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Linux 2.6 - man page for sofs (linux section 3erl)

sofs(3erl)			     Erlang Module Definition			       sofs(3erl)

NAME
       sofs - Functions for Manipulating Sets of Sets

DESCRIPTION
       The  sofs  module  implements operations on finite sets and relations represented as sets.
       Intuitively, a set is a collection of elements; every element belongs to the set, and  the
       set contains every element.

       Given  a set A and a sentence S(x), where x is a free variable, a new set B whose elements
       are exactly those elements of A for which S(x) holds can be formed, this is denoted B = {x
       in  A  :  S(x)}. Sentences are expressed using the logical operators "for some" (or "there
       exists"), "for all", "and", "or", "not". If the existence of  a	set  containing  all  the
       specified elements is known (as will always be the case in this module), we write B = {x :
       S(x)}.

       The unordered set containing the elements a, b and c is denoted {a, b, c}.  This  notation
       is  not	to  be confused with tuples. The ordered pair of a and b, with first coordinate a
       and second coordinate b, is denoted (a, b). An ordered pair is an ordered set of two  ele-
       ments.  In this module ordered sets can contain one, two or more elements, and parentheses
       are used to enclose the elements. Unordered sets and ordered sets are orthogonal, again in
       this module; there is no unordered set equal to any ordered set.

       The  set  that  contains no elements is called the empty set . If two sets A and B contain
       the same elements, then A is equal to B, denoted A = B. Two ordered sets are equal if they
       contain the same number of elements and have equal elements at each coordinate. If a set A
       contains all elements that B contains, then B is a subset of A. The union of  two  sets	A
       and  B  is  the	smallest  set  that contains all elements of A and all elements of B. The
       intersection of two sets A and B is the set that contains all elements of A that belong to
       B.  Two	sets  are  disjoint if their intersection is the empty set. The difference of two
       sets A and B is the set that contains all elements of A that do not belong to B. The  sym-
       metric difference of two sets is the set that contains those element that belong to either
       of the two sets, but not both. The union of a collection of sets is the smallest set  that
       contains all the elements that belong to at least one set of the collection. The intersec-
       tion of a non-empty collection of sets is the set that contains all elements  that  belong
       to every set of the collection.

       The  Cartesian  product of two sets X and Y, denoted X x Y, is the set {a : a = (x, y) for
       some x in X and for some y in Y}. A relation is a subset of X x Y. Let R  be  a	relation.
       The fact that (x, y) belongs to R is written as x R y. Since relations are sets, the defi-
       nitions of the last paragraph (subset, union, and so on) apply to relations as  well.  The
       domain  of  R is the set {x : x R y for some y in Y}. The range of R is the set {y : x R y
       for some x in X}. The converse of R is the set {a : a = (y, x) for some (x, y) in R}. If A
       is  a subset of X, then the image of A under R is the set {y : x R y for some x in A}, and
       if B is a subset of Y, then the inverse image of B is the set {x : x R y for some y in B}.
       If  R is a relation from X to Y and S is a relation from Y to Z, then the relative product
       of R and S is the relation T from X to Z defined so that x T z if and only if there exists
       an  element  y  in  Y  such  that  x R y and y S z. The restriction of R to A is the set S
       defined so that x S y if and only if there exists an element x in A such that x R y. If	S
       is  a  restriction  of R to A, then R is an extension of S to X. If X = Y then we call R a
       relation in X. The field of a relation R in X is the union of the  domain  of  R  and  the
       range  of R. If R is a relation in X, and if S is defined so that x S y if x R y and not x
       = y, then S is the strict relation corresponding to R, and vice versa, if S is a  relation
       in  X,  and  if	R is defined so that x R y if x S y or x = y, then R is the weak relation
       corresponding to S. A relation R in X is reflexive if x R x for every element x of  X;  it
       is  symmetric  if  x R y implies that y R x; and it is transitive if x R y and y R z imply
       that x R z.

       A function F is a relation, a subset of X x Y, such that the domain of F is equal to X and
       such  that  for every x in X there is a unique element y in Y with (x, y) in F. The latter
       condition can be formulated as follows: if x F y and x F z then y = z. In this module,  it
       will  not be required that the domain of F be equal to X for a relation to be considered a
       function. Instead of writing (x, y) in F or x F y, we write F(x) = y when F is a function,
       and  say that F maps x onto y, or that the value of F at x is y. Since functions are rela-
       tions, the definitions of the last paragraph (domain, range, and so on) apply to functions
       as  well.  If the converse of a function F is a function F', then F' is called the inverse
       of F. The relative product of two functions F1 and F2 is called the composite of F1 and F2
       if the range of F1 is a subset of the domain of F2.

       Sometimes,  when  the  range of a function is more important than the function itself, the
       function is called a family . The domain of a family is called the index  set  ,  and  the
       range  is  called  the  indexed	set . If x is a family from I to X, then x[i] denotes the
       value of the function at index i. The notation "a family in X" is used for such a  family.
       When the indexed set is a set of subsets of a set X, then we call x a family of subsets of
       X. If x is a family of subsets of X, then the union of the range of x is called the  union
       of  the	family x. If x is non-empty (the index set is non-empty), the intersection of the
       family x is the intersection of the range of x. In this module,	the  only  families  that
       will  be considered are families of subsets of some set X; in the following the word "fam-
       ily" will be used for such families of subsets.

       A partition of a set X is a collection S of non-empty subsets of X whose union  is  X  and
       whose elements are pairwise disjoint. A relation in a set is an equivalence relation if it
       is reflexive, symmetric and transitive. If R is an equivalence relation in X, and x is  an
       element	of  X,	the equivalence class of x with respect to R is the set of all those ele-
       ments y of X for which x R y holds. The equivalence classes constitute a  partitioning  of
       X. Conversely, if C is a partition of X, then the relation that holds for any two elements
       of X if they belong to the same equivalence class, is an equivalence relation  induced  by
       the partition C. If R is an equivalence relation in X, then the canonical map is the func-
       tion that maps every element of X onto its equivalence class.

       Relations as defined above (as sets of ordered pairs) will from now on be referred  to  as
       binary  relations  . We call a set of ordered sets (x[1], ..., x[n]) an (n-ary) relation ,
       and say that the relation is a subset of the Cartesian product X[1] x  ...  x  X[n]  where
       x[i]  is an element of X[i], 1 <= i <= n. The projection of an n-ary relation R onto coor-
       dinate i is the set {x[i] : (x[1], ..., x[i], ..., x[n]) in R for some x[j] in X[j], 1  <=
       j  <=  n  and not i = j}. The projections of a binary relation R onto the first and second
       coordinates are the domain and the range of R respectively. The relative product of binary
       relations  can  be  generalized	to  n-ary  relations as follows. Let TR be an ordered set
       (R[1], ..., R[n]) of binary relations from X to Y[i] and S a binary relation from (Y[1]	x
       ...  x  Y[n])  to Z. The relative product of TR and S is the binary relation T from X to Z
       defined so that x T z if and only if there exists an element y[i] in Y[i] for each 1 <=	i
       <=  n  such  that  x  R[i]  y[i] and (y[1], ..., y[n]) S z. Now let TR be a an ordered set
       (R[1], ..., R[n]) of binary relations from X[i] to Y[i] and S a subset of  X[1]	x  ...	x
       X[n].  The multiple relative product of TR and S is defined to be the set {z : z = ((x[1],
       ..., x[n]), (y[1],...,y[n])) for some (x[1], ..., x[n]) in S and for some (x[i], y[i])  in
       R[i],  1  <=  i	<= n}. The natural join of an n-ary relation R and an m-ary relation S on
       coordinate i and j is defined to be the set {z : z = (x[1], ..., x[n], y[1], ...,  y[j-1],
       y[j+1],	...,  y[m])  for  some (x[1], ..., x[n]) in R and for some (y[1], ..., y[m]) in S
       such that x[i] = y[j]}.

       The sets recognized by this module will be represented by elements of the  relation  Sets,
       defined as the smallest set such that:

	 * for every atom T except '_' and for every term X, (T, X) belongs to Sets ( atomic sets
	   );

	 * (['_'], []) belongs to Sets (the untyped empty set );

	 * for every tuple T = {T[1], ..., T[n]} and for every tuple X = {X[1],  ...,  X[n]},  if
	   (T[i],  X[i])  belongs  to  Sets  for  every 1 <= i <= n then (T, X) belongs to Sets (
	   ordered sets );

	 * for every term T, if X is the empty list or a non-empty sorted list [X[1], ...,  X[n]]
	   without  duplicates	such  that  (T, X[i]) belongs to Sets for every 1 <= i <= n, then
	   ([T], X) belongs to Sets ( typed unordered sets ).

       An external set is an element of the range of Sets. A type is an element of the domain  of
       Sets.  If S is an element (T, X) of Sets, then T is a valid type of X, T is the type of S,
       and X is the external set of S. from_term/2 creates a set from a type and an  Erlang  term
       turned into an external set.

       The actual sets represented by Sets are the elements of the range of the function Set from
       Sets to Erlang terms and sets of Erlang terms:

	 * Set(T,Term) = Term, where T is an atom;

	 * Set({T[1], ..., T[n]}, {X[1], ..., X[n]}) = (Set(T[1], X[1]), ..., Set(T[n], X[n]));

	 * Set([T], [X[1], ..., X[n]]) = {Set(T, X[1]), ..., Set(T, X[n])};

	 * Set([T], []) = {}.

       When there is no risk of confusion, elements of Sets will be identified with the sets they
       represent.  For	instance,  if  U is the result of calling union/2 with S1 and S2 as argu-
       ments, then U is said to be the union of S1 and S2. A more precise  formulation	would  be
       that Set(U) is the union of Set(S1) and Set(S2).

       The  types  are	used to implement the various conditions that sets need to fulfill. As an
       example, consider the relative product of two sets R and S, and recall that  the  relative
       product	of  R  and S is defined if R is a binary relation to Y and S is a binary relation
       from Y. The function that implements the relative  product,  relative_product/2	,  checks
       that  the arguments represent binary relations by matching [{A,B}] against the type of the
       first argument (Arg1 say), and [{C,D}] against the type of the second argument (Arg2 say).
       The fact that [{A,B}] matches the type of Arg1 is to be interpreted as Arg1 representing a
       binary relation from X to Y, where X is defined as all sets Set(x) for some element  x  in
       Sets  the  type of which is A, and similarly for Y. In the same way Arg2 is interpreted as
       representing a binary relation from W to Z. Finally it is checked that B matches C,  which
       is sufficient to ensure that W is equal to Y. The untyped empty set is handled separately:
       its type, ['_'], matches the type of any unordered set.

       A few functions of this module ( drestriction/3 , family_projection/2 , partition/2 , par-
       tition_family/2	,  projection/2 , restriction/3 , substitution/2 ) accept an Erlang func-
       tion as a means to modify each element of a given unordered set. Such a	function,  called
       SetFun in the following, can be specified as a functional object (fun), a tuple {external,
       Fun} , or an integer. If SetFun is specified as a fun, the fun is applied to each  element
       of  the given set and the return value is assumed to be a set. If SetFun is specified as a
       tuple {external, Fun} , Fun is applied to the external set of each element  of  the  given
       set  and  the  return value is assumed to be an external set. Selecting the elements of an
       unordered set as external sets and assembling a new unordered set from a list of  external
       sets is in the present implementation more efficient than modifying each element as a set.
       However, this optimization can only be utilized when the elements of the unordered set are
       atomic  or  ordered  sets.  It must also be the case that the type of the elements matches
       some clause of Fun (the type of the created set is the result of applying Fun to the  type
       of  the	given  set),  and that Fun does nothing but selecting, duplicating or rearranging
       parts of the elements. Specifying a SetFun as an integer I  is  equivalent  to  specifying
       {external,  fun(X) -> element(I, X) end} , but is to be preferred since it makes it possi-
       ble to handle this case even more efficiently. Examples of SetFuns:

       {sofs, union}
       fun(S) -> sofs:partition(1, S) end
       {external, fun(A) -> A end}
       {external, fun({A,_,C}) -> {C,A} end}
       {external, fun({_,{_,C}}) -> C end}
       {external, fun({_,{_,{_,E}=C}}) -> {E,{E,C}} end}
       2

       The order in which a SetFun is applied to the elements of an unordered set is  not  speci-
       fied, and may change in future versions of sofs.

       The  execution  time  of the functions of this module is dominated by the time it takes to
       sort lists. When no sorting is needed, the execution time is in	the  worst  case  propor-
       tional  to the sum of the sizes of the input arguments and the returned value. A few func-
       tions execute in constant time: from_external , is_empty_set  ,	is_set	,  is_sofs_set	,
       to_external , type .

       The  functions of this module exit the process with a badarg , bad_function , or type_mis-
       match message when given badly formed arguments or sets the types of which are not compat-
       ible.

       When comparing external sets the operator ==/2 is used.

       Types

       anyset() = - an unordered, ordered or atomic set -
       binary_relation() = - a binary relation -
       bool() = true | false
       external_set() = - an external set -
       family() = - a family (of subsets) -
       function() = - a function -
       ordset() = - an ordered set -
       relation() = - an n-ary relation -
       set() = - an unordered set -
       set_of_sets() = - an unordered set of set() -
       set_fun() = integer() >= 1
		 | {external, fun(external_set()) -> external_set()}
		 | fun(anyset()) -> anyset()
       spec_fun() = {external, fun(external_set()) -> bool()}
		  | fun(anyset()) -> bool()
       type() = - a type -


EXPORTS
       a_function(Tuples [, Type]) -> Function

	      Types  Function = function()
		     Tuples = [tuple()]
		     Type = type()

	      Creates  a  function  .  a_function(F, T) is equivalent to from_term(F, T) , if the
	      result is a function. If no type is explicitly given, [{atom,  atom}]  is  used  as
	      type of the function.

       canonical_relation(SetOfSets) -> BinRel

	      Types  BinRel = binary_relation()
		     SetOfSets = set_of_sets()

	      Returns  the binary relation containing the elements (E, Set) such that Set belongs
	      to SetOfSets and E belongs to Set. If SetOfSets is a partition of a set X and R  is
	      the  equivalence	relation in X induced by SetOfSets, then the returned relation is
	      the canonical map from X onto the equivalence classes with respect to R.

	      1> Ss = sofs:from_term([[a,b],[b,c]]),
	      CR = sofs:canonical_relation(Ss),
	      sofs:to_external(CR).
	      [{a,[a,b]},{b,[a,b]},{b,[b,c]},{c,[b,c]}]

       composite(Function1, Function2) -> Function3

	      Types  Function1 = Function2 = Function3 = function()

	      Returns the composite of the functions Function1 and Function2.

	      1> F1 = sofs:a_function([{a,1},{b,2},{c,2}]),
	      F2 = sofs:a_function([{1,x},{2,y},{3,z}]),
	      F = sofs:composite(F1, F2),
	      sofs:to_external(F).
	      [{a,x},{b,y},{c,y}]

       constant_function(Set, AnySet) -> Function

	      Types  AnySet = anyset()
		     Function = function()
		     Set = set()

	      Creates the function that maps each element of the set Set onto AnySet.

	      1> S = sofs:set([a,b]),
	      E = sofs:from_term(1),
	      R = sofs:constant_function(S, E),
	      sofs:to_external(R).
	      [{a,1},{b,1}]

       converse(BinRel1) -> BinRel2

	      Types  BinRel1 = BinRel2 = binary_relation()

	      Returns the converse of the binary relation BinRel1.

	      1> R1 = sofs:relation([{1,a},{2,b},{3,a}]),
	      R2 = sofs:converse(R1),
	      sofs:to_external(R2).
	      [{a,1},{a,3},{b,2}]

       difference(Set1, Set2) -> Set3

	      Types  Set1 = Set2 = Set3 = set()

	      Returns the difference of the sets Set1 and Set2.

       digraph_to_family(Graph [, Type]) -> Family

	      Types  Graph = digraph() - see digraph(3erl) -
		     Family = family()
		     Type = type()

	      Creates a family from the directed graph Graph. Each vertex a of	Graph  is  repre-
	      sented  by a pair (a, {b[1], ..., b[n]}) where the b[i]'s are the out-neighbours of
	      a. If no type is explicitly given, [{atom, [atom]}] is used as type of the  family.
	      It is assumed that Type is a valid type of the external set of the family.

	      If G is a directed graph, it holds that the vertices and edges of G are the same as
	      the vertices and edges of family_to_digraph(digraph_to_family(G)) .

       domain(BinRel) -> Set

	      Types  BinRel = binary_relation()
		     Set = set()

	      Returns the domain of the binary relation BinRel.

	      1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
	      S = sofs:domain(R),
	      sofs:to_external(S).
	      [1,2]

       drestriction(BinRel1, Set) -> BinRel2

	      Types  BinRel1 = BinRel2 = binary_relation()
		     Set = set()

	      Returns the difference between the binary relation BinRel1 and the  restriction  of
	      BinRel1 to Set.

	      1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
	      S = sofs:set([2,4,6]),
	      R2 = sofs:drestriction(R1, S),
	      sofs:to_external(R2).
	      [{1,a},{3,c}]

	      drestriction(R, S) is equivalent to difference(R, restriction(R, S)) .

       drestriction(SetFun, Set1, Set2) -> Set3

	      Types  SetFun = set_fun()
		     Set1 = Set2 = Set3 = set()

	      Returns  a subset of Set1 containing those elements that do not yield an element in
	      Set2 as the result of applying SetFun.

	      1> SetFun = {external, fun({_A,B,C}) -> {B,C} end},
	      R1 = sofs:relation([{a,aa,1},{b,bb,2},{c,cc,3}]),
	      R2 = sofs:relation([{bb,2},{cc,3},{dd,4}]),
	      R3 = sofs:drestriction(SetFun, R1, R2),
	      sofs:to_external(R3).
	      [{a,aa,1}]

	      drestriction(F, S1, S2) is equivalent to difference(S1, restriction(F, S1, S2)) .

       empty_set() -> Set

	      Types  Set = set()

	      Returns the untyped empty set . empty_set() is equivalent to from_term([], ['_']) .

       extension(BinRel1, Set, AnySet) -> BinRel2

	      Types  AnySet = anyset()
		     BinRel1 = BinRel2 = binary_relation()
		     Set = set()

	      Returns the extension of BinRel1 such that for each element E in Set that does  not
	      belong to the domain of BinRel1, BinRel2 contains the pair (E, AnySet).

	      1> S = sofs:set([b,c]),
	      A = sofs:empty_set(),
	      R = sofs:family([{a,[1,2]},{b,[3]}]),
	      X = sofs:extension(R, S, A),
	      sofs:to_external(X).
	      [{a,[1,2]},{b,[3]},{c,[]}]

       family(Tuples [, Type]) -> Family

	      Types  Family = family()
		     Tuples = [tuple()]
		     Type = type()

	      Creates  a  family  of subsets . family(F, T) is equivalent to from_term(F, T) , if
	      the result is a family. If no type is explicitly given, [{atom, [atom]}] is used as
	      type of the family.

       family_difference(Family1, Family2) -> Family3

	      Types  Family1 = Family2 = Family3 = family()

	      If  Family1  and	Family2  are  families , then Family3 is the family such that the
	      index set is equal to the index set of Family1, and Family3[i]  is  the  difference
	      between Family1[i] and Family2[i] if Family2 maps i, Family1[i] otherwise.

	      1> F1 = sofs:family([{a,[1,2]},{b,[3,4]}]),
	      F2 = sofs:family([{b,[4,5]},{c,[6,7]}]),
	      F3 = sofs:family_difference(F1, F2),
	      sofs:to_external(F3).
	      [{a,[1,2]},{b,[3]}]

       family_domain(Family1) -> Family2

	      Types  Family1 = Family2 = family()

	      If Family1 is a family and Family1[i] is a binary relation for every i in the index
	      set of Family1, then Family2 is the family with the same index set as Family1  such
	      that Family2[i] is the domain of Family1[i].

	      1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
	      F = sofs:family_domain(FR),
	      sofs:to_external(F).
	      [{a,[1,2,3]},{b,[]},{c,[4,5]}]

       family_field(Family1) -> Family2

	      Types  Family1 = Family2 = family()

	      If Family1 is a family and Family1[i] is a binary relation for every i in the index
	      set of Family1, then Family2 is the family with the same index set as Family1  such
	      that Family2[i] is the field of Family1[i].

	      1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
	      F = sofs:family_field(FR),
	      sofs:to_external(F).
	      [{a,[1,2,3,a,b,c]},{b,[]},{c,[4,5,d,e]}]

	      family_field(Family1)  is  equivalent  to family_union(family_domain(Family1), fam-
	      ily_range(Family1)) .

       family_intersection(Family1) -> Family2

	      Types  Family1 = Family2 = family()

	      If Family1 is a family and Family1[i] is a set of sets for every i in the index set
	      of Family1, then Family2 is the family with the same index set as Family1 such that
	      Family2[i] is the intersection of Family1[i].

	      If Family1[i] is an empty set for some i, then the process exits with a badarg mes-
	      sage.

	      1> F1 = sofs:from_term([{a,[[1,2,3],[2,3,4]]},{b,[[x,y,z],[x,y]]}]),
	      F2 = sofs:family_intersection(F1),
	      sofs:to_external(F2).
	      [{a,[2,3]},{b,[x,y]}]

       family_intersection(Family1, Family2) -> Family3

	      Types  Family1 = Family2 = Family3 = family()

	      If  Family1  and	Family2  are  families , then Family3 is the family such that the
	      index set is the intersection of Family1's and Family2's index sets, and Family3[i]
	      is the intersection of Family1[i] and Family2[i].

	      1> F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),
	      F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),
	      F3 = sofs:family_intersection(F1, F2),
	      sofs:to_external(F3).
	      [{b,[4]},{c,[]}]

       family_projection(SetFun, Family1) -> Family2

	      Types  SetFun = set_fun()
		     Family1 = Family2 = family()
		     Set = set()

	      If  Family1  is a family then Family2 is the family with the same index set as Fam-
	      ily1 such that Family2[i] is the result of calling SetFun with Family1[i] as  argu-
	      ment.

	      1> F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),
	      F2 = sofs:family_projection({sofs, union}, F1),
	      sofs:to_external(F2).
	      [{a,[1,2,3]},{b,[]}]

       family_range(Family1) -> Family2

	      Types  Family1 = Family2 = family()

	      If Family1 is a family and Family1[i] is a binary relation for every i in the index
	      set of Family1, then Family2 is the family with the same index set as Family1  such
	      that Family2[i] is the range of Family1[i].

	      1> FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),
	      F = sofs:family_range(FR),
	      sofs:to_external(F).
	      [{a,[a,b,c]},{b,[]},{c,[d,e]}]

       family_specification(Fun, Family1) -> Family2

	      Types  Fun = spec_fun()
		     Family1 = Family2 = family()

	      If  Family1  is a family , then Family2 is the restriction of Family1 to those ele-
	      ments i of the index set for which Fun applied to Family1[i] returns true . If  Fun
	      is  a  tuple  {external, Fun2} , Fun2 is applied to the external set of Family1[i],
	      otherwise Fun is applied to Family1[i].

	      1> F1 = sofs:family([{a,[1,2,3]},{b,[1,2]},{c,[1]}]),
	      SpecFun = fun(S) -> sofs:no_elements(S) =:= 2 end,
	      F2 = sofs:family_specification(SpecFun, F1),
	      sofs:to_external(F2).
	      [{b,[1,2]}]

       family_to_digraph(Family [, GraphType]) -> Graph

	      Types  Graph = digraph()
		     Family = family()
		     GraphType = - see digraph(3erl) -

	      Creates a directed graph from the family Family. For  each  pair	(a,  {b[1],  ...,
	      b[n]})  of  Family,  the	vertex	a as well the edges (a, b[i]) for 1 <= i <= n are
	      added to a newly created directed graph.

	      If no graph type is given, digraph:new/1 is used for creating the  directed  graph,
	      otherwise the GraphType argument is passed on as second argument to digraph:new/2 .

	      It  F  is  a  family,  it  holds	that  F  is  a	subset	of digraph_to_family(fam-
	      ily_to_digraph(F), type(F)) . Equality holds if union_of_family(F) is a  subset  of
	      domain(F) .

	      Creating a cycle in an acyclic graph exits the process with a cyclic message.

       family_to_relation(Family) -> BinRel

	      Types  Family = family()
		     BinRel = binary_relation()

	      If Family is a family , then BinRel is the binary relation containing all pairs (i,
	      x) such that i belongs to the index set of Family and x belongs to Family[i].

	      1> F = sofs:family([{a,[]}, {b,[1]}, {c,[2,3]}]),
	      R = sofs:family_to_relation(F),
	      sofs:to_external(R).
	      [{b,1},{c,2},{c,3}]

       family_union(Family1) -> Family2

	      Types  Family1 = Family2 = family()

	      If Family1 is a family and Family1[i] is a set of sets for each i in the index  set
	      of Family1, then Family2 is the family with the same index set as Family1 such that
	      Family2[i] is the union of Family1[i].

	      1> F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),
	      F2 = sofs:family_union(F1),
	      sofs:to_external(F2).
	      [{a,[1,2,3]},{b,[]}]

	      family_union(F) is equivalent to family_projection({sofs,union}, F) .

       family_union(Family1, Family2) -> Family3

	      Types  Family1 = Family2 = Family3 = family()

	      If Family1 and Family2 are families , then Family3 is  the  family  such	that  the
	      index set is the union of Family1's and Family2's index sets, and Family3[i] is the
	      union of Family1[i] and Family2[i] if both maps i, Family1[i] or Family2[i]  other-
	      wise.

	      1> F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),
	      F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),
	      F3 = sofs:family_union(F1, F2),
	      sofs:to_external(F3).
	      [{a,[1,2]},{b,[3,4,5]},{c,[5,6,7,8]},{d,[9,10]}]

       field(BinRel) -> Set

	      Types  BinRel = binary_relation()
		     Set = set()

	      Returns the field of the binary relation BinRel.

	      1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
	      S = sofs:field(R),
	      sofs:to_external(S).
	      [1,2,a,b,c]

	      field(R) is equivalent to union(domain(R), range(R)) .

       from_external(ExternalSet, Type) -> AnySet

	      Types  ExternalSet = external_set()
		     AnySet = anyset()
		     Type = type()

	      Creates  a  set  from the external set ExternalSet and the type Type. It is assumed
	      that Type is a valid type of ExternalSet.

       from_sets(ListOfSets) -> Set

	      Types  Set = set()
		     ListOfSets = [anyset()]

	      Returns the unordered set containing the sets of the list ListOfSets.

	      1> S1 = sofs:relation([{a,1},{b,2}]),
	      S2 = sofs:relation([{x,3},{y,4}]),
	      S = sofs:from_sets([S1,S2]),
	      sofs:to_external(S).
	      [[{a,1},{b,2}],[{x,3},{y,4}]]

       from_sets(TupleOfSets) -> Ordset

	      Types  Ordset = ordset()
		     TupleOfSets = tuple-of(anyset())

	      Returns the ordered set containing the sets of the non-empty tuple TupleOfSets.

       from_term(Term [, Type]) -> AnySet

	      Types  AnySet = anyset()
		     Term = term()
		     Type = type()

	      Creates an element of Sets by traversing the term  Term,	sorting  lists,  removing
	      duplicates and deriving or verifying a valid type for the so obtained external set.
	      An explicitly given type Type can be used to limit the depth of the  traversal;  an
	      atomic  type  stops  the traversal, as demonstrated by this example where "foo" and
	      {"foo"} are left unmodified:

	      1> S = sofs:from_term([{{"foo"},[1,1]},{"foo",[2,2]}], [{atom,[atom]}]),
	      sofs:to_external(S).
	      [{{"foo"},[1]},{"foo",[2]}]

	      from_term can be used for creating atomic or ordered sets. The only purpose of such
	      a  set  is that of later building unordered sets since all functions in this module
	      that do anything operate on unordered sets. Creating unordered sets from a  collec-
	      tion  of ordered sets may be the way to go if the ordered sets are big and one does
	      not want to waste heap by rebuilding the elements of the unordered set. An  example
	      showing that a set can be built "layer by layer":

	      1> A = sofs:from_term(a),
	      S = sofs:set([1,2,3]),
	      P1 = sofs:from_sets({A,S}),
	      P2 = sofs:from_term({b,[6,5,4]}),
	      Ss = sofs:from_sets([P1,P2]),
	      sofs:to_external(Ss).
	      [{a,[1,2,3]},{b,[4,5,6]}]

	      Other  functions	that  create  sets  are from_external/2 and from_sets/1 . Special
	      cases of from_term/2 are a_function/1,2 , empty_set/0 , family/1,2 , relation/1,2 ,
	      and set/1,2 .

       image(BinRel, Set1) -> Set2

	      Types  BinRel = binary_relation()
		     Set1 = Set2 = set()

	      Returns the image of the set Set1 under the binary relation BinRel.

	      1> R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),
	      S1 = sofs:set([1,2]),
	      S2 = sofs:image(R, S1),
	      sofs:to_external(S2).
	      [a,b,c]

       intersection(SetOfSets) -> Set

	      Types  Set = set()
		     SetOfSets = set_of_sets()

	      Returns the intersection of the set of sets SetOfSets.

	      Intersecting an empty set of sets exits the process with a badarg message.

       intersection(Set1, Set2) -> Set3

	      Types  Set1 = Set2 = Set3 = set()

	      Returns the intersection of Set1 and Set2.

       intersection_of_family(Family) -> Set

	      Types  Family = family()
		     Set = set()

	      Returns the intersection of the family Family.

	      Intersecting an empty family exits the process with a badarg message.

	      1> F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),
	      S = sofs:intersection_of_family(F),
	      sofs:to_external(S).
	      [2]

       inverse(Function1) -> Function2

	      Types  Function1 = Function2 = function()

	      Returns the inverse of the function Function1.

	      1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
	      R2 = sofs:inverse(R1),
	      sofs:to_external(R2).
	      [{a,1},{b,2},{c,3}]

       inverse_image(BinRel, Set1) -> Set2

	      Types  BinRel = binary_relation()
		     Set1 = Set2 = set()

	      Returns the inverse image of Set1 under the binary relation BinRel.

	      1> R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),
	      S1 = sofs:set([c,d,e]),
	      S2 = sofs:inverse_image(R, S1),
	      sofs:to_external(S2).
	      [2,3]

       is_a_function(BinRel) -> Bool

	      Types  Bool = bool()
		     BinRel = binary_relation()

	      Returns  true if the binary relation BinRel is a function or the untyped empty set,
	      false otherwise.

       is_disjoint(Set1, Set2) -> Bool

	      Types  Bool = bool()
		     Set1 = Set2 = set()

	      Returns true if Set1 and Set2 are disjoint , false otherwise.

       is_empty_set(AnySet) -> Bool

	      Types  AnySet = anyset()
		     Bool = bool()

	      Returns true if Set is an empty unordered set, false otherwise.

       is_equal(AnySet1, AnySet2) -> Bool

	      Types  AnySet1 = AnySet2 = anyset()
		     Bool = bool()

	      Returns true if the AnySet1 and AnySet2 are equal , false otherwise.  This  example
	      shows that ==/2 is used when comparing sets for equality:

	      1> S1 = sofs:set([1.0]),
	      S2 = sofs:set([1]),
	      sofs:is_equal(S1, S2).
	      true

       is_set(AnySet) -> Bool

	      Types  AnySet = anyset()
		     Bool = bool()

	      Returns  true if AnySet is an unordered set , and false if AnySet is an ordered set
	      or an atomic set.

       is_sofs_set(Term) -> Bool

	      Types  Bool = bool()
		     Term = term()

	      Returns true if Term is an unordered set , an ordered set or an atomic  set,  false
	      otherwise.

       is_subset(Set1, Set2) -> Bool

	      Types  Bool = bool()
		     Set1 = Set2 = set()

	      Returns true if Set1 is a subset of Set2, false otherwise.

       is_type(Term) -> Bool

	      Types  Bool = bool()
		     Term = term()

	      Returns true if the term Term is a type .

       join(Relation1, I, Relation2, J) -> Relation3

	      Types  Relation1 = Relation2 = Relation3 = relation()
		     I = J = integer() > 0

	      Returns  the natural join of the relations Relation1 and Relation2 on coordinates I
	      and J.

	      1> R1 = sofs:relation([{a,x,1},{b,y,2}]),
	      R2 = sofs:relation([{1,f,g},{1,h,i},{2,3,4}]),
	      J = sofs:join(R1, 3, R2, 1),
	      sofs:to_external(J).
	      [{a,x,1,f,g},{a,x,1,h,i},{b,y,2,3,4}]

       multiple_relative_product(TupleOfBinRels, BinRel1) -> BinRel2

	      Types  TupleOfBinRels = tuple-of(BinRel)
		     BinRel = BinRel1 = BinRel2 = binary_relation()

	      If TupleOfBinRels is a non-empty tuple {R[1], ..., R[n]} of  binary  relations  and
	      BinRel1  is a binary relation, then BinRel2 is the multiple relative product of the
	      ordered set (R[i], ..., R[n]) and BinRel1.

	      1> Ri = sofs:relation([{a,1},{b,2},{c,3}]),
	      R = sofs:relation([{a,b},{b,c},{c,a}]),
	      MP = sofs:multiple_relative_product({Ri, Ri}, R),
	      sofs:to_external(sofs:range(MP)).
	      [{1,2},{2,3},{3,1}]

       no_elements(ASet) -> NoElements

	      Types  ASet = set() | ordset()
		     NoElements = integer() >= 0

	      Returns the number of elements of the ordered or unordered set ASet.

       partition(SetOfSets) -> Partition

	      Types  SetOfSets = set_of_sets()
		     Partition = set()

	      Returns the partition of the union of the set of sets SetOfSets such that two  ele-
	      ments are considered equal if they belong to the same elements of SetOfSets.

	      1> Sets1 = sofs:from_term([[a,b,c],[d,e,f],[g,h,i]]),
	      Sets2 = sofs:from_term([[b,c,d],[e,f,g],[h,i,j]]),
	      P = sofs:partition(sofs:union(Sets1, Sets2)),
	      sofs:to_external(P).
	      [[a],[b,c],[d],[e,f],[g],[h,i],[j]]

       partition(SetFun, Set) -> Partition

	      Types  SetFun = set_fun()
		     Partition = set()
		     Set = set()

	      Returns  the  partition  of  Set such that two elements are considered equal if the
	      results of applying SetFun are equal.

	      1> Ss = sofs:from_term([[a],[b],[c,d],[e,f]]),
	      SetFun = fun(S) -> sofs:from_term(sofs:no_elements(S)) end,
	      P = sofs:partition(SetFun, Ss),
	      sofs:to_external(P).
	      [[[a],[b]],[[c,d],[e,f]]]

       partition(SetFun, Set1, Set2) -> {Set3, Set4}

	      Types  SetFun = set_fun()
		     Set1 = Set2 = Set3 = Set4 = set()

	      Returns a pair of sets that, regarded as constituting a set, forms a  partition  of
	      Set1.  If  the result of applying SetFun to an element of Set1 yields an element in
	      Set2, the element belongs to Set3, otherwise the element belongs to Set4.

	      1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
	      S = sofs:set([2,4,6]),
	      {R2,R3} = sofs:partition(1, R1, S),
	      {sofs:to_external(R2),sofs:to_external(R3)}.
	      {[{2,b}],[{1,a},{3,c}]}

	      partition(F, S1, S2) is equivalent to {restriction(F, S1, S2), drestriction(F,  S1,
	      S2)} .

       partition_family(SetFun, Set) -> Family

	      Types  Family = family()
		     SetFun = set_fun()
		     Set = set()

	      Returns the family Family where the indexed set is a partition of Set such that two
	      elements are considered equal if the results of applying SetFun are the same  value
	      i. This i is the index that Family maps onto the equivalence class .

	      1> S = sofs:relation([{a,a,a,a},{a,a,b,b},{a,b,b,b}]),
	      SetFun = {external, fun({A,_,C,_}) -> {A,C} end},
	      F = sofs:partition_family(SetFun, S),
	      sofs:to_external(F).
	      [{{a,a},[{a,a,a,a}]},{{a,b},[{a,a,b,b},{a,b,b,b}]}]

       product(TupleOfSets) -> Relation

	      Types  Relation = relation()
		     TupleOfSets = tuple-of(set())

	      Returns the Cartesian product of the non-empty tuple of sets TupleOfSets. If (x[1],
	      ..., x[n]) is an element of the n-ary relation Relation, then x[i]  is  drawn  from
	      element i of TupleOfSets.

	      1> S1 = sofs:set([a,b]),
	      S2 = sofs:set([1,2]),
	      S3 = sofs:set([x,y]),
	      P3 = sofs:product({S1,S2,S3}),
	      sofs:to_external(P3).
	      [{a,1,x},{a,1,y},{a,2,x},{a,2,y},{b,1,x},{b,1,y},{b,2,x},{b,2,y}]

       product(Set1, Set2) -> BinRel

	      Types  BinRel = binary_relation()
		     Set1 = Set2 = set()

	      Returns the Cartesian product of Set1 and Set2.

	      1> S1 = sofs:set([1,2]),
	      S2 = sofs:set([a,b]),
	      R = sofs:product(S1, S2),
	      sofs:to_external(R).
	      [{1,a},{1,b},{2,a},{2,b}]

	      product(S1, S2) is equivalent to product({S1, S2}) .

       projection(SetFun, Set1) -> Set2

	      Types  SetFun = set_fun()
		     Set1 = Set2 = set()

	      Returns  the  set  created  by  substituting  each element of Set1 by the result of
	      applying SetFun to the element.

	      If SetFun is a number i >= 1 and Set1 is a relation, then the returned set  is  the
	      projection of Set1 onto coordinate i.

	      1> S1 = sofs:from_term([{1,a},{2,b},{3,a}]),
	      S2 = sofs:projection(2, S1),
	      sofs:to_external(S2).
	      [a,b]

       range(BinRel) -> Set

	      Types  BinRel = binary_relation()
		     Set = set()

	      Returns the range of the binary relation BinRel.

	      1> R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),
	      S = sofs:range(R),
	      sofs:to_external(S).
	      [a,b,c]

       relation(Tuples [, Type]) -> Relation

	      Types  N = integer()
		     Type = N | type()
		     Relation = relation()
		     Tuples = [tuple()]

	      Creates  a  relation  . relation(R, T) is equivalent to from_term(R, T) , if T is a
	      type and the result is a relation. If Type is  an  integer  N,  then  [{atom,  ...,
	      atom}])  , where the size of the tuple is N, is used as type of the relation. If no
	      type is explicitly given, the size of the first tuple of Tuples is used if there is
	      such a tuple. relation([]) is equivalent to relation([], 2) .

       relation_to_family(BinRel) -> Family

	      Types  Family = family()
		     BinRel = binary_relation()

	      Returns  the  family  Family  such that the index set is equal to the domain of the
	      binary relation BinRel, and Family[i] is the image of the set of i under BinRel.

	      1> R = sofs:relation([{b,1},{c,2},{c,3}]),
	      F = sofs:relation_to_family(R),
	      sofs:to_external(F).
	      [{b,[1]},{c,[2,3]}]

       relative_product(TupleOfBinRels [, BinRel1]) -> BinRel2

	      Types  TupleOfBinRels = tuple-of(BinRel)
		     BinRel = BinRel1 = BinRel2 = binary_relation()

	      If TupleOfBinRels is a non-empty tuple {R[1], ..., R[n]} of  binary  relations  and
	      BinRel1  is  a binary relation, then BinRel2 is the relative product of the ordered
	      set (R[i], ..., R[n]) and BinRel1.

	      If BinRel1 is omitted, the relation of equality between the elements of the  Carte-
	      sian  product of the ranges of R[i], range R[1] x ... x range R[n], is used instead
	      (intuitively, nothing is "lost").

	      1> TR = sofs:relation([{1,a},{1,aa},{2,b}]),
	      R1 = sofs:relation([{1,u},{2,v},{3,c}]),
	      R2 = sofs:relative_product({TR, R1}),
	      sofs:to_external(R2).
	      [{1,{a,u}},{1,{aa,u}},{2,{b,v}}]

	      Note that relative_product({R1}, R2) is different from relative_product(R1,  R2)	;
	      the tuple of one element is not identified with the element itself.

       relative_product(BinRel1, BinRel2) -> BinRel3

	      Types  BinRel1 = BinRel2 = BinRel3 = binary_relation()

	      Returns the relative product of the binary relations BinRel1 and BinRel2.

       relative_product1(BinRel1, BinRel2) -> BinRel3

	      Types  BinRel1 = BinRel2 = BinRel3 = binary_relation()

	      Returns the relative product of the converse of the binary relation BinRel1 and the
	      binary relation BinRel2.

	      1> R1 = sofs:relation([{1,a},{1,aa},{2,b}]),
	      R2 = sofs:relation([{1,u},{2,v},{3,c}]),
	      R3 = sofs:relative_product1(R1, R2),
	      sofs:to_external(R3).
	      [{a,u},{aa,u},{b,v}]

	      relative_product1(R1, R2) is equivalent to relative_product(converse(R1), R2) .

       restriction(BinRel1, Set) -> BinRel2

	      Types  BinRel1 = BinRel2 = binary_relation()
		     Set = set()

	      Returns the restriction of the binary relation BinRel1 to Set.

	      1> R1 = sofs:relation([{1,a},{2,b},{3,c}]),
	      S = sofs:set([1,2,4]),
	      R2 = sofs:restriction(R1, S),
	      sofs:to_external(R2).
	      [{1,a},{2,b}]

       restriction(SetFun, Set1, Set2) -> Set3

	      Types  SetFun = set_fun()
		     Set1 = Set2 = Set3 = set()

	      Returns a subset of Set1 containing those elements that yield an element in Set2 as
	      the result of applying SetFun.

	      1> S1 = sofs:relation([{1,a},{2,b},{3,c}]),
	      S2 = sofs:set([b,c,d]),
	      S3 = sofs:restriction(2, S1, S2),
	      sofs:to_external(S3).
	      [{2,b},{3,c}]

       set(Terms [, Type]) -> Set

	      Types  Set = set()
		     Terms = [term()]
		     Type = type()

	      Creates  an  unordered  set  .  set(L, T) is equivalent to from_term(L, T) , if the
	      result is an unordered set. If no type is explicitly given, [atom] is used as  type
	      of the set.

       specification(Fun, Set1) -> Set2

	      Types  Fun = spec_fun()
		     Set1 = Set2 = set()

	      Returns  the  set  containing every element of Set1 for which Fun returns true . If
	      Fun is a tuple {external, Fun2} , Fun2 is applied to the external set of each  ele-
	      ment, otherwise Fun is applied to each element.

	      1> R1 = sofs:relation([{a,1},{b,2}]),
	      R2 = sofs:relation([{x,1},{x,2},{y,3}]),
	      S1 = sofs:from_sets([R1,R2]),
	      S2 = sofs:specification({sofs,is_a_function}, S1),
	      sofs:to_external(S2).
	      [[{a,1},{b,2}]]

       strict_relation(BinRel1) -> BinRel2

	      Types  BinRel1 = BinRel2 = binary_relation()

	      Returns the strict relation corresponding to the binary relation BinRel1.

	      1> R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),
	      R2 = sofs:strict_relation(R1),
	      sofs:to_external(R2).
	      [{1,2},{2,1}]

       substitution(SetFun, Set1) -> Set2

	      Types  SetFun = set_fun()
		     Set1 = Set2 = set()

	      Returns  a  function,  the  domain of which is Set1. The value of an element of the
	      domain is the result of applying SetFun to the element.

	      1> L = [{a,1},{b,2}].
	      [{a,1},{b,2}]
	      2> sofs:to_external(sofs:projection(1,sofs:relation(L))).
	      [a,b]
	      3> sofs:to_external(sofs:substitution(1,sofs:relation(L))).
	      [{{a,1},a},{{b,2},b}]
	      4> SetFun = {external, fun({A,_}=E) -> {E,A} end},
	      sofs:to_external(sofs:projection(SetFun,sofs:relation(L))).
	      [{{a,1},a},{{b,2},b}]

	      The relation of equality between the elements of {a,b,c}:

	      1> I = sofs:substitution(fun(A) -> A end, sofs:set([a,b,c])),
	      sofs:to_external(I).
	      [{a,a},{b,b},{c,c}]

	      Let SetOfSets be a set of sets and BinRel a binary relation. The function that maps
	      each  element  Set  of  SetOfSets onto the image of Set under BinRel is returned by
	      this function:

	      images(SetOfSets, BinRel) ->
		 Fun = fun(Set) -> sofs:image(BinRel, Set) end,
		 sofs:substitution(Fun, SetOfSets).

	      Here might be the place to reveal something that was more or  less  stated  before,
	      namely  that  external  unordered sets are represented as sorted lists. As a conse-
	      quence, creating the image of a set under a relation R may traverse all elements of
	      R  (to  that comes the sorting of results, the image). In images/2 , BinRel will be
	      traversed once for each element of SetOfSets, which may take too long. The  follow-
	      ing efficient function could be used instead under the assumption that the image of
	      each element of SetOfSets under BinRel is non-empty:

	      images2(SetOfSets, BinRel) ->
		 CR = sofs:canonical_relation(SetOfSets),
		 R = sofs:relative_product1(CR, BinRel),
		 sofs:relation_to_family(R).

       symdiff(Set1, Set2) -> Set3

	      Types  Set1 = Set2 = Set3 = set()

	      Returns the symmetric difference (or the Boolean sum) of Set1 and Set2.

	      1> S1 = sofs:set([1,2,3]),
	      S2 = sofs:set([2,3,4]),
	      P = sofs:symdiff(S1, S2),
	      sofs:to_external(P).
	      [1,4]

       symmetric_partition(Set1, Set2) -> {Set3, Set4, Set5}

	      Types  Set1 = Set2 = Set3 = Set4 = Set5 = set()

	      Returns a triple of sets: Set3 contains the elements of Set1 that do not belong  to
	      Set2;  Set4  contains  the  elements of Set1 that belong to Set2; Set5 contains the
	      elements of Set2 that do not belong to Set1.

       to_external(AnySet) -> ExternalSet

	      Types  ExternalSet = external_set()
		     AnySet = anyset()

	      Returns the external set of an atomic, ordered or unordered set.

       to_sets(ASet) -> Sets

	      Types  ASet = set() | ordset()
		     Sets = tuple_of(AnySet) | [AnySet]

	      Returns the elements of the ordered set ASet as a tuple of sets, and  the  elements
	      of the unordered set ASet as a sorted list of sets without duplicates.

       type(AnySet) -> Type

	      Types  AnySet = anyset()
		     Type = type()

	      Returns the type of an atomic, ordered or unordered set.

       union(SetOfSets) -> Set

	      Types  Set = set()
		     SetOfSets = set_of_sets()

	      Returns the union of the set of sets SetOfSets.

       union(Set1, Set2) -> Set3

	      Types  Set1 = Set2 = Set3 = set()

	      Returns the union of Set1 and Set2.

       union_of_family(Family) -> Set

	      Types  Family = family()
		     Set = set()

	      Returns the union of the family Family.

	      1> F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),
	      S = sofs:union_of_family(F),
	      sofs:to_external(S).
	      [0,1,2,3,4]

       weak_relation(BinRel1) -> BinRel2

	      Types  BinRel1 = BinRel2 = binary_relation()

	      Returns a subset S of the weak relation W corresponding to the binary relation Bin-
	      Rel1. Let F be the field of BinRel1. The subset S is defined so that x S y if x W y
	      for some x in F and for some y in F.

	      1> R1 = sofs:relation([{1,1},{1,2},{3,1}]),
	      R2 = sofs:weak_relation(R1),
	      sofs:to_external(R2).
	      [{1,1},{1,2},{2,2},{3,1},{3,3}]

SEE ALSO
       dict(3erl) , digraph(3erl) , orddict(3erl) , ordsets(3erl) , sets(3erl)

Ericsson AB				  stdlib 1.17.3 			       sofs(3erl)


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