/*  Part of SWI-Prolog

    Author:        Jan Wielemaker and Richard O'Keefe
    E-mail:        J.Wielemaker@cs.vu.nl
    WWW:           http://www.swi-prolog.org
    Copyright (c)  2002-2023, University of Amsterdam
                              VU University Amsterdam
                              SWI-Prolog Solutions b.v.
    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;
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*/

:- module(lists,
        [ member/2,                     % ?X, ?List
          memberchk/2,                  % ?X, ?List
          append/2,                     % +ListOfLists, -List
          append/3,                     % ?A, ?B, ?AB
          prefix/2,                     % ?Part, ?Whole
          select/3,                     % ?X, ?List, ?Rest
          selectchk/3,                  % ?X, ?List, ?Rest
          select/4,                     % ?X, ?XList, ?Y, ?YList
          selectchk/4,                  % ?X, ?XList, ?Y, ?YList
          nextto/3,                     % ?X, ?Y, ?List
          delete/3,                     % ?List, ?X, ?Rest
          nth0/3,                       % ?N, ?List, ?Elem
          nth1/3,                       % ?N, ?List, ?Elem
          nth0/4,                       % ?N, ?List, ?Elem, ?Rest
          nth1/4,                       % ?N, ?List, ?Elem, ?Rest
          last/2,                       % +List, -Element
          proper_length/2,              % @List, -Length
          same_length/2,                % ?List1, ?List2
          reverse/2,                    % +List, -Reversed
          permutation/2,                % ?List, ?Permutation
          flatten/2,                    % +Nested, -Flat
          clumped/2,                    % +Items,-Pairs
          subseq/3,                     % ?List, ?SubList, ?Complement

                                        % Ordered operations
          max_member/2,                 % -Max, +List
          min_member/2,                 % -Min, +List
          max_member/3,                 % :Pred, -Max, +List
          min_member/3,                 % :Pred, -Min, +List

                                        % Lists of numbers
          sum_list/2,                   % +List, -Sum
          max_list/2,                   % +List, -Max
          min_list/2,                   % +List, -Min
          numlist/3,                    % +Low, +High, -List

                                        % set manipulation
          is_set/1,                     % +List
          list_to_set/2,                % +List, -Set
          intersection/3,               % +List1, +List2, -Intersection
          union/3,                      % +List1, +List2, -Union
          subset/2,                     % +SubSet, +Set
          subtract/3                    % +Set, +Delete, -Remaining
        ]).
:- autoload(library(error), [must_be/2, instantiation_error/1]).
:- autoload(library(pairs), [pairs_keys/2]).

:- meta_predicate
    max_member(2, -, +),
    min_member(2, -, +).

:- set_prolog_flag(generate_debug_info, false).

/** <module> List Manipulation

This library provides  commonly  accepted   basic  predicates  for  list
manipulation in the Prolog community. Some additional list manipulations
are built-in. See e.g., memberchk/2, length/2.

The implementation of this library  is   copied  from many places. These
include: "The Craft of Prolog", the   DEC-10  Prolog library (LISTRO.PL)
and the YAP lists library. Some   predicates  are reimplemented based on
their specification by Quintus and SICStus.

@compat Virtually every Prolog system has library(lists), but the set
        of provided predicates is diverse.  There is a fair agreement
        on the semantics of most of these predicates, although error
        handling may vary.
*/

%!  member(?Elem, ?List)
%
%   True if Elem is a  member   of  List.  The SWI-Prolog definition
%   differs from the classical one.  Our definition avoids unpacking
%   each list element twice and  provides   determinism  on the last
%   element.  E.g. this is deterministic:
%
%       ==
%           member(X, [One]).
%       ==
%
%   @author Gertjan van Noord

member(El, [H|T]) :-
    member_(T, El, H).

member_(_, El, El).
member_([H|T], El, _) :-
    member_(T, El, H).

%!  append(?List1, ?List2, ?List1AndList2)
%
%   List1AndList2 is the concatenation of List1 and List2

append([], L, L).
append([H|T], L, [H|R]) :-
    append(T, L, R).

%!  append(+ListOfLists, ?List)
%
%   Concatenate a list of lists.  Is  true   if  ListOfLists  is a list of
%   lists, and List is the concatenation of these lists.
%
%   @param  ListOfLists must be a list of _possibly_ partial lists

append(ListOfLists, List) :-
    must_be(list, ListOfLists),
    append_(ListOfLists, List).

append_([], []).
append_([L|Ls], As) :-
    append(L, Ws, As),
    append_(Ls, Ws).


%!  prefix(?Part, ?Whole)
%
%   True iff Part is a leading substring of Whole.  This is the same
%   as append(Part, _, Whole).

prefix([], _).
prefix([E|T0], [E|T]) :-
    prefix(T0, T).


%!  select(?Elem, ?List1, ?List2)
%
%   Is true when List1,  with  Elem   removed,  results  in  List2. This
%   implementation is determinsitic if the  last   element  of List1 has
%   been selected.

select(X, [Head|Tail], Rest) :-
    select3_(Tail, Head, X, Rest).

select3_(Tail, Head, Head, Tail).
select3_([Head2|Tail], Head, X, [Head|Rest]) :-
    select3_(Tail, Head2, X, Rest).


%!  selectchk(+Elem, +List, -Rest) is semidet.
%
%   Semi-deterministic removal of first element in List that unifies
%   with Elem.

selectchk(Elem, List, Rest) :-
    select(Elem, List, Rest0),
    !,
    Rest = Rest0.


%!  select(?X, ?XList, ?Y, ?YList) is nondet.
%
%   Select from two lists  at  the  same   position.  True  if  XList is
%   unifiable with YList apart a  single   element  at the same position
%   that is unified with X in XList and   with Y in YList. A typical use
%   for this predicate is to  _replace_  an   element,  as  shown in the
%   example  below.  All  possible  substitutions    are   performed  on
%   backtracking.
%
%     ==
%     ?- select(b, [a,b,c,b], 2, X).
%     X = [a, 2, c, b] ;
%     X = [a, b, c, 2] ;
%     false.
%     ==
%
%   @see selectchk/4 provides a semidet version.

select(X, XList, Y, YList) :-
    select4_(XList, X, Y, YList).

select4_([X|List], X, Y, [Y|List]).
select4_([X0|XList], X, Y, [X0|YList]) :-
    select4_(XList, X, Y, YList).

%!  selectchk(?X, ?XList, ?Y, ?YList) is semidet.
%
%   Semi-deterministic version of select/4.

selectchk(X, XList, Y, YList) :-
    select(X, XList, Y, YList),
    !.

%!  nextto(?X, ?Y, ?List)
%
%   True if Y directly follows X in List.

nextto(X, Y, [X,Y|_]).
nextto(X, Y, [_|Zs]) :-
    nextto(X, Y, Zs).

%!  delete(+List1, @Elem, -List2) is det.
%
%   Delete matching elements from a list. True  when List2 is a list
%   with all elements from List1 except   for  those that unify with
%   Elem. Matching Elem with elements of List1  is uses =|\+ Elem \=
%   H|=, which implies that Elem is not changed.
%
%   @deprecated There are too many ways in which one might want to
%               delete elements from a list to justify the name.
%               Think of matching (= vs. ==), delete first/all,
%               be deterministic or not.
%   @see select/3, subtract/3.

delete([], _, []).
delete([Elem|Tail], Del, Result) :-
    (   \+ Elem \= Del
    ->  delete(Tail, Del, Result)
    ;   Result = [Elem|Rest],
        delete(Tail, Del, Rest)
    ).


/*  nth0/3, nth1/3 are improved versions from
    Martin Jansche <martin@pc03.idf.uni-heidelberg.de>
*/

%!  nth0(?Index, ?List, ?Elem)
%
%   True when Elem is the Index'th  element of List. Counting starts
%   at 0.
%
%   @error  type_error(integer, Index) if Index is not an integer or
%           unbound.
%   @see nth1/3.

nth0(Index, List, Elem) :-
    (   integer(Index)
    ->  '$seek_list'(Index, List, RestIndex, RestList),
        nth0_det(RestIndex, RestList, Elem) % take nth det
    ;   var(Index)
    ->  List = [H|T],
        nth_gen(T, Elem, H, 0, Index)       % match
    ;   must_be(integer, Index)
    ).

nth0_det(0, [Elem|_], Elem) :- !.
nth0_det(N, [_|Tail], Elem) :-
    M is N - 1,
    M >= 0,
    nth0_det(M, Tail, Elem).

nth_gen(_, Elem, Elem, Base, Base).
nth_gen([H|Tail], Elem, _, N, Base) :-
    M is N + 1,
    nth_gen(Tail, Elem, H, M, Base).


%!  nth1(?Index, ?List, ?Elem)
%
%   Is true when Elem is  the   Index'th  element  of List. Counting
%   starts at 1.
%
%   @see nth0/3.

nth1(Index, List, Elem) :-
    (   integer(Index)
    ->  Index0 is Index - 1,
        '$seek_list'(Index0, List, RestIndex, RestList),
        nth0_det(RestIndex, RestList, Elem) % take nth det
    ;   var(Index)
    ->  List = [H|T],
        nth_gen(T, Elem, H, 1, Index)       % match
    ;   must_be(integer, Index)
    ).

%!  nth0(?N, ?List, ?Elem, ?Rest) is det.
%
%   Select/insert element at index.  True  when   Elem  is  the N'th
%   (0-based) element of List and Rest is   the  remainder (as in by
%   select/3) of List.  For example:
%
%     ==
%     ?- nth0(I, [a,b,c], E, R).
%     I = 0, E = a, R = [b, c] ;
%     I = 1, E = b, R = [a, c] ;
%     I = 2, E = c, R = [a, b] ;
%     false.
%     ==
%
%     ==
%     ?- nth0(1, L, a1, [a,b]).
%     L = [a, a1, b].
%     ==

nth0(V, In, Element, Rest) :-
    var(V),
    !,
    generate_nth(0, V, In, Element, Rest).
nth0(V, In, Element, Rest) :-
    must_be(nonneg, V),
    find_nth0(V, In, Element, Rest).

%!  nth1(?N, ?List, ?Elem, ?Rest) is det.
%
%   As nth0/4, but counting starts at 1.

nth1(V, In, Element, Rest) :-
    var(V),
    !,
    generate_nth(1, V, In, Element, Rest).
nth1(V, In, Element, Rest) :-
    must_be(positive_integer, V),
    succ(V0, V),
    find_nth0(V0, In, Element, Rest).

generate_nth(I, I, [Head|Rest], Head, Rest).
generate_nth(I, IN, [H|List], El, [H|Rest]) :-
    I1 is I+1,
    generate_nth(I1, IN, List, El, Rest).

find_nth0(0, [Head|Rest], Head, Rest) :- !.
find_nth0(N, [Head|Rest0], Elem, [Head|Rest]) :-
    M is N-1,
    find_nth0(M, Rest0, Elem, Rest).


%!  last(?List, ?Last)
%
%   Succeeds when Last  is  the  last   element  of  List.  This
%   predicate is =semidet= if List is a  list and =multi= if List is
%   a partial list.
%
%   @compat There is no de-facto standard for the argument order of
%           last/2.  Be careful when porting code or use
%           append(_, [Last], List) as a portable alternative.

last([X|Xs], Last) :-
    last_(Xs, X, Last).

last_([], Last, Last).
last_([X|Xs], _, Last) :-
    last_(Xs, X, Last).


%!  proper_length(@List, -Length) is semidet.
%
%   True when Length is the number of   elements  in the proper list
%   List.  This is equivalent to
%
%     ==
%     proper_length(List, Length) :-
%           is_list(List),
%           length(List, Length).
%     ==

proper_length(List, Length) :-
    '$skip_list'(Length0, List, Tail),
    Tail == [],
    Length = Length0.


%!  same_length(?List1, ?List2)
%
%   Is true when List1 and List2 are   lists with the same number of
%   elements. The predicate is deterministic if  at least one of the
%   arguments is a proper list.  It   is  non-deterministic  if both
%   arguments are partial lists.
%
%   @see length/2

same_length([], []).
same_length([_|T1], [_|T2]) :-
    same_length(T1, T2).


%!  reverse(?List1, ?List2)
%
%   Is true when the elements of List2  are in reverse order compared to
%   List1. This predicate is deterministic if   either  list is a proper
%   list. If both  lists  are   _partial  lists_  backtracking generates
%   increasingly long lists.

reverse(Xs, Ys) :-
    reverse(Xs, Ys, [], Ys).

reverse([], [], Ys, Ys).
reverse([X|Xs], [_|Bound], Rs, Ys) :-
    reverse(Xs, Bound, [X|Rs], Ys).


%!  permutation(?Xs, ?Ys) is nondet.
%
%   True when Xs is a permutation of Ys. This can solve for Ys given
%   Xs or Xs given Ys, or  even   enumerate  Xs and Ys together. The
%   predicate  permutation/2  is  primarily   intended  to  generate
%   permutations. Note that a list of  length N has N! permutations,
%   and  unbounded  permutation  generation   becomes  prohibitively
%   expensive, even for rather short lists (10! = 3,628,800).
%
%   If both Xs and Ys are provided  and both lists have equal length
%   the order is |Xs|^2. Simply testing  whether Xs is a permutation
%   of Ys can be  achieved  in   order  log(|Xs|)  using  msort/2 as
%   illustrated below with the =semidet= predicate is_permutation/2:
%
%     ==
%     is_permutation(Xs, Ys) :-
%       msort(Xs, Sorted),
%       msort(Ys, Sorted).
%     ==
%
%   The example below illustrates that Xs   and Ys being proper lists
%   is not a sufficient condition to use the above replacement.
%
%     ==
%     ?- permutation([1,2], [X,Y]).
%     X = 1, Y = 2 ;
%     X = 2, Y = 1 ;
%     false.
%     ==
%
%   @error  type_error(list, Arg) if either argument is not a proper
%           or partial list.

permutation(Xs, Ys) :-
    '$skip_list'(Xlen, Xs, XTail),
    '$skip_list'(Ylen, Ys, YTail),
    (   XTail == [], YTail == []            % both proper lists
    ->  Xlen == Ylen
    ;   var(XTail), YTail == []             % partial, proper
    ->  length(Xs, Ylen)
    ;   XTail == [], var(YTail)             % proper, partial
    ->  length(Ys, Xlen)
    ;   var(XTail), var(YTail)              % partial, partial
    ->  length(Xs, Len),
        length(Ys, Len)
    ;   must_be(list, Xs),                  % either is not a list
        must_be(list, Ys)
    ),
    perm(Xs, Ys).

perm([], []).
perm(List, [First|Perm]) :-
    select(First, List, Rest),
    perm(Rest, Perm).

%!  flatten(+NestedList, -FlatList) is det.
%
%   Is true if FlatList is a  non-nested version of NestedList. Note
%   that empty lists are removed. In   standard Prolog, this implies
%   that the atom '[]' is removed  too.   In  SWI7, `[]` is distinct
%   from '[]'.
%
%   Ending up needing flatten/2 often   indicates, like append/3 for
%   appending two lists, a bad design. Efficient code that generates
%   lists from generated small  lists   must  use  difference lists,
%   often possible through grammar rules for optimal readability.
%
%   @see append/2

flatten(List, FlatList) :-
    flatten(List, [], FlatList0),
    !,
    FlatList = FlatList0.

flatten(Var, Tl, [Var|Tl]) :-
    var(Var),
    !.
flatten([], Tl, Tl) :- !.
flatten([Hd|Tl], Tail, List) :-
    !,
    flatten(Hd, FlatHeadTail, List),
    flatten(Tl, Tail, FlatHeadTail).
flatten(NonList, Tl, [NonList|Tl]).


		 /*******************************
		 *            CLUMPS		*
		 *******************************/

%!  clumped(+Items, -Pairs)
%
%   Pairs is a list  of  `Item-Count`   pairs  that  represents the _run
%   length encoding_ of Items.  For example:
%
%   ```
%   ?- clumped([a,a,b,a,a,a,a,c,c,c], R).
%   R = [a-2, b-1, a-4, c-3].
%   ```
%
%   @compat SICStus

clumped(Items, Counts) :-
    clump(Items, Counts).

clump([], []).
clump([H|T0], [H-C|T]) :-
    ccount(T0, H, T1, 1, C),
    clump(T1, T).

ccount([H|T0], E, T, C0, C) :-
    E == H,
    !,
    C1 is C0+1,
    ccount(T0, E, T, C1, C).
ccount(List, _, List, C, C).


%!  subseq(+List, -SubList, -Complement) is nondet.
%!  subseq(-List, +SubList, +Complement) is nondet.
%
%   Is true when SubList contains a subset   of  the elements of List in
%   the same order and Complement contains all   elements of List not in
%   SubList, also in the order they appear in List.
%
%   @compat SICStus.  The SWI-Prolog version raises an error for less
%   instantiated modes as these do not terminate.

subseq(L, S, C), is_list(L) => subseq_(L, S, C).
subseq(L, S, C), is_list(S), is_list(C) => subseq_(L, S, C).
subseq(L, S, C) =>
    must_be(list_or_partial_list, L),
    must_be(list_or_partial_list, S),
    must_be(list_or_partial_list, C),
    instantiation_error(L).

subseq_([], [], []).
subseq_([H|T0], T1, [H|C]) :-
    subseq_(T0, T1, C).
subseq_([H|T0], [H|T1], C) :-
    subseq_(T0, T1, C).


                 /*******************************
                 *       ORDER OPERATIONS       *
                 *******************************/

%!  max_member(-Max, +List) is semidet.
%
%   True when Max is the largest  member   in  the standard order of
%   terms.  Fails if List is empty.
%
%   @see compare/3
%   @see max_list/2 for the maximum of a list of numbers.

max_member(Max, [H|T]) =>
    max_member_(T, H, Max).
max_member(_, []) =>
    fail.

max_member_([], Max0, Max) =>
    Max = Max0.
max_member_([H|T], Max0, Max) =>
    (   H @=< Max0
    ->  max_member_(T, Max0, Max)
    ;   max_member_(T, H, Max)
    ).


%!  min_member(-Min, +List) is semidet.
%
%   True when Min is the smallest member   in  the standard order of
%   terms. Fails if List is empty.
%
%   @see compare/3
%   @see min_list/2 for the minimum of a list of numbers.

min_member(Min, [H|T]) =>
    min_member_(T, H, Min).
min_member(_, []) =>
    fail.

min_member_([], Min0, Min) =>
    Min = Min0.
min_member_([H|T], Min0, Min) =>
    (   H @>= Min0
    ->  min_member_(T, Min0, Min)
    ;   min_member_(T, H, Min)
    ).


%!  max_member(:Pred, -Max, +List) is semidet.
%
%   True when Max is the largest member according to Pred, which must be
%   a 2-argument callable that behaves like   (@=<)/2.  Fails if List is
%   empty.  The following call is equivalent to max_member/2:
%
%       ?- max_member(@=<, X, [6,1,8,4]).
%       X = 8.
%
%   @see max_list/2 for the maximum of a list of numbers.

max_member(Pred, Max, [H|T]) =>
    max_member_(T, Pred, H, Max).
max_member(_, _, []) =>
    fail.

max_member_([], _, Max0, Max) =>
    Max = Max0.
max_member_([H|T], Pred, Max0, Max) =>
    (   call(Pred, H, Max0)
    ->  max_member_(T, Pred, Max0, Max)
    ;   max_member_(T, Pred, H, Max)
    ).


%!  min_member(:Pred, -Min, +List) is semidet.
%
%   True when Min is the smallest member   according to Pred, which must
%   be a 2-argument callable that behaves like (@=<)/2. Fails if List is
%   empty. The following call is equivalent to max_member/2:
%
%       ?- min_member(@=<, X, [6,1,8,4]).
%       X = 1.
%
%   @see min_list/2 for the minimum of a list of numbers.

min_member(Pred, Min, [H|T]) =>
    min_member_(T, Pred, H, Min).
min_member(_, _, []) =>
    fail.

min_member_([], _, Min0, Min) =>
    Min = Min0.
min_member_([H|T], Pred, Min0, Min) =>
    (   call(Pred, Min0, H)
    ->  min_member_(T, Pred, Min0, Min)
    ;   min_member_(T, Pred, H, Min)
    ).


                 /*******************************
                 *       LISTS OF NUMBERS       *
                 *******************************/

%!  sum_list(+List, -Sum) is det.
%
%   Sum is the result of adding all numbers in List.

sum_list(Xs, Sum) :-
    sum_list(Xs, 0, Sum).

sum_list([], Sum0, Sum) =>
    Sum = Sum0.
sum_list([X|Xs], Sum0, Sum) =>
    Sum1 is Sum0 + X,
    sum_list(Xs, Sum1, Sum).

%!  max_list(+List:list(number), -Max:number) is semidet.
%
%   True if Max is the largest number in List.  Fails if List is
%   empty.
%
%   @see max_member/2.

max_list([H|T], Max) =>
    max_list(T, H, Max).
max_list([], _) => fail.

max_list([], Max0, Max) =>
    Max = Max0.
max_list([H|T], Max0, Max) =>
    Max1 is max(H, Max0),
    max_list(T, Max1, Max).


%!  min_list(+List:list(number), -Min:number) is semidet.
%
%   True if Min is the smallest  number   in  List. Fails if List is
%   empty.
%
%   @see min_member/2.

min_list([H|T], Min) =>
    min_list(T, H, Min).
min_list([], _) => fail.

min_list([], Min0, Min) =>
    Min = Min0.
min_list([H|T], Min0, Min) =>
    Min1 is min(H, Min0),
    min_list(T, Min1, Min).


%!  numlist(+Low, +High, -List) is semidet.
%
%   List is a list [Low, Low+1, ... High].  Fails if High < Low.
%
%   @error type_error(integer, Low)
%   @error type_error(integer, High)

numlist(L, U, Ns) :-
    must_be(integer, L),
    must_be(integer, U),
    L =< U,
    numlist_(L, U, Ns).

numlist_(U, U, List) :-
    !,
    List = [U].
numlist_(L, U, [L|Ns]) :-
    L2 is L+1,
    numlist_(L2, U, Ns).


                /********************************
                *       SET MANIPULATION        *
                *********************************/

%!  is_set(@Set) is semidet.
%
%   True if Set is a proper  list without duplicates. Equivalence is
%   based on ==/2. The  implementation   uses  sort/2, which implies
%   that the complexity is N*log(N) and   the  predicate may cause a
%   resource-error. There are no other error conditions.

is_set(Set) :-
    '$skip_list'(Len, Set, Tail),
    Tail == [],                             % Proper list
    sort(Set, Sorted),
    length(Sorted, Len).


%!  list_to_set(+List, ?Set) is det.
%
%   True when Set has the same elements   as List in the same order.
%   The left-most copy of duplicate elements   is retained. List may
%   contain  variables.  Elements  _E1_  and   _E2_  are  considered
%   duplicates iff _E1_  ==  _E2_  holds.   The  complexity  of  the
%   implementation is N*log(N).
%
%   @see    sort/2 can be used to create an ordered set.  Many
%           set operations on ordered sets are order N rather than
%           order N**2.  The list_to_set/2 predicate is more
%           expensive than sort/2 because it involves, two sorts
%           and a linear scan.
%   @compat Up to version 6.3.11, list_to_set/2 had complexity
%           N**2 and equality was tested using =/2.
%   @error  List is type-checked.

list_to_set(List, Set) :-
    must_be(list, List),
    number_list(List, 1, Numbered),
    sort(1, @=<, Numbered, ONum),
    remove_dup_keys(ONum, NumSet),
    sort(2, @=<, NumSet, ONumSet),
    pairs_keys(ONumSet, Set).

number_list([], _, []).
number_list([H|T0], N, [H-N|T]) :-
    N1 is N+1,
    number_list(T0, N1, T).

remove_dup_keys([], []).
remove_dup_keys([H|T0], [H|T]) :-
    H = V-_,
    remove_same_key(T0, V, T1),
    remove_dup_keys(T1, T).

remove_same_key([V1-_|T0], V, T) :-
    V1 == V,
    !,
    remove_same_key(T0, V, T).
remove_same_key(L, _, L).


%!  intersection(+Set1, +Set2, -Set3) is det.
%
%   True if Set3 unifies with the  intersection   of  Set1 and Set2. The
%   complexity of this predicate is |Set1|*|Set2|. A _set_ is defined to
%   be an unordered list  without   duplicates.  Elements are considered
%   duplicates if they can be unified.
%
%   @see ord_intersection/3.

intersection([], _, Set) =>
    Set = [].
intersection([X|T], L, Intersect) =>
    (   memberchk(X, L)
    ->  Intersect = [X|R],
        intersection(T, L, R)
    ;   intersection(T, L, Intersect)
    ).

%!  union(+Set1, +Set2, -Set3) is det.
%
%   True if Set3 unifies with the union of  the lists Set1 and Set2. The
%   complexity of this predicate is |Set1|*|Set2|. A _set_ is defined to
%   be an unordered list  without   duplicates.  Elements are considered
%   duplicates if they can be unified.
%
%   @see ord_union/3

union([], L0, L) =>
    L = L0.
union([H|T], L, Union) =>
    (   memberchk(H, L)
    ->  union(T, L, Union)
    ;   Union = [H|R],
        union(T, L, R)
    ).

%!  subset(+SubSet, +Set) is semidet.
%
%   True if all elements of SubSet  belong   to  Set as well. Membership
%   test is based on memberchk/2. The   complexity  is |SubSet|*|Set|. A
%   _set_ is defined  to  be  an   unordered  list  without  duplicates.
%   Elements are considered duplicates if they can be unified.
%
%   @see ord_subset/2.

subset([], _) => true.
subset([E|R], Set) =>
    memberchk(E, Set),
    subset(R, Set).


%!  subtract(+Set, +Delete, -Result) is det.
%
%   Delete all elements  in  Delete  from   Set.  Deletion  is  based on
%   unification using memberchk/2. The complexity   is |Delete|*|Set|. A
%   _set_ is defined  to  be  an   unordered  list  without  duplicates.
%   Elements are considered duplicates if they can be unified.
%
%   @see ord_subtract/3.

subtract([], _, R) =>
    R = [].
subtract([E|T], D, R) =>
    (   memberchk(E, D)
    ->  subtract(T, D, R)
    ;   R = [E|R1],
        subtract(T, D, R1)
    ).
