## A.17 library(ordsets): Ordered set manipulation

Ordered sets are lists with unique elements sorted to the standard order of terms (see sort/2). Exploiting ordering, many of the set operations can be expressed in order N rather than N`^`2 when dealing with unordered sets that may contain duplicates. The library(ordsets) is available in a number of Prolog implementations. Our predicates are designed to be compatible with common practice in the Prolog community. The implementation is incomplete and relies partly on library(oset), an older ordered set library distributed with SWI-Prolog. New applications are advices to use library(ordsets).

Some of these predicates match directly to corresponding list operations. It is adviced to use the versions from this library to make clear you are operating on ordered sets.

[semidet]is_ordset(@Term)
  True if Term is an ordered set. All predicates in this library expect ordered sets as input arguments. Failing to fullfil this assumption results in undefined behaviour. Typically, ordered sets are created by predicates from this library, sort/2 or setof/3.
[semidet]ord_empty(?List)
  True when List is the empty ordered set. Simply unifies list with the empty list. Not part of Quintus.
[semidet]ord_seteq(+Set1, +Set2)
  True if Set1 and Set2 have the same elements. As both are canonical sorted lists, this is the same as ==/2.
  Compatibility
    sicstus
[det]list_to_ord_set(+List, -OrdSet)
  Transform a list into an ordered set. This is the same as sorting the list.
[semidet]ord_intersect(+Set1, +Set2)
  True if both ordered sets have a non-empty intersection.
[semidet]ord_disjoint(+Set1, +Set2)
  True if Set1 and Set2 have no common elements. This is the negation of ord_intersect/2.
ord_intersect(+Set1, +Set2, -Intersection)
  Intersection holds the common elements of Set1 and Set2.
  deprecated
    Use ord_intersection/3
ord_intersection(+PowerSet, -Intersection)
  True if Intersection is an ordered set holding all elements common to all sets in PowerSet.
  Compatibility
    sicstus
[det]ord_intersection(+Set1, +Set2, -Intersection)
  Intersection holds the common elements of Set1 and Set2.
[det]ord_intersection(+Set1, +Set2, ?Intersection, ?Difference)
  Intersection and difference between two ordered sets. Intersection is the intersection between Set1 and Set2, while Difference is defined by ord_subtract(Set2, Set1, Difference).
  See also
    ord_intersection/3 and ord_subtract/3.
[det]ord_add_element(+Set1, +Element, ?Set2)
  Insert an element into the set. This is the same as ord_union(Set1, [Element], Set2).
[det]ord_del_element(+Set, +Element, -NewSet)
  Delete an element from an ordered set. This is the same as ord_subtract(Set, [Element], NewSet).
[semidet]ord_memberchk(+Element, +Set)
  Check membership. This could stop comparing we have passed the right value, saving scanning (on average) half the list if Element is not in Set. Probably the built-in memberchk/2 will be faster.
  Compatibility
    Not part of original Quintus library
[nondet]ord_member(?Element, +Set)
  True if Element is a member of Set. Stops if further elements are behind Element in the standard order of terms.
  Compatibility
    sicstus
[semidet]ord_subset(+Sub, +Super)
  Is true if all element of Sub are in Super
[det]ord_subtract(+InOSet, +NotInOSet, -Diff)
  Diff is the set holding all elements of InOSet that are not in NotInOSet.
[det]ord_union(+SetOfSets, -Union)
  True if Union is the union of all elements in the superset SetOfSets. Each member of SetOfSets must be an ordered set, the sets need not be ordered in any way.
  author
    Copied from YAP, probably originally by Richard O'Keefe.
[det]ord_union(+Set1, +Set2, ?Union)
  Union is the union of Set1 and Set2
[det]ord_union(+Set1, +Set2, -Union, -New)
  True if Union iff ord_union(Set1, Set2, Union) and ord_subtract(Set2, Set1, New).
[det]ord_symdiff(+Set1, +Set2, ?Difference)
  Is true when Difference is the symmetric difference of Set1 and Set2. I.e., Difference contains all elements that are not in the intersection of Set1 and Set2. The semantics is the same as the sequence below (but the actual implementation requires only a single scan).

            ord_union(Set1, Set2, Union),
            ord_intersection(Set1, Set2, Intersection),
            ord_subtract(Union, Intersection, Difference).

  For example:

      ?- ord_symdiff([1,2], [2,3], X).
      X = [1,3].