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    1/*  Part of SWI-Prolog
    2
    3    Author:        Jan Wielemaker and Richard O'Keefe
    4    E-mail:        J.Wielemaker@cs.vu.nl
    5    WWW:           http://www.swi-prolog.org
    6    Copyright (c)  2002-2020, University of Amsterdam
    7                              VU University Amsterdam
    8                              SWI-Prolog Solutions b.v.
    9    All rights reserved.
   10
   11    Redistribution and use in source and binary forms, with or without
   12    modification, are permitted provided that the following conditions
   13    are met:
   14
   15    1. Redistributions of source code must retain the above copyright
   16       notice, this list of conditions and the following disclaimer.
   17
   18    2. Redistributions in binary form must reproduce the above copyright
   19       notice, this list of conditions and the following disclaimer in
   20       the documentation and/or other materials provided with the
   21       distribution.
   22
   23    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
   24    "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
   25    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
   26    FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
   27    COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
   28    INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
   29    BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
   30    LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
   31    CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
   32    LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
   33    ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
   34    POSSIBILITY OF SUCH DAMAGE.
   35*/
   36
   37:- module(lists,
   38        [ member/2,                     % ?X, ?List
   39          memberchk/2,                  % ?X, ?List
   40          append/2,                     % +ListOfLists, -List
   41          append/3,                     % ?A, ?B, ?AB
   42          prefix/2,                     % ?Part, ?Whole
   43          select/3,                     % ?X, ?List, ?Rest
   44          selectchk/3,                  % ?X, ?List, ?Rest
   45          select/4,                     % ?X, ?XList, ?Y, ?YList
   46          selectchk/4,                  % ?X, ?XList, ?Y, ?YList
   47          nextto/3,                     % ?X, ?Y, ?List
   48          delete/3,                     % ?List, ?X, ?Rest
   49          nth0/3,                       % ?N, ?List, ?Elem
   50          nth1/3,                       % ?N, ?List, ?Elem
   51          nth0/4,                       % ?N, ?List, ?Elem, ?Rest
   52          nth1/4,                       % ?N, ?List, ?Elem, ?Rest
   53          last/2,                       % +List, -Element
   54          proper_length/2,              % @List, -Length
   55          same_length/2,                % ?List1, ?List2
   56          reverse/2,                    % +List, -Reversed
   57          permutation/2,                % ?List, ?Permutation
   58          flatten/2,                    % +Nested, -Flat
   59          clumped/2,                    % +Items,-Pairs
   60
   61                                        % Ordered operations
   62          max_member/2,                 % -Max, +List
   63          min_member/2,                 % -Min, +List
   64
   65                                        % Lists of numbers
   66          sum_list/2,                   % +List, -Sum
   67          max_list/2,                   % +List, -Max
   68          min_list/2,                   % +List, -Min
   69          numlist/3,                    % +Low, +High, -List
   70
   71                                        % set manipulation
   72          is_set/1,                     % +List
   73          list_to_set/2,                % +List, -Set
   74          intersection/3,               % +List1, +List2, -Intersection
   75          union/3,                      % +List1, +List2, -Union
   76          subset/2,                     % +SubSet, +Set
   77          subtract/3                    % +Set, +Delete, -Remaining
   78        ]).   79:- autoload(library(error),[must_be/2]).   80:- autoload(library(pairs),[pairs_keys/2]).   81
   82
   83:- set_prolog_flag(generate_debug_info, false).

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.

Compatibility
- 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
  115member(El, [H|T]) :-
  116    member_(T, El, H).
  117
  118member_(_, El, El).
  119member_([H|T], El, _) :-
  120    member_(T, El, H).
 append(?List1, ?List2, ?List1AndList2)
List1AndList2 is the concatenation of List1 and List2
  126append([], L, L).
  127append([H|T], L, [H|R]) :-
  128    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.
Arguments:
ListOfLists- must be a list of possibly partial lists
  137append(ListOfLists, List) :-
  138    must_be(list, ListOfLists),
  139    append_(ListOfLists, List).
  140
  141append_([], []).
  142append_([L|Ls], As) :-
  143    append(L, Ws, As),
  144    append_(Ls, Ws).
 prefix(?Part, ?Whole)
True iff Part is a leading substring of Whole. This is the same as append(Part, _, Whole).
  152prefix([], _).
  153prefix([E|T0], [E|T]) :-
  154    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.
  163select(X, [Head|Tail], Rest) :-
  164    select3_(Tail, Head, X, Rest).
  165
  166select3_(Tail, Head, Head, Tail).
  167select3_([Head2|Tail], Head, X, [Head|Rest]) :-
  168    select3_(Tail, Head2, X, Rest).
 selectchk(+Elem, +List, -Rest) is semidet
Semi-deterministic removal of first element in List that unifies with Elem.
  176selectchk(Elem, List, Rest) :-
  177    select(Elem, List, Rest0),
  178    !,
  179    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 also
- selectchk/4 provides a semidet version.
  200select(X, XList, Y, YList) :-
  201    select4_(XList, X, Y, YList).
  202
  203select4_([X|List], X, Y, [Y|List]).
  204select4_([X0|XList], X, Y, [X0|YList]) :-
  205    select4_(XList, X, Y, YList).
 selectchk(?X, ?XList, ?Y, ?YList) is semidet
Semi-deterministic version of select/4.
  211selectchk(X, XList, Y, YList) :-
  212    select(X, XList, Y, YList),
  213    !.
 nextto(?X, ?Y, ?List)
True if Y directly follows X in List.
  219nextto(X, Y, [X,Y|_]).
  220nextto(X, Y, [_|Zs]) :-
  221    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.
See also
- select/3, subtract/3.
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.
  236delete([], _, []).
  237delete([Elem|Tail], Del, Result) :-
  238    (   \+ Elem \= Del
  239    ->  delete(Tail, Del, Result)
  240    ;   Result = [Elem|Rest],
  241        delete(Tail, Del, Rest)
  242    ).
  243
  244
  245/*  nth0/3, nth1/3 are improved versions from
  246    Martin Jansche <martin@pc03.idf.uni-heidelberg.de>
  247*/
 nth0(?Index, ?List, ?Elem)
True when Elem is the Index'th element of List. Counting starts at 0.
Errors
- type_error(integer, Index) if Index is not an integer or unbound.
See also
- nth1/3.
  258nth0(Index, List, Elem) :-
  259    (   integer(Index)
  260    ->  nth0_det(Index, List, Elem)         % take nth deterministically
  261    ;   var(Index)
  262    ->  List = [H|T],
  263        nth_gen(T, Elem, H, 0, Index)       % match
  264    ;   must_be(integer, Index)
  265    ).
  266
  267nth0_det(0, [Elem|_], Elem) :- !.
  268nth0_det(1, [_,Elem|_], Elem) :- !.
  269nth0_det(2, [_,_,Elem|_], Elem) :- !.
  270nth0_det(3, [_,_,_,Elem|_], Elem) :- !.
  271nth0_det(4, [_,_,_,_,Elem|_], Elem) :- !.
  272nth0_det(5, [_,_,_,_,_,Elem|_], Elem) :- !.
  273nth0_det(N, [_,_,_,_,_,_   |Tail], Elem) :-
  274    M is N - 6,
  275    M >= 0,
  276    nth0_det(M, Tail, Elem).
  277
  278nth_gen(_, Elem, Elem, Base, Base).
  279nth_gen([H|Tail], Elem, _, N, Base) :-
  280    succ(N, M),
  281    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 also
- nth0/3.
  291nth1(Index, List, Elem) :-
  292    (   integer(Index)
  293    ->  Index0 is Index - 1,
  294        nth0_det(Index0, List, Elem)        % take nth deterministically
  295    ;   var(Index)
  296    ->  List = [H|T],
  297        nth_gen(T, Elem, H, 1, Index)       % match
  298    ;   must_be(integer, Index)
  299    ).
 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].
  320nth0(V, In, Element, Rest) :-
  321    var(V),
  322    !,
  323    generate_nth(0, V, In, Element, Rest).
  324nth0(V, In, Element, Rest) :-
  325    must_be(nonneg, V),
  326    find_nth0(V, In, Element, Rest).
 nth1(?N, ?List, ?Elem, ?Rest) is det
As nth0/4, but counting starts at 1.
  332nth1(V, In, Element, Rest) :-
  333    var(V),
  334    !,
  335    generate_nth(1, V, In, Element, Rest).
  336nth1(V, In, Element, Rest) :-
  337    must_be(positive_integer, V),
  338    succ(V0, V),
  339    find_nth0(V0, In, Element, Rest).
  340
  341generate_nth(I, I, [Head|Rest], Head, Rest).
  342generate_nth(I, IN, [H|List], El, [H|Rest]) :-
  343    I1 is I+1,
  344    generate_nth(I1, IN, List, El, Rest).
  345
  346find_nth0(0, [Head|Rest], Head, Rest) :- !.
  347find_nth0(N, [Head|Rest0], Elem, [Head|Rest]) :-
  348    M is N-1,
  349    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.
Compatibility
- 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.
  362last([X|Xs], Last) :-
  363    last_(Xs, X, Last).
  364
  365last_([], Last, Last).
  366last_([X|Xs], _, Last) :-
  367    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).
  381proper_length(List, Length) :-
  382    '$skip_list'(Length0, List, Tail),
  383    Tail == [],
  384    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 also
- length/2
  396same_length([], []).
  397same_length([_|T1], [_|T2]) :-
  398    same_length(T1, T2).
 reverse(?List1, ?List2)
Is true when the elements of List2 are in reverse order compared to List1.
  406reverse(Xs, Ys) :-
  407    reverse(Xs, [], Ys, Ys).
  408
  409reverse([], Ys, Ys, []).
  410reverse([X|Xs], Rs, Ys, [_|Bound]) :-
  411    reverse(Xs, [X|Rs], Ys, Bound).
 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.
Errors
- type_error(list, Arg) if either argument is not a proper or partial list.
  447permutation(Xs, Ys) :-
  448    '$skip_list'(Xlen, Xs, XTail),
  449    '$skip_list'(Ylen, Ys, YTail),
  450    (   XTail == [], YTail == []            % both proper lists
  451    ->  Xlen == Ylen
  452    ;   var(XTail), YTail == []             % partial, proper
  453    ->  length(Xs, Ylen)
  454    ;   XTail == [], var(YTail)             % proper, partial
  455    ->  length(Ys, Xlen)
  456    ;   var(XTail), var(YTail)              % partial, partial
  457    ->  length(Xs, Len),
  458        length(Ys, Len)
  459    ;   must_be(list, Xs),                  % either is not a list
  460        must_be(list, Ys)
  461    ),
  462    perm(Xs, Ys).
  463
  464perm([], []).
  465perm(List, [First|Perm]) :-
  466    select(First, List, Rest),
  467    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 also
- append/2
  483flatten(List, FlatList) :-
  484    flatten(List, [], FlatList0),
  485    !,
  486    FlatList = FlatList0.
  487
  488flatten(Var, Tl, [Var|Tl]) :-
  489    var(Var),
  490    !.
  491flatten([], Tl, Tl) :- !.
  492flatten([Hd|Tl], Tail, List) :-
  493    !,
  494    flatten(Hd, FlatHeadTail, List),
  495    flatten(Tl, Tail, FlatHeadTail).
  496flatten(NonList, Tl, [NonList|Tl]).
  497
  498
  499		 /*******************************
  500		 *            CLUMPS		*
  501		 *******************************/
 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].
Compatibility
- SICStus
  515clumped(Items, Counts) :-
  516    clump(Items, Counts).
  517
  518clump([], []).
  519clump([H|T0], [H-C|T]) :-
  520    ccount(T0, H, T1, 1, C),
  521    clump(T1, T).
  522
  523ccount([H|T0], E, T, C0, C) :-
  524    E == H,
  525    !,
  526    C1 is C0+1,
  527    ccount(T0, E, T, C1, C).
  528ccount(List, _, List, C, C).
  529
  530
  531                 /*******************************
  532                 *       ORDER OPERATIONS       *
  533                 *******************************/
 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 also
- compare/3
- max_list/2 for the maximum of a list of numbers.
  543max_member(Max, [H|T]) =>
  544    max_member_(T, H, Max).
  545max_member(_, []) =>
  546    fail.
  547
  548max_member_([], Max0, Max) =>
  549    Max = Max0.
  550max_member_([H|T], Max0, Max) =>
  551    (   H @=< Max0
  552    ->  max_member_(T, Max0, Max)
  553    ;   max_member_(T, H, Max)
  554    ).
 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 also
- compare/3
- min_list/2 for the minimum of a list of numbers.
  565min_member(Min, [H|T]) =>
  566    min_member_(T, H, Min).
  567min_member(_, []) =>
  568    fail.
  569
  570min_member_([], Min0, Min) =>
  571    Min = Min0.
  572min_member_([H|T], Min0, Min) =>
  573    (   H @>= Min0
  574    ->  min_member_(T, Min0, Min)
  575    ;   min_member_(T, H, Min)
  576    ).
  577
  578
  579                 /*******************************
  580                 *       LISTS OF NUMBERS       *
  581                 *******************************/
 sum_list(+List, -Sum) is det
Sum is the result of adding all numbers in List.
  587sum_list(Xs, Sum) :-
  588    sum_list(Xs, 0, Sum).
  589
  590sum_list([], Sum0, Sum) =>
  591    Sum = Sum0.
  592sum_list([X|Xs], Sum0, Sum) =>
  593    Sum1 is Sum0 + X,
  594    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 also
- max_member/2.
  603max_list([H|T], Max) =>
  604    max_list(T, H, Max).
  605max_list([], _) => fail.
  606
  607max_list([], Max0, Max) =>
  608    Max = Max0.
  609max_list([H|T], Max0, Max) =>
  610    Max1 is max(H, Max0),
  611    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 also
- min_member/2.
  621min_list([H|T], Min) =>
  622    min_list(T, H, Min).
  623min_list([], _) => fail.
  624
  625min_list([], Min0, Min) =>
  626    Min = Min0.
  627min_list([H|T], Min0, Min) =>
  628    Min1 is min(H, Min0),
  629    min_list(T, Min1, Min).
 numlist(+Low, +High, -List) is semidet
List is a list [Low, Low+1, ... High]. Fails if High < Low.
Errors
- type_error(integer, Low)
- type_error(integer, High)
  639numlist(L, U, Ns) :-
  640    must_be(integer, L),
  641    must_be(integer, U),
  642    L =< U,
  643    numlist_(L, U, Ns).
  644
  645numlist_(U, U, List) :-
  646    !,
  647    List = [U].
  648numlist_(L, U, [L|Ns]) :-
  649    L2 is L+1,
  650    numlist_(L2, U, Ns).
  651
  652
  653                /********************************
  654                *       SET MANIPULATION        *
  655                *********************************/
 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.
  664is_set(Set) :-
  665    '$skip_list'(Len, Set, Tail),
  666    Tail == [],                             % Proper list
  667    sort(Set, Sorted),
  668    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).
Errors
- List is type-checked.
See also
- 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.
Compatibility
- Up to version 6.3.11, list_to_set/2 had complexity N**2 and equality was tested using =/2.
  688list_to_set(List, Set) :-
  689    must_be(list, List),
  690    number_list(List, 1, Numbered),
  691    sort(1, @=<, Numbered, ONum),
  692    remove_dup_keys(ONum, NumSet),
  693    sort(2, @=<, NumSet, ONumSet),
  694    pairs_keys(ONumSet, Set).
  695
  696number_list([], _, []).
  697number_list([H|T0], N, [H-N|T]) :-
  698    N1 is N+1,
  699    number_list(T0, N1, T).
  700
  701remove_dup_keys([], []).
  702remove_dup_keys([H|T0], [H|T]) :-
  703    H = V-_,
  704    remove_same_key(T0, V, T1),
  705    remove_dup_keys(T1, T).
  706
  707remove_same_key([V1-_|T0], V, T) :-
  708    V1 == V,
  709    !,
  710    remove_same_key(T0, V, T).
  711remove_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 also
- ord_intersection/3.
  723intersection([], _, Set) =>
  724    Set = [].
  725intersection([X|T], L, Intersect) =>
  726    (   memberchk(X, L)
  727    ->  Intersect = [X|R],
  728        intersection(T, L, R)
  729    ;   intersection(T, L, Intersect)
  730    ).
 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 also
- ord_union/3
  741union([], L0, L) =>
  742    L = L0.
  743union([H|T], L, Union) =>
  744    (   memberchk(H, L)
  745    ->  union(T, L, Union)
  746    ;   Union = [H|R],
  747        union(T, L, R)
  748    ).
 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 also
- ord_subset/2.
  759subset([], _) => true.
  760subset([E|R], Set) =>
  761    memberchk(E, Set),
  762    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 also
- ord_subtract/3.
  774subtract([], _, R) =>
  775    R = [].
  776subtract([E|T], D, R) =>
  777    (   memberchk(E, D)
  778    ->  subtract(T, D, R)
  779    ;   R = [E|R1],
  780        subtract(T, D, R1)
  781    )