        domain_contract(ZD1, X, Contract),
                        domains_intersection(YDom2, Contract, NYDom),
                        fd_put(Y, NYDom, YPs2)
                    ;   kill(MState), Z is X * Y
                    )
                )
            )
        ;   nonvar(Y) -> run_propagator(ptimes(Y,X,Z), MState)
        ;   nonvar(Z) ->
            (   X == Y ->
                kill(MState),
                integer_kth_root(Z, 2, R),
                NR is -R,
                X in NR \/ R
            ;   fd_get(X, XD, XL, XU, XPs),
                fd_get(Y, YD, YL, YU, _),
                min_max_factor(n(Z), n(Z), YL, YU, XL, XU, NXL, NXU),
                update_bounds(X, XD, XPs, XL, XU, NXL, NXU),
                (   fd_get(Y, YD2, YL2, YU2, YPs2) ->
                    min_max_factor(n(Z), n(Z), NXL, NXU, YL2, YU2, NYL, NYU),
                    update_bounds(Y, YD2, YPs2, YL2, YU2, NYL, NYU)
                ;   (   Y =\= 0 -> 0 =:= Z mod Y, kill(MState), X is Z // Y
                    ;   kill(MState), Z = 0
                    )
                ),
                (   Z =:= 0 ->
                    (   \+ domain_contains(XD, 0) -> kill(MState), Y = 0
                    ;   \+ domain_contains(YD, 0) -> kill(MState), X = 0
                    ;   true
                    )
                ;  neq_num(X, 0), neq_num(Y, 0)
                )
            )
        ;   (   X == Y -> kill(MState), X^2 #= Z
            ;   fd_get(X, XD, XL, XU, XPs),
                fd_get(Y, _, YL, YU, _),
                fd_get(Z, ZD, ZL, ZU, _),
                (   Y == Z, \+ domain_contains(ZD, 0) -> kill(MState), X = 1
                ;   X == Z, \+ domain_contains(ZD, 0) -> kill(MState), Y = 1
                ;   min_max_factor(ZL, ZU, YL, YU, XL, XU, NXL, NXU),
                    update_bounds(X, XD, XPs, XL, XU, NXL, NXU),
                    (   fd_get(Y, YD2, YL2, YU2, YPs2) ->
                        min_max_factor(ZL, ZU, NXL, NXU, YL2, YU2, NYL, NYU),
                        update_bounds(Y, YD2, YPs2, YL2, YU2, NYL, NYU)
                    ;   NYL = n(Y), NYU = n(Y)
                    ),
                    (   fd_get(Z, ZD2, ZL2, ZU2, ZPs2) ->
                        min_product(NXL, NXU, NYL, NYU, NZL),
                        max_product(NXL, NXU, NYL, NYU, NZU),
                        (   NZL cis_leq ZL2, NZU cis_geq ZU2 -> ZD3 = ZD2
                        ;   domains_intersection(ZD2, from_to(NZL,NZU), ZD3),
                            fd_put(Z, ZD3, ZPs2)
                        ),
                        (   domain_contains(ZD3, 0) ->  true
                        ;   neq_num(X, 0), neq_num(Y, 0)
                        )
                    ;   true
                    )
                )
            )
        ).

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

% X div Y = Z
run_propagator(pdiv(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Y =\= 0, Z is X div Y
            ;   fd_get(Y, YD, YL, YU, YPs),
                (   nonvar(Z) ->
                    (   Z =:= 0 ->
                        (   X =:= 0 -> NI = split(0, from_to(inf,n(-1)),
                                                  from_to(n(1),sup))
                        ;   NY_ is X+sign(X),
                            (   X > 0 -> NI = from_to(n(NY_), sup)
                            ;   NI = from_to(inf, n(NY_))
                            )
                        ),
                        domains_intersection(YD, NI, NYD),
                        fd_put(Y, NYD, YPs)
                    ;   (   sign(X) =:= 1 ->
                            NYL cis max(div(n(X)*n(Z), n(Z)*(n(Z)+n(1))) + n(1), YL),
                            NYU cis min(div(n(X), n(Z)), YU)
                        ;   NYL cis max(-(div(-n(X), n(Z))), YL),
                            NYU cis min(-(div(-n(X)*n(Z), (n(Z)*(n(Z)+n(1))))) - n(1), YU)
                        ),
                        update_bounds(Y, YD, YPs, YL, YU, NYL, NYU)
                    )
                ;   fd_get(Z, ZD, ZL, ZU, ZPs),
                    (   X >= 0, ( YL cis_gt n(0) ; YU cis_lt n(0) )->
                        NZL cis max(div(n(X), YU), ZL),
                        NZU cis min(div(n(X), YL), ZU)
                    ;   X < 0, ( YL cis_gt n(0) ; YU cis_lt n(0) ) ->
                        NZL cis max(div(n(X), YL), ZL),
                        NZU cis min(div(n(X), YU), ZU)
                    ;   % TODO: more stringent bounds, cover Y
                        NZL cis max(-abs(n(X)), ZL),
                        NZU cis min(abs(n(X)), ZU)
                    ),
                    update_bounds(Z, ZD, ZPs, ZL, ZU, NZL, NZU),
                    (   X >= 0, NZL cis_gt n(0), fd_get(Y, YD1, YPs1) ->
                        NYL cis div(n(X), (NZU + n(1))) + n(1),
                        NYU cis div(n(X), NZL),
                        domains_intersection(YD1, from_to(NYL, NYU), NYD1),
                        fd_put(Y, NYD1, YPs1)
                    ;   % TODO: more cases
                        true
                    )
                )
            )
        ;   nonvar(Y) ->
            Y =\= 0,
            (   Y =:= 1 -> kill(MState), X = Z
            ;   Y =:= -1 -> kill(MState), Z #= -X
            ;   fd_get(X, XD, XL, XU, XPs),
                (   nonvar(Z) ->
                    kill(MState),
                    (   Y > 0 ->
                        NXL cis max(n(Z)*n(Y), XL),
                        NXU cis min((n(Z)+n(1))*n(Y)-n(1), XU)
                    ;   NXL cis max((n(Z)+n(1))*n(Y)+n(1), XL),
                        NXU cis min(n(Z)*n(Y), XU)
                    ),
                    update_bounds(X, XD, XPs, XL, XU, NXL, NXU)
                ;   fd_get(Z, ZD, ZPs),
                    domain_contract_less(XD, Y, div, Contracted),
                    domains_intersection(ZD, Contracted, NZD),
                    fd_put(Z, NZD, ZPs),
                    (   fd_get(X, XD2, XPs2) ->
                        domain_expand_more(NZD, Y, div, Expanded),
                        domains_intersection(XD2, Expanded, NXD2),
                        fd_put(X, NXD2, XPs2)
                    ;   true
                    )
                )
            )
        ;   nonvar(Z) ->
            fd_get(X, XD, XL, XU, XPs),
            fd_get(Y, _, YL, YU, _),
            (   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
            ),
            update_bounds(X, XD, XPs, XL, XU, NXL, NXU)
        ;   (   X == Y -> kill(MState), Z = 1
            ;   fd_get(X, _, XL, XU, _),
                fd_get(Y, _, YL, _, _),
                fd_get(Z, ZD, ZPs),
                NZU cis max(abs(XL), XU),
                NZL cis -NZU,
                domains_intersection(ZD, from_to(NZL,NZU), NZD0),
                (   XL cis_geq n(0), YL cis_geq n(0) ->
                    domain_remove_smaller_than(NZD0, 0, NZD1)
                ;   % TODO: cover more cases
                    NZD1 = NZD0
                ),
                fd_put(Z, NZD1, ZPs)
            )
        ).

% X rdiv Y = Z
run_propagator(prdiv(X,Y,Z), MState) :- kill(MState), Z*Y #= X.

% X // Y = Z (round towards zero)
run_propagator(ptzdiv(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Y =\= 0, Z is X // Y
            ;   fd_get(Y, YD, YL, YU, YPs),
                (   nonvar(Z) ->
                    (   Z =:= 0 ->
                        NYL is -abs(X) - 1,
                        NYU is abs(X) + 1,
                        domains_intersection(YD, split(0, from_to(inf,n(NYL)),
                                                       from_to(n(NYU), sup)),
                                             NYD),
                        fd_put(Y, NYD, YPs)
                    ;   (   sign(X) =:= sign(Z) ->
                            NYL cis max(n(X) // (n(Z)+sign(n(Z))) + n(1), YL),
                            NYU cis min(n(X) // n(Z), YU)
                        ;   NYL cis max(n(X) // n(Z), YL),
                            NYU cis min(n(X) // (n(Z)+sign(n(Z))) - n(1), YU)
                        ),
                        update_bounds(Y, YD, YPs, YL, YU, NYL, NYU)
                    )
                ;   fd_get(Z, ZD, ZL, ZU, ZPs),
                    (   X >= 0, ( YL cis_gt n(0) ; YU cis_lt n(0) )->
                        NZL cis max(n(X)//YU, ZL),
                        NZU cis min(n(X)//YL, ZU)
                    ;   X < 0, ( YL cis_gt n(0) ; YU cis_lt n(0) ) ->
                        NZL cis max(n(X)//YL, ZL),
                        NZU cis min(n(X)//YU, ZU)
                    ;   % TODO: more stringent bounds, cover Y
                        NZL cis max(-abs(n(X)), ZL),
                        NZU cis min(abs(n(X)), ZU)
                    ),
                    update_bounds(Z, ZD, ZPs, ZL, ZU, NZL, NZU),
                    (   X >= 0, NZL cis_gt n(0), fd_get(Y, YD1, YPs1) ->
                        NYL cis n(X) // (NZU + n(1)) + n(1),
                        NYU cis n(X) // NZL,
                        domains_intersection(YD1, from_to(NYL, NYU), NYD1),
                        fd_put(Y, NYD1, YPs1)
                    ;   % TODO: more cases
                        true
                    )
                )
            )
        ;   nonvar(Y) ->
            Y =\= 0,
            (   Y =:= 1 -> kill(MState), X = Z
            ;   Y =:= -1 -> kill(MState), Z #= -X
            ;   fd_get(X, XD, XL, XU, XPs),
                (   nonvar(Z) ->
                    kill(MState),
                    (   sign(Z) =:= sign(Y) ->
                        NXL cis max(n(Z)*n(Y), XL),
                        NXU cis min((abs(n(Z))+n(1))*abs(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)
                    ;   NXL cis max((n(Z)+sign(n(Z)))*n(Y)+n(1), XL),
                        NXU cis min(n(Z)*n(Y), XU)
                    ),
                    update_bounds(X, XD, XPs, XL, XU, NXL, NXU)
                ;   fd_get(Z, ZD, ZPs),
                    domain_contract_less(XD, Y, //, Contracted),
                    domains_intersection(ZD, Contracted, NZD),
                    fd_put(Z, NZD, ZPs),
                    (   fd_get(X, XD2, XPs2) ->
                        domain_expand_more(NZD, Y, //, Expanded),
                        domains_intersection(XD2, Expanded, NXD2),
                        fd_put(X, NXD2, XPs2)
                    ;   true
                    )
                )
            )
        ;   nonvar(Z) ->
            fd_get(X, XD, XL, XU, XPs),
            fd_get(Y, _, YL, YU, _),
            (   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
            ),
            update_bounds(X, XD, XPs, XL, XU, NXL, NXU)
        ;   (   X == Y -> kill(MState), Z = 1
            ;   fd_get(X, _, XL, XU, _),
                fd_get(Y, _, YL, _, _),
                fd_get(Z, ZD, ZPs),
                NZU cis max(abs(XL), XU),
                NZL cis -NZU,
                domains_intersection(ZD, from_to(NZL,NZU), NZD0),
                (   XL cis_geq n(0), YL cis_geq n(0) ->
                    domain_remove_smaller_than(NZD0, 0, NZD1)
                ;   % TODO: cover more cases
                    NZD1 = NZD0
                ),
                fd_put(Z, NZD1, ZPs)
            )
        ).


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

run_propagator(pabs(X,Y), MState) :-
        (   nonvar(X) -> kill(MState), Y is abs(X)
        ;   nonvar(Y) ->
            kill(MState),
            Y >= 0,
            YN is -Y,
            X in YN \/ Y
        ;   fd_get(X, XD, XPs),
            fd_get(Y, YD, _),
            domain_negate(YD, YDNegative),
            domains_union(YD, YDNegative, XD1),
            domains_intersection(XD, XD1, XD2),
            fd_put(X, XD2, XPs),
            (   fd_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),
                fd_put(Y, YD4, YPs1)
            ;   true
            )
        ).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Z = X mod Y

run_propagator(pmod(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Y =\= 0, Z is X mod Y
            ;   true
            )
        ;   nonvar(Y) ->
            Y =\= 0,
            (   abs(Y) =:= 1 -> kill(MState), Z = 0
            ;   var(Z) ->
                YP is abs(Y) - 1,
                (   Y > 0, fd_get(X, _, n(XL), n(XU), _) ->
                    (   XL >= 0, XU < Y ->
                        kill(MState), Z = X, ZL = XL, ZU = XU
                    ;   ZL = 0, ZU = YP
                    )
                ;   Y > 0 -> ZL = 0, ZU = YP
                ;   YN is -YP, ZL = YN, ZU = 0
                ),
                (   fd_get(Z, ZD, ZPs) ->
                    domains_intersection(ZD, from_to(n(ZL), n(ZU)), ZD1),
                    domain_infimum(ZD1, n(ZMin)),
                    domain_supremum(ZD1, n(ZMax)),
                    fd_put(Z, ZD1, ZPs)
                ;   ZMin = Z, ZMax = Z
                ),
                (   fd_get(X, XD, XPs), domain_infimum(XD, n(XMin)) ->
                    Z1 is XMin mod Y,
                    (   between(ZMin, ZMax, Z1) -> true
                    ;   Y > 0 ->
                        Next is ((XMin - ZMin + Y - 1) div Y)*Y + ZMin,
                        domain_remove_smaller_than(XD, Next, XD1),
                        fd_put(X, XD1, XPs)
                    ;   neq_num(X, XMin)
                    )
                ;   true
                ),
                (   fd_get(X, XD2, XPs2), domain_supremum(XD2, n(XMax)) ->
                    Z2 is XMax mod Y,
                    (   between(ZMin, ZMax, Z2) -> true
                    ;   Y > 0 ->
                        Prev is ((XMax - ZMin) div Y)*Y + ZMax,
                        domain_remove_greater_than(XD2, Prev, XD3),
                        fd_put(X, XD3, XPs2)
                    ;   neq_num(X, XMax)
                    )
                ;   true
                )
            ;   fd_get(X, XD, XPs),
                % if possible, propagate at the boundaries
                (   domain_infimum(XD, n(Min)) ->
                    (   Min mod Y =:= Z -> true
                    ;   Y > 0 ->
                        Next is ((Min - Z + Y - 1) div Y)*Y + Z,
                        domain_remove_smaller_than(XD, Next, XD1),
                        fd_put(X, XD1, XPs)
                    ;   neq_num(X, Min)
                    )
                ;   true
                ),
                (   fd_get(X, XD2, XPs2) ->
                    (   domain_supremum(XD2, n(Max)) ->
                        (   Max mod Y =:= Z -> true
                        ;   Y > 0 ->
                            Prev is ((Max - Z) div Y)*Y + Z,
                            domain_remove_greater_than(XD2, Prev, XD3),
                            fd_put(X, XD3, XPs2)
                        ;   neq_num(X, Max)
                        )
                    ;   true
                    )
                ;   true
                )
            )
        ;   X == Y -> kill(MState), Z = 0
        ;   true % TODO: propagate more
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Z = X rem Y

run_propagator(prem(X,Y,Z), MState) :-
        (   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Y =\= 0, Z is X rem Y
            ;   U is abs(X),
                fd_get(Y, YD, _),
                (   X >=0, domain_infimum(YD, n(Min)), Min >= 0 -> L = 0
                ;   L is -U
                ),
                Z in L..U
            )
        ;   nonvar(Y) ->
            Y =\= 0,
            (   abs(Y) =:= 1 -> kill(MState), Z = 0
            ;   var(Z) ->
                YP is abs(Y) - 1,
                YN is -YP,
                (   Y > 0, fd_get(X, _, n(XL), n(XU), _) ->
                    (   abs(XL) < Y, XU < Y -> kill(MState), Z = X, ZL = XL
                    ;   XL < 0, abs(XL) < Y -> ZL = XL
                    ;   XL >= 0 -> ZL = 0
                    ;   ZL = YN
                    ),
                    (   XU > 0, XU < Y -> ZU = XU
                    ;   XU < 0 -> ZU = 0
                    ;   ZU = YP
                    )
                ;   ZL = YN, ZU = YP
                ),
                (   fd_get(Z, ZD, ZPs) ->
                    domains_intersection(ZD, from_to(n(ZL), n(ZU)), ZD1),
                    fd_put(Z, ZD1, ZPs)
                ;   ZD1 = from_to(n(Z), n(Z))
                ),
                (   fd_get(X, XD, _), domain_infimum(XD, n(Min)) ->
                    Z1 is Min rem Y,
                    (   domain_contains(ZD1, Z1) -> true
                    ;   neq_num(X, Min)
                    )
                ;   true
                ),
                (   fd_get(X, XD1, _), domain_supremum(XD1, n(Max)) ->
                    Z2 is Max rem Y,
                    (   domain_contains(ZD1, Z2) -> true
                    ;   neq_num(X, Max)
                    )
                ;   true
                )
            ;   fd_get(X, XD1, XPs1),
                % if possible, propagate at the boundaries
                (   domain_infimum(XD1, n(Min)) ->
                    (   Min rem Y =:= Z -> true
                    ;   Y > 0, Min > 0 ->
                        Next is ((Min - Z + Y - 1) div Y)*Y + Z,
                        domain_remove_smaller_than(XD1, Next, XD2),
                        fd_put(X, XD2, XPs1)
                    ;   % TODO: bigger steps in other cases as well
                        neq_num(X, Min)
                    )
                ;   true
                ),
                (   fd_get(X, XD3, XPs3) ->
                    (   domain_supremum(XD3, n(Max)) ->
                        (   Max rem Y =:= Z -> true
                        ;   Y > 0, Max > 0  ->
                            Prev is ((Max - Z) div Y)*Y + Z,
                            domain_remove_greater_than(XD3, Prev, XD4),
                            fd_put(X, XD4, XPs3)
                        ;   % TODO: bigger steps in other cases as well
                            neq_num(X, Max)
                        )
                    ;   true
                    )
                ;   true
                )
            )
        ;   X == Y -> kill(MState), Z = 0
        ;   fd_get(Z, ZD, ZPs) ->
            fd_get(Y, _, YInf, YSup, _),
            fd_get(X, _, XInf, XSup, _),
            M cis max(abs(YInf),YSup),
            (   XInf cis_geq n(0) -> Inf0 = n(0)
            ;   Inf0 = XInf
            ),
            (   XSup cis_leq n(0) -> Sup0 = n(0)
            ;   Sup0 = XSup
            ),
            NInf cis max(max(Inf0, -M + n(1)), min(XInf,-XSup)),
            NSup cis min(min(Sup0, M - n(1)), max(abs(XInf),XSup)),
            domains_intersection(ZD, from_to(NInf,NSup), ZD1),
            fd_put(Z, ZD1, ZPs)
        ;   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
                ;   false % Z < X
                )
            ;   fd_get(Y, _, YInf, YSup, _),
                (   YInf cis_gt n(X) -> Z = Y
                ;   YSup cis_lt n(X) -> Z = X
                ;   YSup = n(M) ->
                    fd_get(Z, ZD, ZPs),
                    domain_remove_greater_than(ZD, M, ZD1),
                    fd_put(Z, ZD1, ZPs)
                ;   true
                )
            )
        ;   nonvar(Y) -> run_propagator(pmax(Y,X,Z), MState)
        ;   fd_get(Z, ZD, ZPs) ->
            fd_get(X, _, XInf, XSup, _),
            fd_get(Y, _, YInf, YSup, _),
            (   YInf cis_gt YSup -> kill(MState), Z = Y
            ;   YSup cis_lt XInf -> kill(MState), Z = X
            ;   n(M) cis max(XSup, YSup) ->
                domain_remove_greater_than(ZD, M, ZD1),
                fd_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 -> kill(MState), X #=< Y
                ;   Z < X -> Z = Y
                ;   false % Z > X
                )
            ;   fd_get(Y, _, YInf, YSup, _),
                (   YSup cis_lt n(X) -> Z = Y
                ;   YInf cis_gt n(X) -> Z = X
                ;   YInf = n(M) ->
                    fd_get(Z, ZD, ZPs),
                    domain_remove_smaller_than(ZD, M, ZD1),
                    fd_put(Z, ZD1, ZPs)
                ;   true
                )
            )
        ;   nonvar(Y) -> run_propagator(pmin(Y,X,Z), MState)
        ;   fd_get(Z, ZD, ZPs) ->
            fd_get(X, _, XInf, XSup, _),
            fd_get(Y, _, YInf, YSup, _),
            (   YSup cis_lt YInf -> kill(MState), Z = Y
            ;   YInf cis_gt XSup -> kill(MState), Z = X
            ;   n(M) cis min(XInf, YInf) ->
                domain_remove_smaller_than(ZD, M, ZD1),
                fd_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 in 0..1, Z #<==> Y #= 0
        ;   Y == 0 -> kill(MState), Z = 1
        ;   Y == 1 -> kill(MState), Z = X
        ;   nonvar(X) ->
            (   nonvar(Y) ->
                (   Y >= 0 -> true ; X =:= -1 ),
                kill(MState),
                Z is X^Y
            ;   nonvar(Z) ->
                (   Z > 1 ->
                    kill(MState),
                    integer_log_b(Z, X, 1, Y)
                ;   true
                )
            ;   fd_get(Y, _, YL, YU, _),
                fd_get(Z, ZD, ZPs),
                (   X > 0, YL cis_geq n(0) ->
                    NZL cis n(X)^YL,
                    NZU cis n(X)^YU,
                    domains_intersection(ZD, from_to(NZL,NZU), NZD),
                    fd_put(Z, NZD, ZPs)
                ;   true
                ),
                (   X > 0,
                    fd_get(Z, _, _, n(ZMax), _),
                    ZMax > 0 ->
                    floor_integer_log_b(ZMax, X, 1, YCeil),
                    Y in inf..YCeil
                ;   true
                )
            )
        ;   nonvar(Z) ->
            (   nonvar(Y) ->
                integer_kth_root(Z, Y, R),
                kill(MState),
                (   even(Y) ->
                    N is -R,
                    X in N \/ R
                ;   X = R
                )
            ;   fd_get(X, _, n(NXL), _, _), NXL > 1 ->
                (   Z > 1, between(NXL, Z, Exp), NXL^Exp > Z ->
                    Exp1 is Exp - 1,
                    fd_get(Y, YD, YPs),
                    domains_intersection(YD, from_to(n(1),n(Exp1)), YD1),
                    fd_put(Y, YD1, YPs),
                    (   fd_get(X, XD, XPs) ->
                        domain_infimum(YD1, n(YL)),
                        integer_kth_root_leq(Z, YL, RU),
                        domains_intersection(XD, from_to(n(NXL),n(RU)), XD1),
                        fd_put(X, XD1, XPs)
                    ;   true
                    )
                ;   true
                )
            ;   true
            )
        ;   nonvar(Y), Y > 0 ->
            (   even(Y) ->
                geq(Z, 0)
            ;   true
            ),
            (   fd_get(X, XD, XL, XU, _), fd_get(Z, ZD, ZL, ZU, ZPs) ->
                (   domain_contains(ZD, 0) -> XD1 = XD
                ;   domain_remove(XD, 0, XD1)
                ),
                (   domain_contains(XD, 0) -> ZD1 = ZD
                ;   domain_remove(ZD, 0, ZD1)
                ),
                (   even(Y) ->
                    (   XL cis_geq n(0) ->
                        NZL cis XL^n(Y)
                    ;   XU cis_leq n(0) ->
                        NZL cis XU^n(Y)
                    ;   NZL = n(0)
                    ),
                    NZU cis max(abs(XL),abs(XU))^n(Y),
                    domains_intersection(ZD1, from_to(NZL,NZU), ZD2)
                ;   (   finite(XL) ->
                        NZL cis XL^n(Y),
                        NZU cis XU^n(Y),
                        domains_intersection(ZD1, from_to(NZL,NZU), ZD2)
                    ;   ZD2 = ZD1
                    )
                ),
                fd_put(Z, ZD2, ZPs),
                (   even(Y), ZU = n(Num) ->
                    integer_kth_root_leq(Num, Y, RU),
                    (   XL cis_geq n(0), ZL = n(Num1) ->
                        integer_kth_root_leq(Num1, Y, RL0),
                        (   RL0^Y < Num1 -> RL is RL0 + 1
                        ;   RL = RL0
                        )
                    ;   RL is -RU
                    ),
                    RL =< RU,
                    NXD = from_to(n(RL),n(RU))
                ;   odd(Y), ZL cis_geq n(0), ZU = n(Num) ->
                    integer_kth_root_leq(Num, Y, RU),
                    ZL = n(Num1),
                    integer_kth_root_leq(Num1, Y, RL0),
                    (   RL0^Y < Num1 -> RL is RL0 + 1
                    ;   RL = RL0
                    ),
                    RL =< RU,
                    NXD = from_to(n(RL),n(RU))
                ;   NXD = XD1   % TODO: propagate more
                ),
                (   fd_get(X, XD2, XPs) ->
                    domains_intersection(XD2, XD1, XD3),
                    domains_intersection(XD3, NXD, XD4),
                    fd_put(X, XD4, XPs)
                ;   true
                )
            ;   true
            )
        ;   fd_get(X, _, XL, _, _),
            XL cis_gt n(0),
            fd_get(Y, _, YL, _, _),
            YL cis_gt n(0),
            fd_get(Z, ZD, ZPs) ->
            n(NZL) cis XL^YL,
            domain_remove_smaller_than(ZD, NZL, ZD1),
            fd_put(Z, ZD1, ZPs)
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Z = X << (Y*S)

run_propagator(pshift(X,Y,Z,S), MState) :-
        (   Y == 0 -> kill(MState), Z = X
        ;   nonvar(X) ->
            (   nonvar(Y) -> kill(MState), Z is X << (Y*S)
            ;   nonvar(Z) ->
                kill(MState),
                (   X =:= 0 -> Z =:= 0
                ;   abs(Z) > abs(X) -> Z #= X * 2^(Y*S)
                ;   X div (2^(-Y*S)) #= Z
                )
            ;   % TODO: handle these cases
                true
            )
        ;   nonvar(Y) ->
            kill(MState),
            (   Y*S > 0 -> Z #= X * 2^(Y*S)
            ;   X div (2^(-Y*S)) #= Z
            )
        ;   % TODO: handle these cases
            true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(pzcompare(Order, A, B), MState) :-
        (   A == B -> kill(MState), Order = (=)
        ;   (   nonvar(A) ->
                (   nonvar(B) ->
                    kill(MState),
                    (   A > B -> Order = (>)
                    ;   Order = (<)
                    )
                ;   fd_get(B, _, BL, BU, _),
                    (   BL cis_gt n(A) -> kill(MState), Order = (<)
                    ;   BU cis_lt n(A) -> kill(MState), Order = (>)
                    ;   true
                    )
                )
            ;   nonvar(B) ->
                fd_get(A, _, AL, AU, _),
                (   AL cis_gt n(B) -> kill(MState), Order = (>)
                ;   AU cis_lt n(B) -> kill(MState), Order = (<)
                ;   true
                )
            ;   fd_get(A, _, AL, AU, _),
                fd_get(B, _, BL, BU, _),
                (   AL cis_gt BU -> kill(MState), Order = (>)
                ;   AU cis_lt BL -> kill(MState), Order = (<)
                ;   true
                )
            )
        ).

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

% reified constraints

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

run_propagator(reified_in(V,Dom,B), MState) :-
        (   integer(V) ->
            kill(MState),
            (   domain_contains(Dom, V) -> B = 1
            ;   B = 0
            )
        ;   B == 1 -> kill(MState), domain(V, Dom)
        ;   B == 0 -> kill(MState), domain_complement(Dom, C), domain(V, C)
        ;   fd_get(V, VD, _),
            (   domains_intersection(VD, Dom, I) ->
                (   I == VD -> kill(MState), B = 1
                ;   true
                )
            ;   kill(MState), B = 0
            )
        ).

run_propagator(reified_tuple_in(Tuple, R, B), MState) :-
        get_attr(R, clpfd_relation, Relation),
        (   B == 1 -> kill(MState), tuples_in([Tuple], Relation)
        ;   (   ground(Tuple) ->
                kill(MState),
                (   memberchk(Tuple, Relation) -> B = 1
                ;   B = 0
                )
            ;   relation_unifiable(Relation, Tuple, Us, _, _),
                (   Us = [] -> kill(MState), B = 0
                ;   true
                )
            )
        ).

run_propagator(tuples_not_in(Tuples, Relation, B), MState) :-
        (   B == 0 ->
            kill(MState),
            tuples_in_conjunction(Tuples, Relation, Conj),
            #\ Conj
        ;   true
        ).

run_propagator(kill_reified_tuples(B, Ps, Bs), _) :-
        (   B == 0 ->
            maplist(kill_entailed, Ps),
            phrase(as(Bs), As),
            maplist(kill_entailed, As)
        ;   true
        ).

run_propagator(reified_fd(V,B), MState) :-
        (   fd_inf(V, I), I \== inf, fd_sup(V, S), S \== sup ->
            kill(MState),
            B = 1
        ;   B == 0 ->
            (   fd_inf(V, inf) -> true
            ;   fd_sup(V, sup) -> true
            ;   false
            )
        ;   true
        ).

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

run_propagator(pskeleton(X,Y,D,Skel,Z,_), MState) :-
        (   Y == 0 -> kill(MState), D = 0
        ;   D == 1 -> kill(MState), neq_num(Y, 0), skeleton([X,Y,Z], Skel)
        ;   integer(Y), Y =\= 0 -> kill(MState), D = 1, skeleton([X,Y,Z], Skel)
        ;   fd_get(Y, YD, _), \+ domain_contains(YD, 0) ->
            kill(MState),
            D = 1, skeleton([X,Y,Z], Skel)
        ;   true
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Propagators for arithmetic functions that only propagate
   functionally. These are currently the bitwise operations.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

run_propagator(pfunction(Op,A,B,R), MState) :-
        (   integer(A), integer(B) ->
            kill(MState),
            Expr =.. [Op,A,B],
            R is Expr
        ;   true
        ).
run_propagator(pfunction(Op,A,R), MState) :-
        (   integer(A) ->
            kill(MState),
            Expr =.. [Op,A],
            R is Expr
        ;   true
        ).

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

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(reified_eq(DX,X,DY,Y,Ps,B), MState) :-
        (   DX == 0 -> kill(MState, Ps), B = 0
        ;   DY == 0 -> kill(MState, Ps), 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)
                    ;   fd_get(Y, YD, _),
                        (   domain_contains(YD, X) -> true
                        ;   kill(MState, Ps), B = 0
                        )
                    )
                ;   nonvar(Y) -> run_propagator(reified_eq(DY,Y,DX,X,Ps,B), MState)
                ;   X == Y -> kill(MState), B = 1
                ;   fd_get(X, _, XL, XU, _),
                    fd_get(Y, _, YL, YU, _),
                    (   XL cis_gt YU -> kill(MState, Ps), B = 0
                    ;   YL cis_gt XU -> kill(MState, Ps), B = 0
                    ;   true
                    )
                )
            ;   B =:= 0 -> kill(MState), X #\= Y
            ;   true
            )
        ;   true
        ).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(reified_neq(DX,X,DY,Y,Ps,B), MState) :-
        (   DX == 0 -> kill(MState, Ps), B = 0
        ;   DY == 0 -> kill(MState, Ps), 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)
                    ;   fd_get(Y, YD, _),
                        (   domain_contains(YD, X) -> true
                        ;   kill(MState, Ps), B = 1
                        )
                    )
                ;   nonvar(Y) -> run_propagator(reified_neq(DY,Y,DX,X,Ps,B), MState)
                ;   X == Y -> kill(MState), B = 0
                ;   fd_get(X, _, XL, XU, _),
                    fd_get(Y, _, YL, YU, _),
                    (   XL cis_gt YU -> kill(MState, Ps), B = 1
                    ;   YL cis_gt XU -> kill(MState, Ps), B = 1
                    ;   true
                    )
                )
            ;   B =:= 0 -> kill(MState), X = Y
            ;   true
            )
        ;   true
        ).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(reified_and(X,Ps1,Y,Ps2,B), MState) :-
        (   nonvar(X) ->
            kill(MState),
            (   X =:= 0 -> maplist(kill_entailed, Ps2), B = 0
            ;   B = Y
            )
        ;   nonvar(Y) -> run_propagator(reified_and(Y,Ps2,X,Ps1,B), MState)
        ;   B == 1 -> kill(MState), X = 1, Y = 1
        ;   true
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
run_propagator(reified_or(X,Ps1,Y,Ps2,B), MState) :-
        (   nonvar(X) ->
            kill(MState),
            (   X =:= 1 -> maplist(kill_entailed, Ps2), B = 1
            ;   B = Y
            )
        ;   nonvar(Y) -> run_propagator(reified_or(Y,Ps2,X,Ps1,B), MState)
        ;   B == 0 -> kill(MState), X = 0, Y = 0
        ;   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, Ps), MState) :-
        (   nonvar(X) ->
            kill(MState),
            (   X =:= 1 -> Y = 1
            ;   maplist(kill_entailed, Ps)
            )
        ;   nonvar(Y) ->
            kill(MState),
            (   Y =:= 0 -> X = 0
            ;   maplist(kill_entailed, Ps)
            )
        ;   true
        ).

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

update_bounds(X, XD, XPs, XL, XU, NXL, NXU) :-
        (   NXL == XL, NXU == XU -> true
        ;   domains_intersection(XD, from_to(NXL, NXU), NXD),
            fd_put(X, NXD, XPs)
        ).

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

finite(n(_)).

in_(L, U, X) :-
        fd_get(X, XD, XPs),
        domains_intersection(XD, from_to(L,U), NXD),
        fd_put(X, NXD, XPs).

min_max_factor(L1, U1, L2, U2, L3, U3, Min, Max) :-
        (   U1 cis_lt n(0),
            L2 cis_lt n(0), U2 cis_gt n(0),
            L3 cis_lt n(0), U3 cis_gt n(0) ->
            maplist(in_(L1,U1), [Z1,Z2]),
            with_local_attributes([X1,Y1,X2,Y2], [], (
                in_(L2, n(-1), X1), in_(n(1), U3, Y1),
                (   X1*Y1 #= Z1 ->
                    (   fd_get(Y1, _, Inf1, Sup1, _) -> true
                    ;   Inf1 = n(Y1), Sup1 = n(Y1)
                    )
                ;   Inf1 = inf, Sup1 = n(-1)
                ),
                in_(n(1), U2, X2), in_(L3, n(-1), Y2),
                (   X2*Y2 #= Z2 ->
                    (   fd_get(Y2, _, Inf2, Sup2, _) -> true
                    ;   Inf2 = n(Y2), Sup2 = n(Y2)
                    )
                ;   Inf2 = n(1), Sup2 = sup
                )
            ), [Inf1,Sup1,Inf2,Sup2]),
            Min cis max(min(Inf1,Inf2), L3),
            Max cis min(max(Sup1,Sup2), U3)
        ;   L1 cis_gt n(0),
            L2 cis_lt n(0), U2 cis_gt n(0),
            L3 cis_lt n(0), U3 cis_gt n(0) ->
            maplist(in_(L1,U1), [Z1,Z2]),
            with_local_attributes([X1,Y1,X2,Y2], [], (
                in_(L2, n(-1), X1), in_(L3, n(-1), Y1),
                (   X1*Y1 #= Z1 ->
                    (   fd_get(Y1, _, Inf1, Sup1, _) -> true
                    ;   Inf1 = n(Y1), Sup1 = n(Y1)
                    )
                ;   Inf1 = n(1), Sup1 = sup
                ),
                in_(n(1), U2, X2), in_(n(1), U3, Y2),
                (   X2*Y2 #= Z2 ->
                    (   fd_get(Y2, _, Inf2, Sup2, _) -> true
                    ;   Inf2 = n(Y2), Sup2 = n(Y2)
                    )
                ;   Inf2 = inf, Sup2 = n(-1)
                )
            ), [Inf1,Sup1,Inf2,Sup2]),
            Min cis max(min(Inf1,Inf2), L3),
            Max cis min(max(Sup1,Sup2), U3)
        ;   min_factor(L1, U1, L2, U2, Min0),
            Min cis max(L3,Min0),
            max_factor(L1, U1, L2, U2, Max0),
            Max cis min(U3,Max0)
        ).

min_factor(L1, U1, L2, U2, Min) :-
        (   L1 cis_geq n(0), L2 cis_gt n(0), finite(U2) ->
            Min cis (L1+U2-n(1))//U2
        ;   L1 cis_gt n(0), U2 cis_lt n(0) -> Min cis U1//U2
        ;   L1 cis_gt n(0), L2 cis_geq n(0) -> Min = n(1)
        ;   L1 cis_gt n(0) -> Min cis -U1
        ;   U1 cis_lt n(0), U2 cis_leq n(0) ->
            (   finite(L2) -> Min cis (U1+L2+n(1))//L2
            ;   Min = n(1)
            )
        ;   U1 cis_lt n(0), L2 cis_geq n(0) -> Min cis L1//L2
        ;   U1 cis_lt n(0) -> Min = L1
        ;   L2 cis_leq n(0), U2 cis_geq n(0) -> Min = inf
        ;   Min cis min(min(L1//L2,L1//U2),min(U1//L2,U1//U2))
        ).
max_factor(L1, U1, L2, U2, Max) :-
        (   L1 cis_geq n(0), L2 cis_geq n(0) -> Max cis U1//L2
        ;   L1 cis_gt n(0), U2 cis_leq n(0) ->
            (   finite(L2) -> Max cis (L1-L2-n(1))//L2
            ;   Max = n(-1)
            )
        ;   L1 cis_gt n(0) -> Max = U1
        ;   U1 cis_lt n(0), U2 cis_lt n(0) -> Max cis L1//U2
        ;   U1 cis_lt n(0), L2 cis_geq n(0) ->
            (   finite(U2) -> Max cis (U1-U2+n(1))//U2
            ;   Max = n(-1)
            )
        ;   U1 cis_lt n(0) -> Max cis -L1
        ;   L2 cis_leq n(0), U2 cis_geq n(0) -> Max = sup
        ;   Max cis max(max(L1//L2,L1//U2),max(U1//L2,U1//U2))
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   J-C. Régin: "A filtering algorithm for constraints of difference in
   CSPs", AAAI-94, Seattle, WA, USA, pp 362--367, 1994
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

distinct_attach([], _, _).
distinct_attach([X|Xs], Prop, Right) :-
        (   var(X) ->
            init_propagator(X, Prop),
            make_propagator(pexclude(Xs,Right,X), P1),
            init_propagator(X, P1),
            trigger_prop(P1)
        ;   exclude_fire(Xs, Right, X)
        ),
        distinct_attach(Xs, Prop, [X|Right]).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   For each integer of the union of domains, an attributed variable is
   introduced, to benefit from constant-time access. Attributes are:

   value ... integer corresponding to the node
   free  ... whether this (right) node is still free
   edges ... [flow_from(F,From)] and [flow_to(F,To)] where F has an
             attribute "flow" that is either 0 or 1 and an attribute "used"
             if it is part of a maximum matching
   parent ... used in breadth-first search
   g0_edges ... [flow_to(F,To)] as above
   visited ... true if node was visited in DFS
   index, in_stack, lowlink ... used in Tarjan's SCC algorithm
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

difference_arcs(Vars, FreeLeft, FreeRight) :-
        empty_assoc(E),
        phrase(difference_arcs(Vars, FreeLeft), [E], [NumVar]),
        assoc_to_list(NumVar, LsNumVar),
        pairs_values(LsNumVar, FreeRight).

domain_to_list(Domain, List) :- phrase(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) }, list(Ns).

difference_arcs([], []) --> [].
difference_arcs([V|Vs], FL0) -->
        (   { fd_get(V, Dom, _),
              finite_domain(Dom) } ->
            { FL0 = [V|FL],
              domain_to_list(Dom, Ns) },
            enumerate(Ns, V),
            difference_arcs(Vs, FL)
        ;   difference_arcs(Vs, FL0)
        ).

enumerate([], _) --> [].
enumerate([N|Ns], V) -->
        state(NumVar0, NumVar),
        { (   get_assoc(N, NumVar0, Y) -> NumVar0 = NumVar
          ;   put_assoc(N, NumVar0, Y, NumVar),
              put_attr(Y, value, N)
          ),
          put_attr(F, flow, 0),
          append_edge(Y, edges, flow_from(F,V)),
          append_edge(V, edges, flow_to(F,Y)) },
        enumerate(Ns, V).

append_edge(V, Attr, E) :-
        (   get_attr(V, Attr, Es) ->
            put_attr(V, Attr, [E|Es])
        ;   put_attr(V, Attr, [E])
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Strategy: Breadth-first search until we find a free right vertex in
   the value graph, then find an augmenting path in reverse.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

clear_parent(V) :- del_attr(V, parent).

maximum_matching([]).
maximum_matching([FL|FLs]) :-
        augmenting_path_to([[FL]], Levels, To),
        phrase(augmenting_path(FL, To), Path),
        maplist(maplist(clear_parent), Levels),
        del_attr(To, free),
        adjust_alternate_1(Path),
        maximum_matching(FLs).

reachables([]) --> [].
reachables([V|Vs]) -->
        { get_attr(V, edges, Es) },
        reachables_(Es, V),
        reachables(Vs).

reachables_([], _) --> [].
reachables_([E|Es], V) -->
        edge_reachable(E, V),
        reachables_(Es, V).

edge_reachable(flow_to(F,To), V) -->
        (   { get_attr(F, flow, 0),
              \+ get_attr(To, parent, _) } ->
            { put_attr(To, parent, V-F) },
            [To]
        ;   []
        ).
edge_reachable(flow_from(F,From), V) -->
        (   { get_attr(F, flow, 1),
              \+ get_attr(From, parent, _) } ->
            { put_attr(From, parent, V-F) },
            [From]
        ;   []
        ).

augmenting_path_to(Levels0, Levels, Right) :-
        Levels0 = [Vs|_],
        Levels1 = [Tos|Levels0],
        phrase(reachables(Vs), Tos),
        Tos = [_|_],
        (   member(Right, Tos), get_attr(Right, free, true) ->
            Levels = Levels1
        ;   augmenting_path_to(Levels1, Levels, Right)
        ).

augmenting_path(S, V) -->
        (   { V == S } -> []
        ;   { get_attr(V, parent, V1-Augment) },
            [Augment],
            augmenting_path(S, V1)
        ).

adjust_alternate_1([A|Arcs]) :-
        put_attr(A, flow, 1),
        adjust_alternate_0(Arcs).

adjust_alternate_0([]).
adjust_alternate_0([A|Arcs]) :-
        put_attr(A, flow, 0),
        adjust_alternate_1(Arcs).

% Instead of applying Berge's property directly, we can translate the
% problem in such a way, that we have to search for the so-called
% strongly connected components of the graph.

g_g0(V) :-
        get_attr(V, edges, Es),
        maplist(g_g0_(V), Es).

g_g0_(V, flow_to(F,To)) :-
        (   get_attr(F, flow, 1) ->
            append_edge(V, g0_edges, flow_to(F,To))
        ;   append_edge(To, g0_edges, flow_to(F,V))
        ).


g0_successors(V, Tos) :-
        (   get_attr(V, g0_edges, Tos0) ->
            maplist(arg(2), Tos0, Tos)
        ;   Tos = []
        ).

put_free(F) :- put_attr(F, free, true).

free_node(F) :- get_attr(F, free, true).

del_vars_attr(Vars, Attr) :- maplist(del_attr_(Attr), Vars).

del_attr_(Attr, Var) :- del_attr(Var, Attr).

with_local_attributes(Vars, Attrs, Goal, Result) :-
        catch((maplist(del_vars_attr(Vars), Attrs),
               Goal,
               maplist(del_attrs, Vars),
               % reset all attributes, only the result matters
               throw(local_attributes(Result,Vars))),
              local_attributes(Result,Vars),
              true).

distinct(Vars) :-
        with_local_attributes(Vars, [edges,parent,g0_edges,index,visited],
              (difference_arcs(Vars, FreeLeft, FreeRight0),
               length(FreeLeft, LFL),
               length(FreeRight0, LFR),
               LFL =< LFR,
               maplist(put_free, FreeRight0),
               maximum_matching(FreeLeft),
               include(free_node, FreeRight0, FreeRight),
               maplist(g_g0, FreeLeft),
               scc(FreeLeft, g0_successors),
               maplist(dfs_used, FreeRight),
               phrase(distinct_goals(FreeLeft), Gs)), Gs),
        disable_queue,
        maplist(call, Gs),
        enable_queue.

distinct_goals([]) --> [].
distinct_goals([V|Vs]) -->
        { get_attr(V, edges, Es) },
        distinct_goals_(Es, V),
        distinct_goals(Vs).

distinct_goals_([], _) --> [].
distinct_goals_([flow_to(F,To)|Es], V) -->
        (   { get_attr(F, flow, 0),
              \+ get_attr(F, used, true),
              get_attr(V, lowlink, L1),
              get_attr(To, lowlink, L2),
              L1 =\= L2 } ->
            { get_attr(To, value, N) },
            [neq_num(V, N)]
        ;   []
        ),
        distinct_goals_(Es, V).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Mark used edges.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

dfs_used(V) :-
        (   get_attr(V, visited, true) -> true
        ;   put_attr(V, visited, true),
            (   get_attr(V, g0_edges, Es) ->
                dfs_used_edges(Es)
            ;   true
            )
        ).

dfs_used_edges([]).
dfs_used_edges([flow_to(F,To)|Es]) :-
        put_attr(F, used, true),
        dfs_used(To),
        dfs_used_edges(Es).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Tarjan's strongly connected components algorithm.

   DCGs are used to implicitly pass around the global index, stack
   and the predicate relating a vertex to its successors.

   For more information about this technique, see:

                 https://www.metalevel.at/prolog/dcg
                 ===================================

   A Prolog implementation of this algorithm is also available as a
   standalone library from:

                   https://www.metalevel.at/scc.pl
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

scc(Vs, Succ) :- phrase(scc(Vs), [s(0,[],Succ)], _).

scc([])     --> [].
scc([V|Vs]) -->
        (   vindex_defined(V) -> scc(Vs)
        ;   scc_(V), scc(Vs)
        ).

vindex_defined(V) --> { get_attr(V, index, _) }.

vindex_is_index(V) -->
        state(s(Index,_,_)),
        { put_attr(V, index, Index) }.

vlowlink_is_index(V) -->
        state(s(Index,_,_)),
        { put_attr(V, lowlink, Index) }.

index_plus_one -->
        state(s(I,Stack,Succ), s(I1,Stack,Succ)),
        { I1 is I+1 }.

s_push(V)  -->
        state(s(I,Stack,Succ), s(I,[V|Stack],Succ)),
        { put_attr(V, in_stack, true) }.

vlowlink_min_lowlink(V, VP) -->
        { get_attr(V, lowlink, VL),
          get_attr(VP, lowlink, VPL),
          VL1 is min(VL, VPL),
          put_attr(V, lowlink, VL1) }.

successors(V, Tos) --> state(s(_,_,Succ)), { call(Succ, V, Tos) }.

scc_(V) -->
        vindex_is_index(V),
        vlowlink_is_index(V),
        index_plus_one,
        s_push(V),
        successors(V, Tos),
        each_edge(Tos, V),
        (   { get_attr(V, index, VI),
              get_attr(V, lowlink, VI) } -> pop_stack_to(V, VI)
        ;   []
        ).

pop_stack_to(V, N) -->
        state(s(I,[First|Stack],Succ), s(I,Stack,Succ)),
        { del_attr(First, in_stack) },
        (   { First == V } -> []
        ;   { put_attr(First, lowlink, N) },
            pop_stack_to(V, N)
        ).

each_edge([], _) --> [].
each_edge([VP|VPs], V) -->
        (   vindex_defined(VP) ->
            (   v_in_stack(VP) ->
                vlowlink_min_lowlink(V, VP)
            ;   []
            )
        ;   scc_(VP),
            vlowlink_min_lowlink(V, VP)
        ),
        each_edge(VPs, V).

state(S), [S] --> [S].

state(S0, S), [S] --> [S0].

v_in_stack(V) --> { get_attr(V, in_stack, true) }.

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Weak arc consistent constraint of difference, currently only
   available internally. Candidate for all_different/2 option.

   See Neng-Fa Zhou: "Programming Finite-Domain Constraint Propagators
   in Action Rules", Theory and Practice of Logic Programming, Vol.6,
   No.5, pp 483-508, 2006
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

weak_arc_all_distinct(Ls) :-
        must_be(list, Ls),
        Orig = original_goal(_, weak_arc_all_distinct(Ls)),
        all_distinct(Ls, [], Orig),
        do_queue.

all_distinct([], _, _).
all_distinct([X|Right], Left, Orig) :-
        %\+ list_contains(Right, X),
        (   var(X) ->
            make_propagator(weak_distinct(Left,Right,X,Orig), Prop),
            init_propagator(X, Prop),
            trigger_prop(Prop)
%             make_propagator(check_distinct(Left,Right,X), Prop2),
%             init_propagator(X, Prop2),
%             trigger_prop(Prop2)
        ;   exclude_fire(Left, Right, X)
        ),
        outof_reducer(Left, Right, X),
        all_distinct(Right, [X|Left], Orig).

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

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

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

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

outof_reducer(Left, Right, Var) :-
        (   fd_get(Var, Dom, _) ->
            append(Left, Right, Others),
            domain_num_elements(Dom, N),
            num_subsets(Others, Dom, 0, Num, NonSubs),
            (   n(Num) cis_geq N -> false
            ;   n(Num) cis 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) :-
        (   fd_get(X, XDom, XPs) ->
            domain_subtract(XDom, Dom, NXDom),
            fd_put(X, NXDom, XPs)
        ;   true
        ),
        reduce_from_others(Xs, Dom).

num_subsets([], _Dom, Num, Num, []).
num_subsets([S|Ss], Dom, Num0, Num, NonSubs) :-
        (   fd_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)
%
%   Describes a set of non-overlapping tasks.
%   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. Example:
%
%   ==
%   ?- length(Vs, 3),
%      Vs ins 0..3,
%      serialized(Vs, [1,2,3]),
%      label(Vs).
%   Vs = [0, 1, 3] ;
%   Vs = [2, 0, 3] ;
%   false.
%   ==
%
%  @see Dorndorf et al. 2000, "Constraint Propagation Techniques for the
%       Disjunctive Scheduling Problem"

serialized(Starts, Durations) :-
        must_be(list(integer), Durations),
        pairs_keys_values(SDs, Starts, Durations),
        Orig = original_goal(_, serialized(Starts, Durations)),
        serialize(SDs, Orig).

serialize([], _).
serialize([S-D|SDs], Orig) :-
        D >= 0,
        serialize(SDs, S, D, Orig),
        serialize(SDs, Orig).

serialize([], _, _, _).
serialize([S-D|Rest], S0, D0, Orig) :-
        D >= 0,
        propagator_init_trigger([S0,S], pserialized(S,D,S0,D0,Orig)),
        serialize(Rest, S0, D0, Orig).

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

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

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

serialize_lower_upper(S_I, D_I, S_J, D_J, MState) :-
        (   var(S_I) ->
            serialize_lower_bound(S_I, D_I, S_J, D_J, MState),
            (   var(S_I) -> serialize_upper_bound(S_I, D_I, S_J, D_J, MState)
            ;   true
            )
        ;   true
        ).

serialize_lower_bound(I, D_I, J, D_J, MState) :-
        fd_get(I, DomI, Ps),
        (   domain_infimum(DomI, n(EST_I)),
            latest_start_time(J, n(LST_J)),
            EST_I + D_I > LST_J,
            earliest_start_time(J, n(EST_J)) ->
            (   nonvar(J) -> kill(MState)
            ;   true
            ),
            EST is EST_J+D_J,
            domain_remove_smaller_than(DomI, EST, DomI1),
            fd_put(I, DomI1, Ps)
        ;   true
        ).

serialize_upper_bound(I, D_I, J, D_J, MState) :-
        fd_get(I, DomI, Ps),
        (   domain_supremum(DomI, n(LST_I)),
            earliest_start_time(J, n(EST_J)),
            EST_J + D_J > LST_I,
            latest_start_time(J, n(LST_J)) ->
            (   nonvar(J) -> kill(MState)
            ;   true
            ),
            LST is LST_J-D_I,
            domain_remove_greater_than(DomI, LST, DomI1),
            fd_put(I, DomI1, Ps)
        ;   true
        ).

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

%%    element(?N, +Vs, ?V)
%
%     The N-th element of the list of finite domain variables Vs is V.
%     Analogous to nth1/3.

element(N, Is, V) :-
        must_be(list, Is),
        length(Is, L),
        N in 1..L,
        element_(Is, 1, N, V),
        propagator_init_trigger([N|Is], pelement(N,Is,V)).

element_domain(V, VD) :-
        (   fd_get(V, VD, _) -> true
        ;   VD = from_to(n(V), n(V))
        ).

element_([], _, _, _).
element_([I|Is], N0, N, V) :-
        ?(I) #\= ?(V) #==> ?(N) #\= N0,
        N1 is N0 + 1,
        element_(Is, N1, N, V).

integers_remaining([], _, _, D, D).
integers_remaining([V|Vs], N0, Dom, D0, D) :-
        (   domain_contains(Dom, N0) ->
            element_domain(V, VD),
            domains_union(D0, VD, D1)
        ;   D1 = D0
        ),
        N1 is N0 + 1,
        integers_remaining(Vs, N1, Dom, D1, D).

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

%%    global_cardinality(+Vs, +Pairs)
%
%     Global Cardinality constraint. Equivalent to
%     global_cardinality(Vs, Pairs, []). See global_cardinality/3.
%
%     Example:
%
%     ==
%     ?- Vs = [_,_,_], global_cardinality(Vs, [1-2,3-_]), label(Vs).
%     Vs = [1, 1, 3] ;
%     Vs = [1, 3, 1] ;
%     Vs = [3, 1, 1].
%     ==

global_cardinality(Xs, Pairs) :- global_cardinality(Xs, Pairs, []).

%%    global_cardinality(+Vs, +Pairs, +Options)
%
%     Global Cardinality constraint. Vs  is  a   list  of  finite domain
%     variables, Pairs is a list  of  Key-Num   pairs,  where  Key is an
%     integer and Num is a finite  domain variable. The constraint holds
%     iff each V in Vs is equal to   some key, and for each Key-Num pair
%     in Pairs, the number of occurrences of   Key in Vs is Num. Options
%     is a list of options. Supported options are:
%
%     * consistency(value)
%     A weaker form of consistency is used.
%
%     * cost(Cost, Matrix)
%     Matrix is a list of rows, one for each variable, in the order
%     they occur in Vs. Each of these rows is a list of integers, one
%     for each key, in the order these keys occur in Pairs. When
%     variable v_i is assigned the value of key k_j, then the
%     associated cost is Matrix_{ij}. Cost is the sum of all costs.

global_cardinality(Xs, Pairs, Options) :-
        must_be(list(list), [Xs,Pairs,Options]),
        maplist(fd_variable, Xs),
        maplist(gcc_pair, Pairs),
        pairs_keys_values(Pairs, Keys, Nums),
        (   sort(Keys, Keys1), same_length(Keys, Keys1) -> true
        ;   domain_error(gcc_unique_key_pairs, Pairs)
        ),
        length(Xs, L),
        Nums ins 0..L,
        list_to_drep(Keys, Drep),
        Xs ins Drep,
        gcc_pairs(Pairs, Xs, Pairs1),
        % pgcc_check must be installed before triggering other
        % propagators
        propagator_init_trigger(Xs, pgcc_check(Pairs1)),
        propagator_init_trigger(Nums, pgcc_check_single(Pairs1)),
        (   member(OD, Options), OD == consistency(value) -> true
        ;   propagator_init_trigger(Nums, pgcc_single(Xs, Pairs1)),
            propagator_init_trigger(Xs, pgcc(Xs, Pairs, Pairs1))
        ),
        (   member(OC, Options), functor(OC, cost, 2) ->
            OC = cost(Cost, Matrix),
            must_be(list(list(integer)), Matrix),
            maplist(keys_costs(Keys), Xs, Matrix, Costs),
            sum(Costs, #=, Cost)
        ;   true
        ).

keys_costs(Keys, X, Row, C) :-
        element(N, Keys, X),
        element(N, Row, C).

gcc_pair(Pair) :-
        (   Pair = Key-Val ->
            must_be(integer, Key),
            fd_variable(Val)
        ;   domain_error(gcc_pair, Pair)
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   For each Key-Num0 pair, we introduce an auxiliary variable Num and
   attach the following attributes to it:

   clpfd_gcc_num: equal Num0, the user-visible counter variable
   clpfd_gcc_vs: the remaining variables in the constraint that can be
   equal Key.
   clpfd_gcc_occurred: stores how often Key already occurred in vs.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

gcc_pairs([], _, []).
gcc_pairs([Key-Num0|KNs], Vs, [Key-Num|Rest]) :-
        put_attr(Num, clpfd_gcc_num, Num0),
        put_attr(Num, clpfd_gcc_vs, Vs),
        put_attr(Num, clpfd_gcc_occurred, 0),
        gcc_pairs(KNs, Vs, Rest).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    J.-C. Régin: "Generalized Arc Consistency for Global Cardinality
    Constraint", AAAI-96 Portland, OR, USA, pp 209--215, 1996
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

gcc_global(Vs, KNs) :-
        gcc_check(KNs),
        % reach fix-point: all elements of clpfd_gcc_vs must be variables
        do_queue,
        with_local_attributes(Vs, [edges,parent,index],
              (gcc_arcs(KNs, S, Vals),
               variables_with_num_occurrences(Vs, VNs),
               maplist(target_to_v(T), VNs),
               (   get_attr(S, edges, Es) ->
                   put_attr(S, parent, none), % Mark S as seen to avoid going back to S.
                   feasible_flow(Es, S, T), % First construct a feasible flow (if any)
                   maximum_flow(S, T),      % only then, maximize it.
                   gcc_consistent(T),
                   scc(Vals, gcc_successors),
                   phrase(gcc_goals(Vals), Gs)
               ;   Gs = [] )), Gs),
        disable_queue,
        maplist(call, Gs),
        enable_queue.

gcc_consistent(T) :-
        get_attr(T, edges, Es),
        maplist(saturated_arc, Es).

saturated_arc(arc_from(_,U,_,Flow)) :- get_attr(Flow, flow, U).

gcc_goals([]) --> [].
gcc_goals([Val|Vals]) -->
        { get_attr(Val, edges, Es) },
        gcc_edges_goals(Es, Val),
        gcc_goals(Vals).

gcc_edges_goals([], _) --> [].
gcc_edges_goals([E|Es], Val) -->
        gcc_edge_goal(E, Val),
        gcc_edges_goals(Es, Val).

gcc_edge_goal(arc_from(_,_,_,_), _) --> [].
gcc_edge_goal(arc_to(_,_,V,F), Val) -->
        (   { get_attr(F, flow, 0),
              get_attr(V, lowlink, L1),
              get_attr(Val, lowlink, L2),
              L1 =\= L2,
              get_attr(Val, value, Value) } ->
            [neq_num(V, Value)]
        ;   []
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Like in all_distinct/1, first use breadth-first search, then
   construct an augmenting path in reverse.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

maximum_flow(S, T) :-
        (   gcc_augmenting_path([[S]], Levels, T) ->
            phrase(augmenting_path(S, T), Path),
            Path = [augment(_,First,_)|Rest],
            path_minimum(Rest, First, Min),
            maplist(gcc_augment(Min), Path),
            maplist(maplist(clear_parent), Levels),
            maximum_flow(S, T)
        ;   true
        ).

feasible_flow([], _, _).
feasible_flow([A|As], S, T) :-
        make_arc_feasible(A, S, T),
        feasible_flow(As, S, T).

make_arc_feasible(A, S, T) :-
        A = arc_to(L,_,V,F),
        get_attr(F, flow, Flow),
        (   Flow >= L -> true
        ;   Diff is L - Flow,
            put_attr(V, parent, S-augment(F,Diff,+)),
            gcc_augmenting_path([[V]], Levels, T),
            phrase(augmenting_path(S, T), Path),
            path_minimum(Path, Diff, Min),
            maplist(gcc_augment(Min), Path),
            maplist(maplist(clear_parent), Levels),
            make_arc_feasible(A, S, T)
        ).

gcc_augmenting_path(Levels0, Levels, T) :-
        Levels0 = [Vs|_],
        Levels1 = [Tos|Levels0],
        phrase(gcc_reachables(Vs), Tos),
        Tos = [_|_],
        (   member(To, Tos), To == T -> Levels = Levels1
        ;   gcc_augmenting_path(Levels1, Levels, T)
        ).

gcc_reachables([])     --> [].
gcc_reachables([V|Vs]) -->
        { get_attr(V, edges, Es) },
        gcc_reachables_(Es, V),
        gcc_reachables(Vs).

gcc_reachables_([], _)     --> [].
gcc_reachables_([E|Es], V) -->
        gcc_reachable(E, V),
        gcc_reachables_(Es, V).

gcc_reachable(arc_from(_,_,V,F), P) -->
        (   { \+ get_attr(V, parent, _),
              get_attr(F, flow, Flow),
              Flow > 0 } ->
            { put_attr(V, parent, P-augment(F,Flow,-)) },
            [V]
        ;   []
        ).
gcc_reachable(arc_to(_L,U,V,F), P) -->
        (   { \+ get_attr(V, parent, _),
              get_attr(F, flow, Flow),
              Flow < U } ->
            { Diff is U - Flow,
              put_attr(V, parent, P-augment(F,Diff,+)) },
            [V]
        ;   []
        ).


path_minimum([], Min, Min).
path_minimum([augment(_,A,_)|As], Min0, Min) :-
        Min1 is min(Min0,A),
        path_minimum(As, Min1, Min).

gcc_augment(Min, augment(F,_,Sign)) :-
        get_attr(F, flow, Flow0),
        gcc_flow_(Sign, Flow0, Min, Flow),
        put_attr(F, flow, Flow).

gcc_flow_(+, F0, A, F) :- F is F0 + A.
gcc_flow_(-, F0, A, F) :- F is F0 - A.

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Build value network for global cardinality constraint.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

gcc_arcs([], _, []).
gcc_arcs([Key-Num0|KNs], S, Vals) :-
        (   get_attr(Num0, clpfd_gcc_vs, Vs) ->
            get_attr(Num0, clpfd_gcc_num, Num),
            get_attr(Num0, clpfd_gcc_occurred, Occ),
            (   nonvar(Num) -> U is Num - Occ, U = L
            ;   fd_get(Num, _, n(L0), n(U0), _),
                L is L0 - Occ, U is U0 - Occ
            ),
            put_attr(Val, value, Key),
            Vals = [Val|Rest],
            put_attr(F, flow, 0),
            append_edge(S, edges, arc_to(L, U, Val, F)),
            put_attr(Val, edges, [arc_from(L, U, S, F)]),
            variables_with_num_occurrences(Vs, VNs),
            maplist(val_to_v(Val), VNs)
        ;   Vals = Rest
        ),
        gcc_arcs(KNs, S, Rest).

variables_with_num_occurrences(Vs0, VNs) :-
        include(var, Vs0, Vs1),
        msort(Vs1, Vs),
        (   Vs == [] -> VNs = []
        ;   Vs = [V|Rest],
            variables_with_num_occurrences(Rest, V, 1, VNs)
        ).

variables_with_num_occurrences([], Prev, Count, [Prev-Count]).
variables_with_num_occurrences([V|Vs], Prev, Count0, VNs) :-
        (   V == Prev ->
            Count1 is Count0 + 1,
            variables_with_num_occurrences(Vs, Prev, Count1, VNs)
        ;   VNs = [Prev-Count0|Rest],
            variables_with_num_occurrences(Vs, V, 1, Rest)
        ).


target_to_v(T, V-Count) :-
        put_attr(F, flow, 0),
        append_edge(V, edges, arc_to(0, Count, T, F)),
        append_edge(T, edges, arc_from(0, Count, V, F)).

val_to_v(Val, V-Count) :-
        put_attr(F, flow, 0),
        append_edge(V, edges, arc_from(0, Count, Val, F)),
        append_edge(Val, edges, arc_to(0, Count, V, F)).


gcc_successors(V, Tos) :-
        get_attr(V, edges, Tos0),
        phrase(gcc_successors_(Tos0), Tos).

gcc_successors_([])     --> [].
gcc_successors_([E|Es]) --> gcc_succ_edge(E), gcc_successors_(Es).

gcc_succ_edge(arc_to(_,U,V,F)) -->
        (   { get_attr(F, flow, Flow),
              Flow < U } -> [V]
        ;   []
        ).
gcc_succ_edge(arc_from(_,_,V,F)) -->
        (   { get_attr(F, flow, Flow),
              Flow > 0 } -> [V]
        ;   []
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Simple consistency check, run before global propagation.
   Importantly, it removes all ground values from clpfd_gcc_vs.

   The pgcc_check/1 propagator in itself suffices to ensure
   consistency.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

gcc_check(Pairs) :-
        disable_queue,
        gcc_check_(Pairs),
        enable_queue.

gcc_done(Num) :-
        del_attr(Num, clpfd_gcc_vs),
        del_attr(Num, clpfd_gcc_num),
        del_attr(Num, clpfd_gcc_occurred).

gcc_check_([]).
gcc_check_([Key-Num0|KNs]) :-
        (   get_attr(Num0, clpfd_gcc_vs, Vs) ->
            get_attr(Num0, clpfd_gcc_num, Num),
            get_attr(Num0, clpfd_gcc_occurred, Occ0),
            vs_key_min_others(Vs, Key, 0, Min, Os),
            put_attr(Num0, clpfd_gcc_vs, Os),
            put_attr(Num0, clpfd_gcc_occurred, Occ1),
            Occ1 is Occ0 + Min,
            geq(Num, Occ1),
            % The queue is disabled for efficiency here in any case.
            % If it were enabled, make sure to retain the invariant
            % that gcc_global is never triggered during an
            % inconsistent state (after gcc_done/1 but before all
            % relevant constraints are posted).
            (   Occ1 == Num -> all_neq(Os, Key), gcc_done(Num0)
            ;   Os == [] -> gcc_done(Num0), Num = Occ1
            ;   length(Os, L),
                Max is Occ1 + L,
                geq(Max, Num),
                (   nonvar(Num) -> Diff is Num - Occ1
                ;   fd_get(Num, ND, _),
                    domain_infimum(ND, n(NInf)),
                    Diff is NInf - Occ1
                ),
                L >= Diff,
                (   L =:= Diff ->
                    Num is Occ1 + Diff,
                    maplist(=(Key), Os),
                    gcc_done(Num0)
                ;   true
                )
            )
        ;   true
        ),
        gcc_check_(KNs).

vs_key_min_others([], _, Min, Min, []).
vs_key_min_others([V|Vs], Key, Min0, Min, Others) :-
        (   fd_get(V, VD, _) ->
            (   domain_contains(VD, Key) ->
                Others = [V|Rest],
                vs_key_min_others(Vs, Key, Min0, Min, Rest)
            ;   vs_key_min_others(Vs, Key, Min0, Min, Others)
            )
        ;   (   V =:= Key ->
                Min1 is Min0 + 1,
                vs_key_min_others(Vs, Key, Min1, Min, Others)
            ;   vs_key_min_others(Vs, Key, Min0, Min, Others)
            )
        ).

all_neq([], _).
all_neq([X|Xs], C) :-
        neq_num(X, C),
        all_neq(Xs, C).

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

%%    circuit(+Vs)
%
%     True iff the list Vs of finite domain variables induces a
%     Hamiltonian circuit. The k-th element of Vs denotes the
%     successor of node k. Node indexing starts with 1. Examples:
%
%     ==
%     ?- length(Vs, _), circuit(Vs), label(Vs).
%     Vs = [] ;
%     Vs = [1] ;
%     Vs = [2, 1] ;
%     Vs = [2, 3, 1] ;
%     Vs = [3, 1, 2] ;
%     Vs = [2, 3, 4, 1] .
%     ==

circuit(Vs) :-
        must_be(list, Vs),
        maplist(fd_variable, Vs),
        length(Vs, L),
        Vs ins 1..L,
        (   L =:= 1 -> true
        ;   neq_index(Vs, 1),
            make_propagator(pcircuit(Vs), Prop),
            distinct_attach(Vs, Prop, []),
            trigger_once(Prop)
        ).

neq_index([], _).
neq_index([X|Xs], N) :-
        neq_num(X, N),
        N1 is N + 1,
        neq_index(Xs, N1).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Necessary condition for existence of a Hamiltonian circuit: The
   graph has a single strongly connected component. If the list is
   ground, the condition is also sufficient.

   Ts are used as temporary variables to attach attributes:

   lowlink, index: used for SCC
   [arc_to(V)]: possible successors
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

propagate_circuit(Vs) :-
        with_local_attributes([], [],
            (same_length(Vs, Ts),
             circuit_graph(Vs, Ts, Ts),
             scc(Ts, circuit_successors),
             maplist(single_component, Ts)), _).

single_component(V) :- get_attr(V, lowlink, 0).

circuit_graph([], _, _).
circuit_graph([V|Vs], Ts0, [T|Ts]) :-
        (   nonvar(V) -> Ns = [V]
        ;   fd_get(V, Dom, _),
            domain_to_list(Dom, Ns)
        ),
        phrase(circuit_edges(Ns, Ts0), Es),
        put_attr(T, edges, Es),
        circuit_graph(Vs, Ts0, Ts).

circuit_edges([], _) --> [].
circuit_edges([N|Ns], Ts) -->
        { nth1(N, Ts, T) },
        [arc_to(T)],
        circuit_edges(Ns, Ts).

circuit_successors(V, Tos) :-
        get_attr(V, edges, Tos0),
        maplist(arg(1), Tos0, Tos).

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

%% cumulative(+Tasks)
%
%  Equivalent to cumulative(Tasks, [limit(1)]). See cumulative/2.

cumulative(Tasks) :- cumulative(Tasks, [limit(1)]).

%% cumulative(+Tasks, +Options)
%
%  Schedule with a limited resource. Tasks is a list of tasks, each of
%  the form task(S_i, D_i, E_i, C_i, T_i). S_i denotes the start time,
%  D_i the positive duration, E_i the end time, C_i the non-negative
%  resource consumption, and T_i the task identifier. Each of these
%  arguments must be a finite domain variable with bounded domain, or
%  an integer. The constraint holds iff at each time slot during the
%  start and end of each task, the total resource consumption of all
%  tasks running at that time does not exceed the global resource
%  limit. Options is a list of options. Currently, the only supported
%  option is:
%
%    * limit(L)
%      The integer L is the global resource limit. Default is 1.
%
%  For example, given the following predicate that relates three tasks
%  of durations 2 and 3 to a list containing their starting times:
%
%  ==
%  tasks_starts(Tasks, [S1,S2,S3]) :-
%          Tasks = [task(S1,3,_,1,_),
%                   task(S2,2,_,1,_),
%                   task(S3,2,_,1,_)].
%  ==
%
%  We can use cumulative/2 as follows, and obtain a schedule:
%
%  ==
%  ?- tasks_starts(Tasks, Starts), Starts ins 0..10,
%     cumulative(Tasks, [limit(2)]), label(Starts).
%  Tasks = [task(0, 3, 3, 1, _G36), task(0, 2, 2, 1, _G45), ...],
%  Starts = [0, 0, 2] .
%  ==

cumulative(Tasks, Options) :-
        must_be(list(list), [Tasks,Options]),
        (   Options = [] -> L = 1
        ;   Options = [limit(L)] -> must_be(integer, L)
        ;   domain_error(cumulative_options_empty_or_limit, Options)
        ),
        (   Tasks = [] -> true
        ;   fully_elastic_relaxation(Tasks, L),
            maplist(task_bs, Tasks, Bss),
            maplist(arg(1), Tasks, Starts),
            maplist(fd_inf, Starts, MinStarts),
            maplist(arg(3), Tasks, Ends),
            maplist(fd_sup, Ends, MaxEnds),
            min_list(MinStarts, Start),
            max_list(MaxEnds, End),
            resource_limit(Start, End, Tasks, Bss, L)
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Trivial lower and upper bounds, assuming no gaps and not necessarily
   retaining the rectangular shape of each task.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

fully_elastic_relaxation(Tasks, Limit) :-
        maplist(task_duration_consumption, Tasks, Ds, Cs),
        maplist(area, Ds, Cs, As),
        sum(As, #=, ?(Area)),
        ?(MinTime) #= (Area + Limit - 1) // Limit,
        tasks_minstart_maxend(Tasks, MinStart, MaxEnd),
        MaxEnd #>= MinStart + MinTime.

task_duration_consumption(task(_,D,_,C,_), D, C).

area(X, Y, Area) :- ?(Area) #= ?(X) * ?(Y).

tasks_minstart_maxend(Tasks, Start, End) :-
        maplist(task_start_end, Tasks, [Start0|Starts], [End0|Ends]),
        foldl(min_, Starts, Start0, Start),
        foldl(max_, Ends, End0, End).

max_(E, M0, M) :- ?(M) #= max(E, M0).

min_(E, M0, M) :- ?(M) #= min(E, M0).

task_start_end(task(Start,_,End,_,_), ?(Start), ?(End)).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   All time slots must respect the resource limit.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

resource_limit(T, T, _, _, _) :- !.
resource_limit(T0, T, Tasks, Bss, L) :-
        maplist(contribution_at(T0), Tasks, Bss, Cs),
        sum(Cs, #=<, L),
        T1 is T0 + 1,
        resource_limit(T1, T, Tasks, Bss, L).

task_bs(Task, InfStart-Bs) :-
        Task = task(Start,D,End,_,_Id),
        ?(D) #> 0,
        ?(End) #= ?(Start) + ?(D),
        maplist(must_be_finite_fdvar, [End,Start,D]),
        fd_inf(Start, InfStart),
        fd_sup(End, SupEnd),
        L is SupEnd - InfStart,
        length(Bs, L),
        task_running(Bs, Start, End, InfStart).

task_running([], _, _, _).
task_running([B|Bs], Start, End, T) :-
        ((T #>= Start) #/\ (T #< End)) #<==> ?(B),
        T1 is T + 1,
        task_running(Bs, Start, End, T1).

contribution_at(T, Task, Offset-Bs, Contribution) :-
        Task = task(Start,_,End,C,_),
        ?(C) #>= 0,
        fd_inf(Start, InfStart),
        fd_sup(End, SupEnd),
        (   T < InfStart -> Contribution = 0
        ;   T >= SupEnd -> Contribution = 0
        ;   Index is T - Offset,
            nth0(Index, Bs, B),
            ?(Contribution) #= B*C
        ).

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

%% disjoint2(+Rectangles)
%
%  True iff Rectangles are not overlapping. Rectangles is a list of
%  terms of the form F(X_i, W_i, Y_i, H_i), where F is any functor,
%  and the arguments are finite domain variables or integers that
%  denote, respectively, the X coordinate, width, Y coordinate and
%  height of each rectangle.

disjoint2(Rs0) :-
        must_be(list, Rs0),
        maplist(=.., Rs0, Rs),
        non_overlapping(Rs).

non_overlapping([]).
non_overlapping([R|Rs]) :-
        maplist(non_overlapping_(R), Rs),
        non_overlapping(Rs).

non_overlapping_(A, B) :-
        a_not_in_b(A, B),
        a_not_in_b(B, A).

a_not_in_b([_,AX,AW,AY,AH], [_,BX,BW,BY,BH]) :-
        ?(AX) #=< ?(BX) #/\ ?(BX) #< ?(AX) + ?(AW) #==>
                   ?(AY) + ?(AH) #=< ?(BY) #\/ ?(BY) + ?(BH) #=< ?(AY),
        ?(AY) #=< ?(BY) #/\ ?(BY) #< ?(AY) + ?(AH) #==>
                   ?(AX) + ?(AW) #=< ?(BX) #\/ ?(BX) + ?(BW) #=< ?(AX).

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

%% automaton(+Vs, +Nodes, +Arcs)
%
%  Describes a list of finite domain variables with a finite
%  automaton. Equivalent to automaton(Vs, _, Vs, Nodes, Arcs,
%  [], [], _), a common use case of automaton/8. In the following
%  example, a list of binary finite domain variables is constrained to
%  contain at least two consecutive ones:
%
%  ==
%  two_consecutive_ones(Vs) :-
%          automaton(Vs, [source(a),sink(c)],
%                    [arc(a,0,a), arc(a,1,b),
%                     arc(b,0,a), arc(b,1,c),
%                     arc(c,0,c), arc(c,1,c)]).
%  ==
%
%  Example query:
%
%  ==
%  ?- length(Vs, 3), two_consecutive_ones(Vs), label(Vs).
%  Vs = [0, 1, 1] ;
%  Vs = [1, 1, 0] ;
%  Vs = [1, 1, 1].
%  ==

automaton(Sigs, Ns, As) :- automaton(_, _, Sigs, Ns, As, [], [], _).


%% automaton(+Sequence, ?Template, +Signature, +Nodes, +Arcs, +Counters, +Initials, ?Finals)
%
%  Describes a list of finite domain variables with a finite
%  automaton. True iff the finite automaton induced by Nodes and Arcs
%  (extended with Counters) accepts Signature. Sequence is a list of
%  terms, all of the same shape. Additional constraints must link
%  Sequence to Signature, if necessary. Nodes is a list of
%  source(Node) and sink(Node) terms. Arcs is a list of
%  arc(Node,Integer,Node) and arc(Node,Integer,Node,Exprs) terms that
%  denote the automaton's transitions. Each node is represented by an
%  arbitrary term. Transitions that are not mentioned go to an
%  implicit failure node. `Exprs` is a list of arithmetic expressions,
%  of the same length as Counters. In each expression, variables
%  occurring in Counters symbolically refer to previous counter
%  values, and variables occurring in Template refer to the current
%  element of Sequence. When a transition containing arithmetic
%  expressions is taken, each counter is updated according to the
%  result of the corresponding expression. When a transition without
%  arithmetic expressions is taken, all counters remain unchanged.
%  Counters is a list of variables. Initials is a list of finite
%  domain variables or integers denoting, in the same order, the
%  initial value of each counter. These values are related to Finals
%  according to the arithmetic expressions of the taken transitions.
%
%  The following example is taken from Beldiceanu, Carlsson, Debruyne
%  and Petit: "Reformulation of Global Constraints Based on
%  Constraints Checkers", Constraints 10(4), pp 339-362 (2005). It
%  relates a sequence of integers and finite domain variables to its
%  number of inflexions, which are switches between strictly ascending
%  and strictly descending subsequences:
%
%  ==
%  sequence_inflexions(Vs, N) :-
%          variables_signature(Vs, Sigs),
%          automaton(Sigs, _, Sigs,
%                    [source(s),sink(i),sink(j),sink(s)],
%                    [arc(s,0,s), arc(s,1,j), arc(s,2,i),
%                     arc(i,0,i), arc(i,1,j,[C+1]), arc(i,2,i),
%                     arc(j,0,j), arc(j,1,j),
%                     arc(j,2,i,[C+1])],
%                    [C], [0], [N]).
%
%  variables_signature([], []).
%  variables_signature([V|Vs], Sigs) :-
%          variables_signature_(Vs, V, Sigs).
%
%  variables_signature_([], _, []).
%  variables_signature_([V|Vs], Prev, [S|Sigs]) :-
%          V #= Prev #<==> S #= 0,
%          Prev #< V #<==> S #= 1,
%          Prev #> V #<==> S #= 2,
%          variables_signature_(Vs, V, Sigs).
%  ==
%
%  Example queries:
%
%  ==
%  ?- sequence_inflexions([1,2,3,3,2,1,3,0], N).
%  N = 3.
%
%  ?- length(Ls, 5), Ls ins 0..1,
%     sequence_inflexions(Ls, 3), label(Ls).
%  Ls = [0, 1, 0, 1, 0] ;
%  Ls = [1, 0, 1, 0, 1].
%  ==

template_var_path(V, Var, []) :- var(V), !, V == Var.
template_var_path(T, Var, [N|Ns]) :-
        arg(N, T, Arg),
        template_var_path(Arg, Var, Ns).

path_term_variable([], V, V).
path_term_variable([P|Ps], T, V) :-
        arg(P, T, Arg),
        path_term_variable(Ps, Arg, V).

initial_expr(_, []-1).

automaton(Seqs, Template, Sigs, Ns, As0, Cs, Is, Fs) :-
        must_be(list(list), [Sigs,Ns,As0,Cs,Is]),
        (   var(Seqs) ->
            (   current_prolog_flag(clpfd_monotonic, true) ->
                instantiation_error(Seqs)
            ;   Seqs = Sigs
            )
        ;   must_be(list, Seqs)
        ),
        maplist(monotonic, Cs, CsM),
        maplist(arc_normalized(CsM), As0, As),
        include_args1(sink, Ns, Sinks),
        include_args1(source, Ns, Sources),
        maplist(initial_expr, Cs, Exprs0),
        phrase((arcs_relation(As, Relation),
                nodes_nums(Sinks, SinkNums0),
                nodes_nums(Sources, SourceNums0)),
               [s([]-0, Exprs0)], [s(_,Exprs1)]),
        maplist(expr0_expr, Exprs1, Exprs),
        phrase(transitions(Seqs, Template, Sigs, Start, End, Exprs, Cs, Is, Fs), Tuples),
        list_to_drep(SourceNums0, SourceDrep),
        Start in SourceDrep,
        list_to_drep(SinkNums0, SinkDrep),
        End in SinkDrep,
        tuples_in(Tuples, Relation).

expr0_expr(Es0-_, Es) :-
        pairs_keys(Es0, Es1),
        reverse(Es1, Es).

transitions([], _, [], S, S, _, _, Cs, Cs) --> [].
transitions([Seq|Seqs], Template, [Sig|Sigs], S0, S, Exprs, Counters, Cs0, Cs) -->
        [[S0,Sig,S1|Is]],
        { phrase(exprs_next(Exprs, Is, Cs1), [s(Seq,Template,Counters,Cs0)], _) },
        transitions(Seqs, Template, Sigs, S1, S, Exprs, Counters, Cs1, Cs).

exprs_next([], [], []) --> [].
exprs_next([Es|Ess], [I|Is], [C|Cs]) -->
        exprs_values(Es, Vs),
        { element(I, Vs, C) },
        exprs_next(Ess, Is, Cs).

exprs_values([], []) --> [].
exprs_values([E0|Es], [V|Vs]) -->
        { term_variables(E0, EVs0),
          copy_term(E0, E),
          term_variables(E, EVs),
          ?(V) #= E },
        match_variables(EVs0, EVs),
        exprs_values(Es, Vs).

match_variables([], _) --> [].
match_variables([V0|Vs0], [V|Vs]) -->
        state(s(Seq,Template,Counters,Cs0)),
        { (   template_var_path(Template, V0, Ps) ->
              path_term_variable(Ps, Seq, V)
          ;   template_var_path(Counters, V0, Ps) ->
              path_term_variable(Ps, Cs0, V)
          ;   domain_error(variable_from_template_or_counters, V0)
          ) },
        match_variables(Vs0, Vs).

nodes_nums([], []) --> [].
nodes_nums([Node|Nodes], [Num|Nums]) -->
        node_num(Node, Num),
        nodes_nums(Nodes, Nums).

arcs_relation([], []) --> [].
arcs_relation([arc(S0,L,S1,Es)|As], [[From,L,To|Ns]|Rs]) -->
        node_num(S0, From),
        node_num(S1, To),
        state(s(Nodes, Exprs0), s(Nodes, Exprs)),
        { exprs_nums(Es, Ns, Exprs0, Exprs) },
        arcs_relation(As, Rs).

exprs_nums([], [], [], []).
exprs_nums([E|Es], [N|Ns], [Ex0-C0|Exs0], [Ex-C|Exs]) :-
        (   member(Exp-N, Ex0), Exp == E -> C = C0, Ex = Ex0
        ;   N = C0, C is C0 + 1, Ex = [E-C0|Ex0]
        ),
        exprs_nums(Es, Ns, Exs0, Exs).

node_num(Node, Num) -->
        state(s(Nodes0-C0, Exprs), s(Nodes-C, Exprs)),
        { (   member(N-Num, Nodes0), N == Node -> C = C0, Nodes = Nodes0
          ;   Num = C0, C is C0 + 1, Nodes = [Node-C0|Nodes0]
          )
        }.

include_args1(Goal, Ls0, As) :-
        include(Goal, Ls0, Ls),
        maplist(arg(1), Ls, As).

source(source(_)).

sink(sink(_)).

monotonic(Var, ?(Var)).

arc_normalized(Cs, Arc0, Arc) :- arc_normalized_(Arc0, Cs, Arc).

arc_normalized_(arc(S0,L,S,Cs), _, arc(S0,L,S,Cs)).
arc_normalized_(arc(S0,L,S), Cs, arc(S0,L,S,Cs)).

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

%% transpose(+Matrix, ?Transpose)
%
%  Transpose a list of lists of the same length. Example:
%
%  ==
%  ?- transpose([[1,2,3],[4,5,6],[7,8,9]], Ts).
%  Ts = [[1, 4, 7], [2, 5, 8], [3, 6, 9]].
%  ==
%
%  This predicate is useful in many constraint programs. Consider for
%  instance Sudoku:
%
%  ==
%  sudoku(Rows) :-
%          length(Rows, 9), maplist(same_length(Rows), Rows),
%          append(Rows, Vs), Vs ins 1..9,
%          maplist(all_distinct, Rows),
%          transpose(Rows, Columns),
%          maplist(all_distinct, Columns),
%          Rows = [As,Bs,Cs,Ds,Es,Fs,Gs,Hs,Is],
%          blocks(As, Bs, Cs), blocks(Ds, Es, Fs), blocks(Gs, Hs, Is).
%
%  blocks([], [], []).
%  blocks([N1,N2,N3|Ns1], [N4,N5,N6|Ns2], [N7,N8,N9|Ns3]) :-
%          all_distinct([N1,N2,N3,N4,N5,N6,N7,N8,N9]),
%          blocks(Ns1, Ns2, Ns3).
%
%  problem(1, [[_,_,_,_,_,_,_,_,_],
%              [_,_,_,_,_,3,_,8,5],
%              [_,_,1,_,2,_,_,_,_],
%              [_,_,_,5,_,7,_,_,_],
%              [_,_,4,_,_,_,1,_,_],
%              [_,9,_,_,_,_,_,_,_],
%              [5,_,_,_,_,_,_,7,3],
%              [_,_,2,_,1,_,_,_,_],
%              [_,_,_,_,4,_,_,_,9]]).
%  ==
%
%  Sample query:
%
%  ==
%  ?- problem(1, Rows), sudoku(Rows), maplist(portray_clause, Rows).
%  [9, 8, 7, 6, 5, 4, 3, 2, 1].
%  [2, 4, 6, 1, 7, 3, 9, 8, 5].
%  [3, 5, 1, 9, 2, 8, 7, 4, 6].
%  [1, 2, 8, 5, 3, 7, 6, 9, 4].
%  [6, 3, 4, 8, 9, 2, 1, 5, 7].
%  [7, 9, 5, 4, 6, 1, 8, 3, 2].
%  [5, 1, 9, 2, 8, 6, 4, 7, 3].
%  [4, 7, 2, 3, 1, 9, 5, 6, 8].
%  [8, 6, 3, 7, 4, 5, 2, 1, 9].
%  Rows = [[9, 8, 7, 6, 5, 4, 3, 2|...], ... , [...|...]].
%  ==

transpose(Ls, Ts) :-
        must_be(list(list), Ls),
        lists_transpose(Ls, Ts).

lists_transpose([], []).
lists_transpose([L|Ls], Ts) :-
        maplist(same_length(L), Ls),
        foldl(transpose_, L, Ts, [L|Ls], _).

transpose_(_, Fs, Lists0, Lists) :-
        maplist(list_first_rest, Lists0, Fs, Lists).

list_first_rest([L|Ls], L, Ls).

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

%% zcompare(?Order, ?A, ?B)
%
% Analogous to compare/3, with finite domain variables A and B.
%
% Think of zcompare/3 as _reifying_ an arithmetic comparison of two
% integers. This means that we can explicitly reason about the
% different cases _within_ our programs. As in compare/3, the atoms
% =|<|=, =|>|= and =|=|= denote the different cases of the
% trichotomy. In contrast to compare/3 though, zcompare/3 works
% correctly for _all modes_, also if only a subset of the arguments is
% instantiated. This allows you to make several predicates over
% integers deterministic while preserving their generality and
% completeness.  For example:
%
% ==
% n_factorial(N, F) :-
%         zcompare(C, N, 0),
%         n_factorial_(C, N, F).
%
% n_factorial_(=, _, 1).
% n_factorial_(>, N, F) :-
%         F #= F0*N,
%         N1 #= N - 1,
%         n_factorial(N1, F0).
% ==
%
% This version of n_factorial/2 is deterministic if the first argument
% is instantiated, because argument indexing can distinguish the
% different clauses that reflect the possible and admissible outcomes
% of a comparison of `N` against 0. Example:
%
% ==
% ?- n_factorial(30, F).
% F = 265252859812191058636308480000000.
% ==
%
% Since there is no clause for =|<|=, the predicate automatically
% _fails_ if `N` is less than 0. The predicate can still be used in
% all directions, including the most general query:
%
% ==
% ?- n_factorial(N, F).
% N = 0,
% F = 1 ;
% N = F, F = 1 ;
% N = F, F = 2 .
% ==
%
% In this case, all clauses are tried on backtracking, and zcompare/3
% ensures that the respective ordering between N and 0 holds in each
% case.
%
% The truth value of a comparison can also be reified with (#<==>)/2
% in combination with one of the [_arithmetic
% constraints_](<#clpfd-arith-constraints>). See
% [reification](<#clpfd-reification>). However, zcompare/3 lets you
% more conveniently distinguish the cases.

zcompare(Order, A, B) :-
        (   nonvar(Order) ->
            zcompare_(Order, A, B)
        ;   integer(A), integer(B) ->
            compare(Order, A, B)
        ;   freeze(Order, zcompare_(Order, A, B)),
            fd_variable(A),
            fd_variable(B),
            propagator_init_trigger([A,B], pzcompare(Order, A, B))
        ).

zcompare_(=, A, B) :- ?(A) #= ?(B).
zcompare_(<, A, B) :- ?(A) #< ?(B).
zcompare_(>, A, B) :- ?(A) #> ?(B).

%% chain(+Zs, +Relation)
%
% Zs form a chain with respect to Relation. Zs is a list of finite
% domain variables that are a chain with respect to the partial order
% Relation, in the order they appear in the list. Relation must be #=,
% #=<, #>=, #< or #>. For example:
%
% ==
% ?- chain([X,Y,Z], #>=).
% X#>=Y,
% Y#>=Z.
% ==

chain(Zs, Relation) :-
        must_be(list, Zs),
        maplist(fd_variable, Zs),
        must_be(ground, Relation),
        (   chain_relation(Relation) -> true
        ;   domain_error(chain_relation, Relation)
        ),
        chain_(Zs, Relation).

chain_([], _).
chain_([X|Xs], Relation) :- foldl(chain(Relation), Xs, X, _).

chain_relation(#=).
chain_relation(#<).
chain_relation(#=<).
chain_relation(#>).
chain_relation(#>=).

chain(Relation, X, Prev, X) :- call(Relation, ?(Prev), ?(X)).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Reflection predicates
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

%% fd_var(+Var)
%
%  True iff Var is a CLP(FD) variable.

fd_var(X) :- get_attr(X, clpfd, _).

%% fd_inf(+Var, -Inf)
%
%  Inf is the infimum of the current domain of Var.

fd_inf(X, Inf) :-
        (   fd_get(X, XD, _) ->
            domain_infimum(XD, Inf0),
            bound_portray(Inf0, Inf)
        ;   must_be(integer, X),
            Inf = X
        ).

%% fd_sup(+Var, -Sup)
%
%  Sup is the supremum of the current domain of Var.

fd_sup(X, Sup) :-
        (   fd_get(X, XD, _) ->
            domain_supremum(XD, Sup0),
            bound_portray(Sup0, Sup)
        ;   must_be(integer, X),
            Sup = X
        ).

%% fd_size(+Var, -Size)
%
%  Reflect the current size of a domain. Size is the number of
%  elements of the current domain of Var, or the atom *sup* if the
%  domain is unbounded.

fd_size(X, S) :-
        (   fd_get(X, XD, _) ->
            domain_num_elements(XD, S0),
            bound_portray(S0, S)
        ;   must_be(integer, X),
            S = 1
        ).

%% fd_dom(+Var, -Dom)
%
%  Dom is the current domain (see in/2) of Var. This predicate is
%  useful if you want to reason about domains. It is _not_ needed if
%  you only want to display remaining domains; instead, separate your
%  model from the search part and let the toplevel display this
%  information via residual goals.
%
%  For example, to implement a custom labeling strategy, you may need
%  to inspect the current domain of a finite domain variable. With the
%  following code, you can convert a _finite_ domain to a list of
%  integers:
%
%  ==
%  dom_integers(D, Is) :- phrase(dom_integers_(D), Is).
%
%  dom_integers_(I)      --> { integer(I) }, [I].
%  dom_integers_(L..U)   --> { numlist(L, U, Is) }, Is.
%  dom_integers_(D1\/D2) --> dom_integers_(D1), dom_integers_(D2).
%  ==
%
%  Example:
%
%  ==
%  ?- X in 1..5, X #\= 4, fd_dom(X, D), dom_integers(D, Is).
%  D = 1..3\/5,
%  Is = [1,2,3,5],
%  X in 1..3\/5.
%  ==

fd_dom(X, Drep) :-
        (   fd_get(X, XD, _) ->
            domain_to_drep(XD, Drep)
        ;   must_be(integer, X),
            Drep = X..X
        ).

%% fd_degree(+Var, -Degree) is det.
%
%  Degree is the number of constraints currently attached to Var.

fd_degree(X, Degree) :-
        (   fd_get(X, _, Ps) ->
            props_number(Ps, Degree)
        ;   must_be(integer, X),
            Degree = 0
        ).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   FD set predicates
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Note: The predicate names and "FD set" terminology are used for
   compatibility/consistency with SICStus Prolog's library(clpfd).
   Outside of these predicates, the SWI-Prolog CLP(FD) implementation
   refers to an "FD set" as simply a "domain". The human-readable domain
   notation used by (is)/2, fd_dom/2, etc. is called a "ConstantRange"
   by SICStus and a "drep" internally by SWI.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

%% (?Var in_set +Set) is nondet.
%
%  Var is an element of the FD set Set.

X in_set Set :- domain(X, Set).

%% fd_set(?Var, -Set) is det.
%
%  Set is the FD set representation of the current domain of Var.

fd_set(X, Set) :-
        (   fd_get(X, Set, _) ->
            true
        ;   must_be(integer, X),
            Set = from_to(n(X), n(X))
        ).

%% is_fdset(@Set) is semidet.
%
%  Set is currently bound to a valid FD set.

is_fdset(Set) :-
        nonvar(Set),
        is_domain(Set).

%% empty_fdset(-Set) is det.
%
%  Set is the empty FD set.

empty_fdset(empty).

%% fdset_parts(?Set, ?Min, ?Max, ?Rest) is semidet.
%
%  Set is a non-empty FD set representing the domain Min..Max \/ Rest,
%  where Min..Max is a non-empty interval (see fdset_interval/3)
%  and Rest is another FD set (possibly empty).
%
%  If Max is *sup*, then Rest is the empty FD set. Otherwise, if Rest
%  is non-empty, all elements of Rest are greater than Max+1.
%
%  This predicate should only be called with either Set or all other
%  arguments being ground.

% Single interval case for both modes.
fdset_parts(from_to(CMin, CMax), Min, Max, empty) :-
        !,
        fdset_interval(from_to(CMin, CMax), Min, Max).
% Split domain case for mode (-,+,+,+).
fdset_parts(Set, Min, Max, Rest) :-
        var(Set),
        !,
        Set = split(Hole, Left, Rest),
        fdset_interval(Left, Min, Max),
        % Rest is not empty, so Max cannot be sup, because all elements
        % of Rest must be greater than Max.
        Max \== sup,
        Hole is Max + 1,
        % Ensure that Min..Max is less than and not adjacent to Rest.
        all_greater_than(Rest, Hole).
% Special case for mode (+,-,-,-) for split domain with empty left side.
% (The code for the common case would silently fail here.)
fdset_parts(split(_, empty, Right), Min, Max, Rest) :-
        !,
        fdset_parts(Right, Min, Max, Rest).
% Finally, handle all other split domains for mode (+,-,-,-).
fdset_parts(split(Hole, Left, Right), Min, Max, Rest) :-
        fdset_parts(Left, Min, Max, LeftRest),
        (   LeftRest == empty
        ->  Rest = Right
        ;   Rest = split(Hole, LeftRest, Right)
        ).

%% empty_interval(+Min, +Max) is semidet.
%
%  Min..Max is an empty interval. Min and Max are integers or one of the
%  atoms *inf* or *sup*.

empty_interval(inf, inf) :- !.
empty_interval(sup, inf) :- !.
empty_interval(sup, sup) :- !.
empty_interval(Min, Max) :-
        Min \== inf,
        Max \== sup,
        Min > Max.

%% fdset_interval(?Interval, ?Min, ?Max) is semidet.
%
%  Interval is a non-empty FD set consisting of the single interval
%  Min..Max.
%  Min is an integer or the atom *inf* to denote negative infinity.
%  Max is an integer or the atom *sup* to denote positive infinity.
%
%  Either Interval or Min and Max must be ground.

fdset_interval(from_to(inf, sup), inf, sup) :- !.
fdset_interval(from_to(inf, n(Max)), inf, Max) :-
        !,
        integer(Max).
fdset_interval(from_to(n(Min), sup), Min, sup) :-
        !,
        integer(Min).
fdset_interval(from_to(n(Min), n(Max)), Min, Max) :-
        integer(Min),
        integer(Max),
        Min =< Max.

%% fdset_singleton(?Set, ?Elt) is semidet.
%
%  Set is the FD set containing the single integer Elt.
%
%  Either Set or Elt must be ground.

fdset_singleton(Set, Elt) :- fdset_interval(Set, Elt, Elt).

%% fdset_min(+Set, -Min) is semidet.
%
%  Min is the lower bound (infimum) of the non-empty FD set Set.
%  Min is an integer or the atom *inf* if Set has no lower bound.

fdset_min(Set, Min) :-
        domain_infimum(Set, CMin),
        bound_portray(CMin, Min).

%% fdset_max(+Set, -Max) is semidet.
%
%  Max is the upper bound (supremum) of the non-empty FD set Set.
%  Max is an integer or the atom *sup* if Set has no upper bound.

fdset_max(Set, Max) :-
        domain_supremum(Set, CMax),
        bound_portray(CMax, Max).

%% fdset_size(+Set, -Size) is det.
%
%  Size is the number of elements of the FD set Set, or the atom *sup*
%  if Set is infinite.

fdset_size(Set, Size) :-
        domain_num_elements(Set, CSize),
        bound_portray(CSize, Size).

%% list_to_fdset(+List, -Set) is det.
%
%  Set is an FD set containing all elements of List, which must be a
%  list of integers.

list_to_fdset(List, Set) :- list_to_domain(List, Set).

%% fdset_to_list(+Set, -List) is det.
%
%  List is a list containing all elements of the finite FD set Set,
%  in ascending order.

fdset_to_list(Set, List) :- domain_to_list(Set, List).

%% range_to_fdset(+Domain, -Set) is det.
%
%  Set is an FD set equivalent to the domain Domain. Domain uses the
%  same syntax as accepted by (in)/2.

range_to_fdset(Domain, Set) :- drep_to_domain(Domain, Set).

%% fdset_to_range(+Set, -Domain) is det.
%
%  Domain is a domain equivalent to the FD set Set. Domain is returned
%  in the same format as by fd_dom/2.

fdset_to_range(empty, 1..0) :- !.
fdset_to_range(Set, Domain) :- domain_to_drep(Set, Domain).

%% fdset_add_element(+Set1, +Elt, -Set2) is det.
%
%  Set2 is the same FD set as Set1, but with the integer Elt added.
%  If Elt is already in Set1, the set is returned unchanged.

fdset_add_element(Set1, Elt, Set2) :-
        fdset_singleton(EltSet, Elt),
        domains_union(Set1, EltSet, Set2).

%% fdset_del_element(+Set1, +Elt, -Set2) is det.
%
%  Set2 is the same FD set as Set1, but with the integer Elt removed.
%  If Elt is not in Set1, the set returned unchanged.

fdset_del_element(Set1, Elt, Set2) :- domain_remove(Set1, Elt, Set2).

%% fdset_disjoint(+Set1, +Set2) is semidet.
%
%  The FD sets Set1 and Set2 have no elements in common.

fdset_disjoint(Set1, Set2) :- \+ fdset_intersect(Set1, Set2).

%% fdset_intersect(+Set1, +Set2) is semidet.
%
%  The FD sets Set1 and Set2 have at least one element in common.

fdset_intersect(Set1, Set2) :- domains_intersection(Set1, Set2, _).

%% fdset_intersection(+Set1, +Set2, -Intersection) is det.
%
%  Intersection is an FD set (possibly empty) of all elements that the
%  FD sets Set1 and Set2 have in common.

fdset_intersection(Set1, Set2, Intersection) :-
        domains_intersection_(Set1, Set2, Intersection).

%% fdset_member(?Elt, +Set) is nondet.
%
%  The integer Elt is a member of the FD set Set. If Elt is unbound,
%  Set must be finite and all elements are enumerated on backtracking.

fdset_member(Elt, Set) :-
        (   var(Elt)
        ->  domain_direction_element(Set, up, Elt)
        ;   integer(Elt),
            domain_contains(Set, Elt)
        ).

%% fdset_eq(+Set1, +Set2) is semidet.
%
%  True if the FD sets Set1 and Set2 are equal, i. e. contain exactly
%  the same elements. This is not necessarily the same as unification or
%  a term equality check, because some FD sets have multiple possible
%  term representations.

fdset_eq(empty, empty) :- !.
fdset_eq(Set1, Set2) :-
        fdset_parts(Set1, Min, Max, Rest1),
        fdset_parts(Set2, Min, Max, Rest2),
        fdset_eq(Rest1, Rest2).

%% fdset_subset(+Set1, +Set2) is semidet.
%
%  The FD set Set1 is a (non-strict) subset of Set2, i. e. every element
%  of Set1 is also in Set2.

fdset_subset(Set1, Set2) :- domain_subdomain(Set2, Set1).

%% fdset_subtract(+Set1, +Set2, -Difference) is det.
%
%  The FD set Difference is Set1 with all elements of Set2 removed,
%  i. e. the set difference of Set1 and Set2.

fdset_subtract(Set1, Set2, Difference) :-
        domain_subtract(Set1, Set2, Difference).

%% fdset_union(+Set1, +Set2, -Union) is det.
%
%  The FD set Union is the union of FD sets Set1 and Set2.

fdset_union(Set1, Set2, Union) :- domains_union(Set1, Set2, Union).

%% fdset_union(+Sets, -Union) is det.
%
%  The FD set Union is the n-ary union of all FD sets in the list Sets.
%  If Sets is empty, Union is the empty FD set.

fdset_union([], empty).
fdset_union([Set|Sets], Union) :- fdset_union_(Sets, Set, Union).

fdset_union_([], Set, Set).
fdset_union_([Set2|Sets], Set1, Union) :-
        domains_union(Set1, Set2, SetTemp),
        fdset_union_(Sets, SetTemp, Union).

%% fdset_complement(+Set, -Complement) is det.
%
%  The FD set Complement is the complement of the FD set Set.
%  Equivalent to fdset_subtract(inf..sup, Set, Complement).

fdset_complement(Set, Complement) :- domain_complement(Set, Complement).


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Entailment detection. Subject to change.

   Currently, Goals entail E if posting ({#\ E} U Goals), then
   labeling all variables, fails. E must be reifiable. Examples:

   %?- clpfd:goals_entail([X#>2], X #> 3).
   %@ false.

   %?- clpfd:goals_entail([X#>1, X#<3], X #= 2).
   %@ true.

   %?- clpfd:goals_entail([X#=Y+1], X #= Y+1).
   %@ ERROR: Arguments are not sufficiently instantiated
   %@    Exception: (15) throw(error(instantiation_error, _G2680)) ?

   %?- clpfd:goals_entail([[X,Y] ins 0..10, X#=Y+1], X #= Y+1).
   %@ true.

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

goals_entail(Goals, E) :-
        must_be(list, Goals),
        \+ (   maplist(call, Goals), #\ E,
               term_variables(Goals-E, Vs),
               label(Vs)
           ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Unification hook and constraint projection
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

attr_unify_hook(clpfd_attr(_,_,_,Dom,Ps), Other) :-
        (   nonvar(Other) ->
            (   integer(Other) -> true
            ;   type_error(integer, Other)
            ),
            domain_contains(Dom, Other),
            trigger_props(Ps),
            do_queue
        ;   fd_get(Other, OD, OPs),
            domains_intersection(OD, Dom, Dom1),
            append_propagators(Ps, OPs, Ps1),
            fd_put(Other, Dom1, Ps1),
            trigger_props(Ps1),
            do_queue
        ).

append_propagators(fd_props(Gs0,Bs0,Os0), fd_props(Gs1,Bs1,Os1), fd_props(Gs,Bs,Os)) :-
        maplist(append, [Gs0,Bs0,Os0], [Gs1,Bs1,Os1], [Gs,Bs,Os]).

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

list_to_drep(List, Drep) :-
        list_to_domain(List, Dom),
        domain_to_drep(Dom, Drep).

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).

attribute_goals(X) -->
        % { get_attr(X, clpfd, Attr), format("A: ~w\n", [Attr]) },
        { get_attr(X, clpfd, clpfd_attr(_,_,_,Dom,fd_props(Gs,Bs,Os))),
          append(Gs, Bs, Ps0),
          append(Ps0, Os, Ps),
          domain_to_drep(Dom, Drep) },
        (   { default_domain(Dom), \+ all_dead_(Ps) } -> []
        ;   [clpfd:(X in Drep)]
        ),
        attributes_goals(Ps).

clpfd_aux:attribute_goals(_) --> [].
clpfd_aux:attr_unify_hook(_,_) :- false.

clpfd_gcc_vs:attribute_goals(_) --> [].
clpfd_gcc_vs:attr_unify_hook(_,_) :- false.

clpfd_gcc_num:attribute_goals(_) --> [].
clpfd_gcc_num:attr_unify_hook(_,_) :- false.

clpfd_gcc_occurred:attribute_goals(_) --> [].
clpfd_gcc_occurred:attr_unify_hook(_,_) :- false.

clpfd_relation:attribute_goals(_) --> [].
clpfd_relation:attr_unify_hook(_,_) :- false.

attributes_goals([]) --> [].
attributes_goals([propagator(P, State)|As]) -->
        (   { ground(State) } -> []
        ;   { phrase(attribute_goal_(P), Gs) } ->
            { del_attr(State, clpfd_aux), State = processed,
              (   current_prolog_flag(clpfd_monotonic, true) ->
                  maplist(unwrap_with(bare_integer), Gs, Gs1)
              ;   maplist(unwrap_with(=), Gs, Gs1)
              ),
              maplist(with_clpfd, Gs1, Gs2) },
            list(Gs2)
        ;   [P] % possibly user-defined constraint
        ),
        attributes_goals(As).

with_clpfd(G, clpfd:G).

unwrap_with(_, V, V)           :- var(V), !.
unwrap_with(Goal, ?(V0), V)    :- !, call(Goal, V0, V).
unwrap_with(Goal, Term0, Term) :-
        Term0 =.. [F|Args0],
        maplist(unwrap_with(Goal), Args0, Args),
        Term =.. [F|Args].

bare_integer(V0, V)    :- ( integer(V0) -> V = V0 ; V = #(V0) ).

attribute_goal_(presidual(Goal))       --> [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_(absdiff_neq(X,Y,C))    --> [abs(?(X) - ?(Y)) #\= C].
attribute_goal_(absdiff_geq(X,Y,C))    --> [abs(?(X) - ?(Y)) #>= C].
attribute_goal_(x_neq_y_plus_z(X,Y,Z)) --> [?(X) #\= ?(Y) + ?(Z)].
attribute_goal_(x_leq_y_plus_c(X,Y,C)) --> [?(X) #=< ?(Y) + C].
attribute_goal_(ptzdiv(X,Y,Z))         --> [?(X) // ?(Y) #= ?(Z)].
attribute_goal_(pdiv(X,Y,Z))           --> [?(X) div ?(Y) #= ?(Z)].
attribute_goal_(prdiv(X,Y,Z))          --> [?(X) rdiv ?(Y) #= ?(Z)].
attribute_goal_(pshift(X,Y,Z,1))       --> [?(X) << ?(Y) #= ?(Z)].
attribute_goal_(pshift(X,Y,Z,-1))      --> [?(X) >> ?(Y) #= ?(Z)].
attribute_goal_(pexp(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_(prem(X,Y,Z))           --> [?(X) rem ?(Y) #= ?(Z)].
attribute_goal_(pmax(X,Y,Z))           --> [?(Z) #= max(?(X),?(Y))].
attribute_goal_(pmin(X,Y,Z))           --> [?(Z) #= min(?(X),?(Y))].
attribute_goal_(scalar_product_neq(Cs,Vs,C)) -->
        [Left #\= Right],
        { scalar_product_left_right([-1|Cs], [C|Vs], Left, Right) }.
attribute_goal_(scalar_product_eq(Cs,Vs,C)) -->
        [Left #= Right],
        { scalar_product_left_right([-1|Cs], [C|Vs], Left, Right) }.
attribute_goal_(scalar_product_leq(Cs,Vs,C)) -->
        [Left #=< Right],
        { scalar_product_left_right([-1|Cs], [C|Vs], Left, Right) }.
attribute_goal_(pdifferent(_,_,_,O))    --> original_goal(O).
attribute_goal_(weak_distinct(_,_,_,O)) --> original_goal(O).
attribute_goal_(pdistinct(Vs))          --> [all_distinct(Vs)].
attribute_goal_(pexclude(_,_,_))  --> [].
attribute_goal_(pelement(N,Is,V)) --> [element(N, Is, V)].
attribute_goal_(pgcc(Vs, Pairs, _))   --> [global_cardinality(Vs, Pairs)].
attribute_goal_(pgcc_single(_,_))     --> [].
attribute_goal_(pgcc_check_single(_)) --> [].
attribute_goal_(pgcc_check(_))        --> [].
attribute_goal_(pcircuit(Vs))       --> [circuit(Vs)].
attribute_goal_(pserialized(_,_,_,_,O)) --> original_goal(O).
attribute_goal_(rel_tuple(R, Tuple)) -->
        { get_attr(R, clpfd_relation, Rel) },
        [tuples_in([Tuple], Rel)].
attribute_goal_(pzcompare(O,A,B)) --> [zcompare(O,A,B)].
% reified constraints
attribute_goal_(reified_in(V, D, B)) -->
        [V in Drep #<==> ?(B)],
        { domain_to_drep(D, Drep) }.
attribute_goal_(reified_tuple_in(Tuple, R, B)) -->
        { get_attr(R, clpfd_relation, Rel) },
        [tuples_in([Tuple], Rel) #<==> ?(B)].
attribute_goal_(kill_reified_tuples(_,_,_)) --> [].
attribute_goal_(tuples_not_in(_,_,_)) --> [].
attribute_goal_(reified_fd(V,B)) --> [finite_domain(V) #<==> ?(B)].
attribute_goal_(pskeleton(X,Y,D,_,Z,F)) -->
        { Prop =.. [F,X,Y,Z],
          phrase(attribute_goal_(Prop), Goals), list_goal(Goals, Goal) },
        [?(D) #= 1 #==> Goal, ?(Y) #\= 0 #==> ?(D) #= 1].
attribute_goal_(reified_neq(DX,X,DY,Y,_,B)) -->
        conjunction(DX, DY, ?(X) #\= ?(Y), B).
attribute_goal_(reified_eq(DX,X,DY,Y,_,B))  -->
        conjunction(DX, DY, ?(X) #= ?(Y), B).
attribute_goal_(reified_geq(DX,X,DY,Y,_,B)) -->
        conjunction(DX, DY, ?(X) #>= ?(Y), B).
attribute_goal_(reified_and(X,_,Y,_,B))    --> [?(X) #/\ ?(Y) #<==> ?(B)].
attribute_goal_(reified_or(X, _, Y, _, B)) --> [?(X) #\/ ?(Y) #<==> ?(B)].
attribute_goal_(reified_not(X, Y))         --> [#\ ?(X) #<==> ?(Y)].
attribute_goal_(pimpl(X, Y, _))            --> [?(X) #==> ?(Y)].
attribute_goal_(pfunction(Op, A, B, R)) -->
        { Expr =.. [Op,?(A),?(B)] },
        [?(R) #= Expr].
attribute_goal_(pfunction(Op, A, R)) -->
        { Expr =.. [Op,?(A)] },
        [?(R) #= Expr].

conjunction(A, B, G, D) -->
        (   { A == 1, B == 1 } -> [G #<==> ?(D)]
        ;   { A == 1 } -> [(?(B) #/\ G) #<==> ?(D)]
        ;   { B == 1 } -> [(?(A) #/\ G) #<==> ?(D)]
        ;   [(?(A) #/\ ?(B) #/\ G) #<==> ?(D)]
        ).

original_goal(original_goal(State, Goal)) -->
        (   { var(State) } ->
            { State = processed },
            [Goal]
        ;   []
        ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Projection of scalar product.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

scalar_product_left_right(Cs, Vs, Left, Right) :-
        pairs_keys_values(Pairs0, Cs, Vs),
        partition(ground, Pairs0, Grounds, Pairs),
        maplist(pair_product, Grounds, Prods),
        sum_list(Prods, Const),
        NConst is -Const,
        partition(compare_coeff0, Pairs, Negatives, _, Positives),
        maplist(negate_coeff, Negatives, Rights),
        scalar_plusterm(Rights, Right0),
        scalar_plusterm(Positives, Left0),
        (   Const =:= 0 -> Left = Left0, Right = Right0
        ;   Right0 == 0 -> Left = Left0, Right = NConst
        ;   Left0 == 0 ->  Left = Const, Right = Right0
        ;   (   Const < 0 ->
                Left = Left0,       Right = Right0+NConst
            ;   Left = Left0+Const, Right = Right0
            )
        ).

negate_coeff(A0-B, A-B) :- A is -A0.

pair_product(A-B, Prod) :- Prod is A*B.

compare_coeff0(Coeff-_, Compare) :- compare(Compare, Coeff, 0).

scalar_plusterm([], 0).
scalar_plusterm([CV|CVs], T) :-
        coeff_var_term(CV, T0),
        foldl(plusterm_, CVs, T0, T).

plusterm_(CV, T0, T0+T) :- coeff_var_term(CV, T).

coeff_var_term(C-V, T) :- ( C =:= 1 -> T = ?(V) ; T = C * ?(V) ).

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Generated predicates
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

:- discontiguous term_expansion/2.

term_expansion(make_parse_clpfd, Clauses)   :- make_parse_clpfd(Clauses).
term_expansion(make_parse_reified, Clauses) :- make_parse_reified(Clauses).
term_expansion(make_matches, Clauses)       :- make_matches(Clauses).

make_parse_clpfd.
make_parse_reified.
make_matches.

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Global variables
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

make_clpfd_var('$clpfd_queue') :-
        make_queue.
make_clpfd_var('$clpfd_current_propagator') :-
        nb_setval('$clpfd_current_propagator', []).
make_clpfd_var('$clpfd_queue_status') :-
        nb_setval('$clpfd_queue_status', enabled).

:- multifile user:exception/3.

user:exception(undefined_global_variable, Name, retry) :-
        make_clpfd_var(Name), !.

warn_if_bounded_arithmetic :-
        (   current_prolog_flag(bounded, true) ->
            print_message(warning, clpfd(bounded))
        ;   true
        ).

:- initialization(warn_if_bounded_arithmetic).


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Messages
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

:- multifile prolog:message//1.

prolog:message(clpfd(bounded)) -->
        ['Using CLP(FD) with bounded arithmetic may yield wrong results.'-[]].


		 /*******************************
		 *	      SANDBOX		*
		 *******************************/

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
The clpfd library cannot  be   analysed  completely by library(sandbox).
However, the API does not provide any  meta predicates. It provides some
unification hooks, but put_attr/3 does not  allow injecting in arbitrary
attributes.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

:- multifile
	sandbox:safe_primitive/1.

safe_api(Name/Arity, sandbox:safe_primitive(clpfd:Head)) :-
	functor(Head, Name, Arity).

term_expansion(safe_api, Clauses) :-
	module_property(clpfd, exports(API)),
	maplist(safe_api, API, Clauses).

safe_api.
% Support clpfd goal expansion.
sandbox:safe_primitive(clpfd:clpfd_equal(_,_)).
sandbox:safe_primitive(clpfd:clpfd_geq(_,_)).
sandbox:safe_primitive(clpfd:clpfd_in(_,_)).
% Enabling monotonic CLP(FD) is safe.
sandbox:safe_primitive(set_prolog_flag(clpfd_monotonic, _)).
