diff options
Diffstat (limited to 'lib/stdlib/doc/src/sofs.xml')
| -rw-r--r-- | lib/stdlib/doc/src/sofs.xml | 1023 |
1 files changed, 429 insertions, 594 deletions
diff --git a/lib/stdlib/doc/src/sofs.xml b/lib/stdlib/doc/src/sofs.xml index 8c8ae51262..2e7768a1df 100644 --- a/lib/stdlib/doc/src/sofs.xml +++ b/lib/stdlib/doc/src/sofs.xml @@ -4,7 +4,7 @@ <erlref> <header> <copyright> - <year>2001</year><year>2010</year> + <year>2001</year><year>2011</year> <holder>Ericsson AB. All Rights Reserved.</holder> </copyright> <legalnotice> @@ -181,10 +181,11 @@ the <marker id="canonical_map"></marker><em>canonical map</em> is the function that maps every element of X onto its equivalence class. </p> - <p>Relations as defined above (as sets of ordered pairs) will from - now on be referred to as <em>binary relations</em>. We call a - set of ordered sets (x[1], ..., x[n]) - an <em>(n-ary) relation</em>, and say that the relation is a subset of + <p><marker id="binary_relation"></marker>Relations as defined above + (as sets of ordered pairs) will from now on be referred to as + <em>binary relations</em>. We call a set of ordered sets + (x[1], ..., x[n]) an <marker id="n_ary_relation"></marker> + <em>(n-ary) relation</em>, and say that the relation is a subset of the <marker id="Cartesian_product_tuple"></marker>Cartesian product X[1] × ... × X[n] where x[i] is an element of X[i], 1 <= i <= n. @@ -210,7 +211,7 @@ X[i] to Y[i] and S a subset of X[1] × ... × X[n]. The <marker id="multiple_relative_product"></marker><em>multiple - relative product</em> of TR and and S is defined to be the + relative product</em> 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], @@ -293,7 +294,8 @@ <c>partition_family/2</c>, <c>projection/2</c>, <c>restriction/3</c>, <c>substitution/2</c>) accept an Erlang function as a means to modify each element of a given unordered - set. Such a function, called SetFun in the following, can be + set. <marker id="set_fun"></marker>Such a function, called + SetFun in the following, can be specified as a functional object (fun), a tuple <c>{external, Fun}</c>, or an integer. If SetFun is specified as a fun, the fun is applied to each element of the @@ -337,34 +339,73 @@ fun(S) -> sofs:partition(1, S) end message when given badly formed arguments or sets the types of which are not compatible.</p> <p>When comparing external sets the operator <c>==/2</c> is used.</p> - <p><em>Types</em></p> - <pre> -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 - </pre> </description> + <datatypes> + <datatype> + <name name="anyset"></name> + <desc><p>Any kind of set (also included are the atomic sets).</p></desc> + </datatype> + <datatype> + <name name="binary_relation"></name> + <desc><p>A <seealso marker="#binary_relation">binary + relation</seealso>.</p></desc> + </datatype> + <datatype> + <name name="external_set"></name> + <desc><p>An <seealso marker="#external_set">external + set</seealso>.</p></desc> + </datatype> + <datatype> + <name name="family"></name> + <desc><p>A <seealso marker="#family">family</seealso> (of subsets).</p> + </desc> + </datatype> + <datatype> + <name name="a_function"></name> + <desc><p>A <seealso marker="#function">function</seealso>.</p></desc> + </datatype> + <datatype> + <name name="ordset"></name> + <desc><p>An <seealso marker="#sets_definition">ordered + set</seealso>.</p></desc> + </datatype> + <datatype> + <name name="relation"></name> + <desc><p>An <seealso marker="#n_ary_relation">n-ary relation</seealso>. + </p></desc> + </datatype> + <datatype> + <name name="a_set"></name> + <desc><p>An <seealso marker="#sets_definition">unordered + set</seealso>.</p></desc> + </datatype> + <datatype> + <name name="set_of_sets"></name> + <desc><p>An <seealso marker="#sets_definition">unordered + set</seealso> of unordered sets.</p></desc> + </datatype> + <datatype> + <name name="set_fun"></name> + <desc><p>A <seealso marker="#set_fun">SetFun</seealso>.</p></desc> + </datatype> + <datatype> + <name name="spec_fun"></name> + </datatype> + <datatype> + <name name="type"></name> + <desc><p>A <seealso marker="#type">type</seealso>.</p></desc> + </datatype> + <datatype> + <!-- Parameterized opaque types are NYI: --> + <name><marker id="type-tuple_of">tuple_of(T)</marker></name> + <desc><p>A tuple where the elements are of type <c>T</c>.</p></desc> + </datatype> + </datatypes> <funcs> <func> - <name>a_function(Tuples [, Type]) -> Function</name> + <name name="a_function" arity="1"/> + <name name="a_function" arity="2"/> <fsummary>Create a function.</fsummary> - <type> - <v>Function = function()</v> - <v>Tuples = [tuple()]</v> - <v>Type = type()</v> - </type> <desc> <p>Creates a <seealso marker="#function">function</seealso>. <c>a_function(F, T)</c> is equivalent to @@ -375,16 +416,12 @@ type() = - a type - </pre> </desc> </func> <func> - <name>canonical_relation(SetOfSets) -> BinRel</name> + <name name="canonical_relation" arity="1"/> <fsummary>Return the canonical map.</fsummary> - <type> - <v>BinRel = binary_relation()</v> - <v>SetOfSets = set_of_sets()</v> - </type> <desc> <p>Returns the binary relation containing the elements - (E, Set) such that Set belongs to SetOfSets and E - belongs to Set. If SetOfSets is + (E, Set) such that Set belongs to <anno>SetOfSets</anno> and E + belongs to Set. If SetOfSets is a <seealso marker="#partition">partition</seealso> of a set X and R is the equivalence relation in X induced by SetOfSets, then the returned relation is @@ -398,14 +435,12 @@ type() = - a type - </pre> </desc> </func> <func> - <name>composite(Function1, Function2) -> Function3</name> + <name name="composite" arity="2"/> <fsummary>Return the composite of two functions.</fsummary> - <type> - <v>Function1 = Function2 = Function3 = function()</v> - </type> <desc> <p>Returns the <seealso marker="#composite">composite</seealso> of - the functions Function1 and Function2.</p> + the functions <anno>Function1</anno> and + <anno>Function2</anno>.</p> <pre> 1> <input>F1 = sofs:a_function([{a,1},{b,2},{c,2}]),</input> <input>F2 = sofs:a_function([{1,x},{2,y},{3,z}]),</input> @@ -415,14 +450,9 @@ type() = - a type - </pre> </desc> </func> <func> - <name>constant_function(Set, AnySet) -> Function</name> + <name name="constant_function" arity="2"/> <fsummary>Create the function that maps each element of a set onto another set.</fsummary> - <type> - <v>AnySet = anyset()</v> - <v>Function = function()</v> - <v>Set = set()</v> - </type> <desc> <p>Creates the <seealso marker="#function">function</seealso> that maps each element of the set Set onto AnySet.</p> @@ -435,14 +465,11 @@ type() = - a type - </pre> </desc> </func> <func> - <name>converse(BinRel1) -> BinRel2</name> + <name name="converse" arity="1"/> <fsummary>Return the converse of a binary relation.</fsummary> - <type> - <v>BinRel1 = BinRel2 = binary_relation()</v> - </type> <desc> <p>Returns the <seealso marker="#converse">converse</seealso> - of the binary relation BinRel1.</p> + of the binary relation <anno>BinRel1</anno>.</p> <pre> 1> <input>R1 = sofs:relation([{1,a},{2,b},{3,a}]),</input> <input>R2 = sofs:converse(R1),</input> @@ -451,31 +478,25 @@ type() = - a type - </pre> </desc> </func> <func> - <name>difference(Set1, Set2) -> Set3</name> + <name name="difference" arity="2"/> <fsummary>Return the difference of two sets.</fsummary> - <type> - <v>Set1 = Set2 = Set3 = set()</v> - </type> <desc> <p>Returns the <seealso marker="#difference">difference</seealso> of - the sets Set1 and Set2.</p> + the sets <anno>Set1</anno> and <anno>Set2</anno>.</p> </desc> </func> <func> - <name>digraph_to_family(Graph [, Type]) -> Family</name> + <name name="digraph_to_family" arity="1"/> + <name name="digraph_to_family" arity="2"/> <fsummary>Create a family from a directed graph.</fsummary> - <type> - <v>Graph = digraph() - see digraph(3) -</v> - <v>Family = family()</v> - <v>Type = type()</v> - </type> <desc> <p>Creates a <seealso marker="#family">family</seealso> from - the directed graph Graph. Each vertex a of Graph is + the directed graph <anno>Graph</anno>. Each vertex a of + <anno>Graph</anno> is represented 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 + the family. It is assumed that <anno>Type</anno> is a <seealso marker="#valid_type">valid type</seealso> of the external set of the family.</p> <p>If G is a directed graph, it holds that the vertices and @@ -484,15 +505,11 @@ type() = - a type - </pre> </desc> </func> <func> - <name>domain(BinRel) -> Set</name> + <name name="domain" arity="1"/> <fsummary>Return the domain of a binary relation.</fsummary> - <type> - <v>BinRel = binary_relation()</v> - <v>Set = set()</v> - </type> <desc> <p>Returns the <seealso marker="#domain">domain</seealso> of - the binary relation BinRel.</p> + the binary relation <anno>BinRel</anno>.</p> <pre> 1> <input>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</input> <input>S = sofs:domain(R),</input> @@ -501,16 +518,13 @@ type() = - a type - </pre> </desc> </func> <func> - <name>drestriction(BinRel1, Set) -> BinRel2</name> + <name name="drestriction" arity="2"/> <fsummary>Return a restriction of a binary relation.</fsummary> - <type> - <v>BinRel1 = BinRel2 = binary_relation()</v> - <v>Set = set()</v> - </type> <desc> - <p>Returns the difference between the binary relation BinRel1 + <p>Returns the difference between the binary relation + <anno>BinRel1</anno> and the <seealso marker="#restriction">restriction</seealso> - of BinRel1 to Set.</p> + of <anno>BinRel1</anno> to <anno>Set</anno>.</p> <pre> 1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input> <input>S = sofs:set([2,4,6]),</input> @@ -522,16 +536,13 @@ type() = - a type - </pre> </desc> </func> <func> - <name>drestriction(SetFun, Set1, Set2) -> Set3</name> + <name name="drestriction" arity="3"/> <fsummary>Return a restriction of a relation.</fsummary> - <type> - <v>SetFun = set_fun()</v> - <v>Set1 = Set2 = Set3 = set()</v> - </type> - <desc> - <p>Returns a subset of Set1 containing those elements that do - not yield an element in Set2 as the result of applying - SetFun.</p> + <desc> + <p>Returns a subset of <anno>Set1</anno> containing those elements + that do + not yield an element in <anno>Set2</anno> as the result of applying + <anno>SetFun</anno>.</p> <pre> 1> <input>SetFun = {external, fun({_A,B,C}) -> {B,C} end},</input> <input>R1 = sofs:relation([{a,aa,1},{b,bb,2},{c,cc,3}]),</input> @@ -544,11 +555,8 @@ type() = - a type - </pre> </desc> </func> <func> - <name>empty_set() -> Set</name> + <name name="empty_set" arity="0"/> <fsummary>Return the untyped empty set.</fsummary> - <type> - <v>Set = set()</v> - </type> <desc> <p>Returns the <seealso marker="#sets_definition">untyped empty set</seealso>. <c>empty_set()</c> is equivalent to @@ -556,19 +564,14 @@ type() = - a type - </pre> </desc> </func> <func> - <name>extension(BinRel1, Set, AnySet) -> BinRel2</name> + <name name="extension" arity="3"/> <fsummary>Extend the domain of a binary relation.</fsummary> - <type> - <v>AnySet = anyset()</v> - <v>BinRel1 = BinRel2 = binary_relation()</v> - <v>Set = set()</v> - </type> <desc> <p>Returns the <seealso marker="#extension">extension</seealso> of - BinRel1 such that - for each element E in Set that does not belong to the - <seealso marker="#domain">domain</seealso> of BinRel1, - BinRel2 contains the pair (E, AnySet).</p> + <anno>BinRel1</anno> such that + for each element E in <anno>Set</anno> that does not belong to the + <seealso marker="#domain">domain</seealso> of <anno>BinRel1</anno>, + <anno>BinRel2</anno> contains the pair (E, AnySet).</p> <pre> 1> <input>S = sofs:set([b,c]),</input> <input>A = sofs:empty_set(),</input> @@ -579,13 +582,9 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family(Tuples [, Type]) -> Family</name> + <name name="family" arity="1"/> + <name name="family" arity="2"/> <fsummary>Create a family of subsets.</fsummary> - <type> - <v>Family = family()</v> - <v>Tuples = [tuple()]</v> - <v>Type = type()</v> - </type> <desc> <p>Creates a <seealso marker="#family">family of subsets</seealso>. <c>family(F, T)</c> is equivalent to @@ -596,18 +595,17 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_difference(Family1, Family2) -> Family3</name> + <name name="family_difference" arity="2"/> <fsummary>Return the difference of two families.</fsummary> - <type> - <v>Family1 = Family2 = Family3 = family()</v> - </type> <desc> - <p>If Family1 and Family2 + <p>If <anno>Family1</anno> and <anno>Family2</anno> are <seealso marker="#family">families</seealso>, then - Family3 is the family + <anno>Family3</anno> 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.</p> + <anno>Family1</anno>, and <anno>Family3</anno>[i] is the + difference between <anno>Family1</anno>[i] + and <anno>Family2</anno>[i] if <anno>Family2</anno> maps i, + <anno>Family1</anno>[i] otherwise.</p> <pre> 1> <input>F1 = sofs:family([{a,[1,2]},{b,[3,4]}]),</input> <input>F2 = sofs:family([{b,[4,5]},{c,[6,7]}]),</input> @@ -617,17 +615,18 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_domain(Family1) -> Family2</name> + <name name="family_domain" arity="1"/> <fsummary>Return a family of domains.</fsummary> - <type> - <v>Family1 = Family2 = family()</v> - </type> - <desc> - <p>If Family1 is a <seealso marker="#family">family</seealso> - 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 <seealso marker="#domain">domain</seealso> of Family1[i].</p> + <desc> + <p>If <anno>Family1</anno> is + a <seealso marker="#family">family</seealso> + and <anno>Family1</anno>[i] is a binary relation for every i + in the index set of <anno>Family1</anno>, + then <anno>Family2</anno> is the family with the same index + set as <anno>Family1</anno> such + that <anno>Family2</anno>[i] is + the <seealso marker="#domain">domain</seealso> of + <anno>Family1</anno>[i].</p> <pre> 1> <input>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</input> <input>F = sofs:family_domain(FR),</input> @@ -636,17 +635,18 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_field(Family1) -> Family2</name> + <name name="family_field" arity="1"/> <fsummary>Return a family of fields.</fsummary> - <type> - <v>Family1 = Family2 = family()</v> - </type> - <desc> - <p>If Family1 is a <seealso marker="#family">family</seealso> - 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 <seealso marker="#field">field</seealso> of Family1[i].</p> + <desc> + <p>If <anno>Family1</anno> is + a <seealso marker="#family">family</seealso> + and <anno>Family1</anno>[i] is a binary relation for every i + in the index set of <anno>Family1</anno>, + then <anno>Family2</anno> is the family with the same index + set as <anno>Family1</anno> such + that <anno>Family2</anno>[i] is + the <seealso marker="#field">field</seealso> of + <anno>Family1</anno>[i].</p> <pre> 1> <input>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</input> <input>F = sofs:family_field(FR),</input> @@ -657,21 +657,21 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_intersection(Family1) -> Family2</name> + <name name="family_intersection" arity="1"/> <fsummary>Return the intersection of a family of sets of sets.</fsummary> - <type> - <v>Family1 = Family2 = family()</v> - </type> - <desc> - <p>If Family1 is a <seealso marker="#family">family</seealso> - 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 <seealso marker="#intersection_n">intersection</seealso> of - Family1[i].</p> - <p>If Family1[i] is an empty set for some i, then the process - exits with a <c>badarg</c> message.</p> + <desc> + <p>If <anno>Family1</anno> is + a <seealso marker="#family">family</seealso> + and <anno>Family1</anno>[i] is a set of sets for every i in + the index set of <anno>Family1</anno>, + then <anno>Family2</anno> is the family with the same index + set as <anno>Family1</anno> such + that <anno>Family2</anno>[i] is + the <seealso marker="#intersection_n">intersection</seealso> + of <anno>Family1</anno>[i].</p> + <p>If <anno>Family1</anno>[i] is an empty set for some i, then + the process exits with a <c>badarg</c> message.</p> <pre> 1> <input>F1 = sofs:from_term([{a,[[1,2,3],[2,3,4]]},{b,[[x,y,z],[x,y]]}]),</input> <input>F2 = sofs:family_intersection(F1),</input> @@ -680,17 +680,16 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_intersection(Family1, Family2) -> Family3</name> + <name name="family_intersection" arity="2"/> <fsummary>Return the intersection of two families.</fsummary> - <type> - <v>Family1 = Family2 = Family3 = family()</v> - </type> - <desc> - <p>If Family1 and Family2 - are <seealso marker="#family">families</seealso>, 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].</p> + <desc> + <p>If <anno>Family1</anno> and <anno>Family2</anno> + are <seealso marker="#family">families</seealso>, + then <anno>Family3</anno> is the family such that the index + set is the intersection of <anno>Family1</anno>'s and + <anno>Family2</anno>'s index sets, + and <anno>Family3</anno>[i] is the intersection of + <anno>Family1</anno>[i] and <anno>Family2</anno>[i].</p> <pre> 1> <input>F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),</input> <input>F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),</input> @@ -700,18 +699,16 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_projection(SetFun, Family1) -> Family2</name> + <name name="family_projection" arity="2"/> <fsummary>Return a family of modified subsets.</fsummary> - <type> - <v>SetFun = set_fun()</v> - <v>Family1 = Family2 = family()</v> - <v>Set = set()</v> - </type> - <desc> - <p>If Family1 is a <seealso marker="#family">family</seealso> - then Family2 is the family with the same index set as - Family1 such that Family2[i] is the result of calling SetFun - with Family1[i] as argument.</p> + <desc> + <p>If <anno>Family1</anno> is + a <seealso marker="#family">family</seealso> + then <anno>Family2</anno> is the family with the same index + set as <anno>Family1</anno> such + that <anno>Family2</anno>[i] is the result of + calling <anno>SetFun</anno> with <anno>Family1</anno>[i] as + argument.</p> <pre> 1> <input>F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</input> <input>F2 = sofs:family_projection({sofs, union}, F1),</input> @@ -720,17 +717,18 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_range(Family1) -> Family2</name> + <name name="family_range" arity="1"/> <fsummary>Return a family of ranges.</fsummary> - <type> - <v>Family1 = Family2 = family()</v> - </type> - <desc> - <p>If Family1 is a <seealso marker="#family">family</seealso> - 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 <seealso marker="#range">range</seealso> of Family1[i].</p> + <desc> + <p>If <anno>Family1</anno> is + a <seealso marker="#family">family</seealso> + and <anno>Family1</anno>[i] is a binary relation for every i + in the index set of <anno>Family1</anno>, + then <anno>Family2</anno> is the family with the same index + set as <anno>Family1</anno> such + that <anno>Family2</anno>[i] is + the <seealso marker="#range">range</seealso> of + <anno>Family1</anno>[i].</p> <pre> 1> <input>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</input> <input>F = sofs:family_range(FR),</input> @@ -739,22 +737,21 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_specification(Fun, Family1) -> Family2</name> + <name name="family_specification" arity="2"/> <fsummary>Select a subset of a family using a predicate.</fsummary> - <type> - <v>Fun = spec_fun()</v> - <v>Family1 = Family2 = family()</v> - </type> - <desc> - <p>If Family1 is a <seealso marker="#family">family</seealso>, - then Family2 is - the <seealso marker="#restriction">restriction</seealso> of - Family1 to those elements i of the - index set for which Fun applied to Family1[i] returns - <c>true</c>. If Fun is a tuple <c>{external, Fun2}</c>, - Fun2 is applied to - the <seealso marker="#external_set">external set</seealso> of - Family1[i], otherwise Fun is applied to Family1[i].</p> + <desc> + <p>If <anno>Family1</anno> is + a <seealso marker="#family">family</seealso>, + then <anno>Family2</anno> is + the <seealso marker="#restriction">restriction</seealso> of + <anno>Family1</anno> to those elements i of the index set + for which <anno>Fun</anno> applied + to <anno>Family1</anno>[i] returns + <c>true</c>. If <anno>Fun</anno> is a + tuple <c>{external, Fun2}</c>, Fun2 is applied to + the <seealso marker="#external_set">external set</seealso> + of <anno>Family1</anno>[i], otherwise <anno>Fun</anno> is + applied to <anno>Family1</anno>[i].</p> <pre> 1> <input>F1 = sofs:family([{a,[1,2,3]},{b,[1,2]},{c,[1]}]),</input> <input>SpecFun = fun(S) -> sofs:no_elements(S) =:= 2 end,</input> @@ -764,24 +761,22 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_to_digraph(Family [, GraphType]) -> Graph</name> + <name name="family_to_digraph" arity="1"/> + <name name="family_to_digraph" arity="2"/> <fsummary>Create a directed graph from a family.</fsummary> - <type> - <v>Graph = digraph()</v> - <v>Family = family()</v> - <v>GraphType = - see digraph(3) -</v> - </type> <desc> <p>Creates a directed graph from - the <seealso marker="#family">family</seealso> Family. For each - pair (a, {b[1], ..., b[n]}) of Family, the vertex + the <seealso marker="#family">family</seealso> <anno>Family</anno>. + For each pair (a, {b[1], ..., b[n]}) + of <anno>Family</anno>, the vertex a as well the edges (a, b[i]) for 1 <= i <= n are added to a newly created directed graph.</p> - <p>If no graph type is given, <c>digraph:new/1</c> is used for - creating the directed graph, otherwise the GraphType + <p>If no graph type is given <seealso marker="digraph#new/0"> + digraph:new/0</seealso> is used for + creating the directed graph, otherwise the <anno>GraphType</anno> argument is passed on as second argument to - <c>digraph:new/2</c>.</p> + <seealso marker="digraph#new/1">digraph:new/1</seealso>.</p> <p>It F is a family, it holds that F is a subset of <c>digraph_to_family(family_to_digraph(F), type(F))</c>. Equality holds if <c>union_of_family(F)</c> is a subset of @@ -791,17 +786,15 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_to_relation(Family) -> BinRel</name> + <name name="family_to_relation" arity="1"/> <fsummary>Create a binary relation from a family.</fsummary> - <type> - <v>Family = family()</v> - <v>BinRel = binary_relation()</v> - </type> - <desc> - <p>If Family is a <seealso marker="#family">family</seealso>, - 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].</p> + <desc> + <p>If <anno>Family</anno> is + a <seealso marker="#family">family</seealso>, + then <anno>BinRel</anno> is the binary relation containing + all pairs (i, x) such that i belongs to the index set + of <anno>Family</anno> and x belongs + to <anno>Family</anno>[i].</p> <pre> 1> <input>F = sofs:family([{a,[]}, {b,[1]}, {c,[2,3]}]),</input> <input>R = sofs:family_to_relation(F),</input> @@ -810,17 +803,18 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_union(Family1) -> Family2</name> + <name name="family_union" arity="1"/> <fsummary>Return the union of a family of sets of sets.</fsummary> - <type> - <v>Family1 = Family2 = family()</v> - </type> - <desc> - <p>If Family1 is a <seealso marker="#family">family</seealso> - 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 <seealso marker="#union_n">union</seealso> of Family1[i].</p> + <desc> + <p>If <anno>Family1</anno> is + a <seealso marker="#family">family</seealso> + and <anno>Family1</anno>[i] is a set of sets for each i in + the index set of <anno>Family1</anno>, + then <anno>Family2</anno> is the family with the same index + set as <anno>Family1</anno> such + that <anno>Family2</anno>[i] is + the <seealso marker="#union_n">union</seealso> of + <anno>Family1</anno>[i].</p> <pre> 1> <input>F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</input> <input>F2 = sofs:family_union(F1),</input> @@ -831,18 +825,18 @@ type() = - a type - </pre> </desc> </func> <func> - <name>family_union(Family1, Family2) -> Family3</name> + <name name="family_union" arity="2"/> <fsummary>Return the union of two families.</fsummary> - <type> - <v>Family1 = Family2 = Family3 = family()</v> - </type> - <desc> - <p>If Family1 and Family2 - are <seealso marker="#family">families</seealso>, 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] otherwise.</p> + <desc> + <p>If <anno>Family1</anno> and <anno>Family2</anno> + are <seealso marker="#family">families</seealso>, + then <anno>Family3</anno> is the family such that the index + set is the union of <anno>Family1</anno>'s + and <anno>Family2</anno>'s index sets, + and <anno>Family3</anno>[i] is the union + of <anno>Family1</anno>[i] and <anno>Family2</anno>[i] if + both maps i, <anno>Family1</anno>[i] + or <anno>Family2</anno>[i] otherwise.</p> <pre> 1> <input>F1 = sofs:family([{a,[1,2]},{b,[3,4]},{c,[5,6]}]),</input> <input>F2 = sofs:family([{b,[4,5]},{c,[7,8]},{d,[9,10]}]),</input> @@ -852,15 +846,11 @@ type() = - a type - </pre> </desc> </func> <func> - <name>field(BinRel) -> Set</name> + <name name="field" arity="1"/> <fsummary>Return the field of a binary relation.</fsummary> - <type> - <v>BinRel = binary_relation()</v> - <v>Set = set()</v> - </type> <desc> <p>Returns the <seealso marker="#field">field</seealso> of the - binary relation BinRel.</p> + binary relation <anno>BinRel</anno>.</p> <pre> 1> <input>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</input> <input>S = sofs:field(R),</input> @@ -871,31 +861,24 @@ type() = - a type - </pre> </desc> </func> <func> - <name>from_external(ExternalSet, Type) -> AnySet</name> + <name name="from_external" arity="2"/> <fsummary>Create a set.</fsummary> - <type> - <v>ExternalSet = external_set()</v> - <v>AnySet = anyset()</v> - <v>Type = type()</v> - </type> <desc> <p>Creates a set from the <seealso marker="#external_set">external - set</seealso> ExternalSet - and the <seealso marker="#type">type</seealso> Type. It is - assumed that Type is a <seealso marker="#valid_type">valid - type</seealso> of ExternalSet.</p> + set</seealso> <anno>ExternalSet</anno> + and the <seealso marker="#type">type</seealso> <anno>Type</anno>. + It is assumed that <anno>Type</anno> is + a <seealso marker="#valid_type">valid + type</seealso> of <anno>ExternalSet</anno>.</p> </desc> </func> <func> - <name>from_sets(ListOfSets) -> Set</name> + <name name="from_sets" arity="1" clause_i="1"/> <fsummary>Create a set out of a list of sets.</fsummary> - <type> - <v>Set = set()</v> - <v>ListOfSets = [anyset()]</v> - </type> <desc> <p>Returns the <seealso marker="#sets_definition">unordered - set</seealso> containing the sets of the list ListOfSets.</p> + set</seealso> containing the sets of the list + <anno>ListOfSets</anno>.</p> <pre> 1> <input>S1 = sofs:relation([{a,1},{b,2}]),</input> <input>S2 = sofs:relation([{x,3},{y,4}]),</input> @@ -905,38 +888,33 @@ type() = - a type - </pre> </desc> </func> <func> - <name>from_sets(TupleOfSets) -> Ordset</name> + <name name="from_sets" arity="1" clause_i="2"/> <fsummary>Create an ordered set out of a tuple of sets.</fsummary> - <type> - <v>Ordset = ordset()</v> - <v>TupleOfSets = tuple-of(anyset())</v> - </type> <desc> <p>Returns the <seealso marker="#sets_definition">ordered set</seealso> containing the sets of the non-empty tuple - TupleOfSets.</p> + <anno>TupleOfSets</anno>.</p> </desc> </func> <func> - <name>from_term(Term [, Type]) -> AnySet</name> + <name name="from_term" arity="1"/> + <name name="from_term" arity="2"/> <fsummary>Create a set.</fsummary> - <type> - <v>AnySet = anyset()</v> - <v>Term = term()</v> - <v>Type = type()</v> - </type> <desc> <p><marker id="from_term"></marker>Creates an element of <seealso marker="#sets_definition">Sets</seealso> by - traversing the term Term, sorting lists, removing duplicates and + traversing the term <anno>Term</anno>, sorting lists, + removing duplicates and deriving or verifying a <seealso marker="#valid_type">valid type</seealso> for the so obtained external set. An - explicitly given <seealso marker="#type">type</seealso> Type + explicitly given <seealso marker="#type">type</seealso> + <anno>Type</anno> 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:</p> <pre> -1> <input>S = sofs:from_term([{{"foo"},[1,1]},{"foo",[2,2]}], [{atom,[atom]}]),</input> +1> <input>S = sofs:from_term([{{"foo"},[1,1]},{"foo",[2,2]}], +[{atom,[atom]}]),</input> <input>sofs:to_external(S).</input> [{{"foo"},[1]},{"foo",[2]}]</pre> <p><c>from_term</c> can be used for creating atomic or ordered @@ -963,15 +941,12 @@ type() = - a type - </pre> </desc> </func> <func> - <name>image(BinRel, Set1) -> Set2</name> + <name name="image" arity="2"/> <fsummary>Return the image of a set under a binary relation.</fsummary> - <type> - <v>BinRel = binary_relation()</v> - <v>Set1 = Set2 = set()</v> - </type> <desc> <p>Returns the <seealso marker="#image">image</seealso> of the - set Set1 under the binary relation BinRel.</p> + set <anno>Set1</anno> under the binary + relation <anno>BinRel</anno>.</p> <pre> 1> <input>R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),</input> <input>S1 = sofs:set([1,2]),</input> @@ -981,42 +956,32 @@ type() = - a type - </pre> </desc> </func> <func> - <name>intersection(SetOfSets) -> Set</name> + <name name="intersection" arity="1"/> <fsummary>Return the intersection of a set of sets.</fsummary> - <type> - <v>Set = set()</v> - <v>SetOfSets = set_of_sets()</v> - </type> <desc> <p>Returns the <seealso marker="#intersection_n">intersection</seealso> of - the set of sets SetOfSets.</p> + the set of sets <anno>SetOfSets</anno>.</p> <p>Intersecting an empty set of sets exits the process with a <c>badarg</c> message.</p> </desc> </func> <func> - <name>intersection(Set1, Set2) -> Set3</name> + <name name="intersection" arity="2"/> <fsummary>Return the intersection of two sets.</fsummary> - <type> - <v>Set1 = Set2 = Set3 = set()</v> - </type> <desc> <p>Returns the <seealso marker="#intersection">intersection</seealso> of - Set1 and Set2.</p> + <anno>Set1</anno> and <anno>Set2</anno>.</p> </desc> </func> <func> - <name>intersection_of_family(Family) -> Set</name> + <name name="intersection_of_family" arity="1"/> <fsummary>Return the intersection of a family.</fsummary> - <type> - <v>Family = family()</v> - <v>Set = set()</v> - </type> <desc> <p>Returns the intersection of - the <seealso marker="#family">family</seealso> Family.</p> + the <seealso marker="#family">family</seealso> <anno>Family</anno>. + </p> <p>Intersecting an empty family exits the process with a <c>badarg</c> message.</p> <pre> @@ -1027,14 +992,11 @@ type() = - a type - </pre> </desc> </func> <func> - <name>inverse(Function1) -> Function2</name> + <name name="inverse" arity="1"/> <fsummary>Return the inverse of a function.</fsummary> - <type> - <v>Function1 = Function2 = function()</v> - </type> <desc> <p>Returns the <seealso marker="#inverse">inverse</seealso> - of the function Function1.</p> + of the function <anno>Function1</anno>.</p> <pre> 1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input> <input>R2 = sofs:inverse(R1),</input> @@ -1043,16 +1005,13 @@ type() = - a type - </pre> </desc> </func> <func> - <name>inverse_image(BinRel, Set1) -> Set2</name> + <name name="inverse_image" arity="2"/> <fsummary>Return the inverse image of a set under a binary relation.</fsummary> - <type> - <v>BinRel = binary_relation()</v> - <v>Set1 = Set2 = set()</v> - </type> <desc> - <p>Returns the <seealso marker="#inverse_image">inverse - image</seealso> of Set1 under the binary relation BinRel.</p> + <p>Returns the <seealso marker="#inverse_image">inverse + image</seealso> of <anno>Set1</anno> under the binary + relation <anno>BinRel</anno>.</p> <pre> 1> <input>R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),</input> <input>S1 = sofs:set([c,d,e]),</input> @@ -1062,52 +1021,38 @@ type() = - a type - </pre> </desc> </func> <func> - <name>is_a_function(BinRel) -> Bool</name> + <name name="is_a_function" arity="1"/> <fsummary>Test for a function.</fsummary> - <type> - <v>Bool = bool()</v> - <v>BinRel = binary_relation()</v> - </type> <desc> - <p>Returns <c>true</c> if the binary relation BinRel is a - <seealso marker="#function">function</seealso> or the + <p>Returns <c>true</c> if the binary relation <anno>BinRel</anno> + is a <seealso marker="#function">function</seealso> or the untyped empty set, <c>false</c> otherwise.</p> </desc> </func> <func> - <name>is_disjoint(Set1, Set2) -> Bool</name> + <name name="is_disjoint" arity="2"/> <fsummary>Test for disjoint sets.</fsummary> - <type> - <v>Bool = bool()</v> - <v>Set1 = Set2 = set()</v> - </type> <desc> - <p>Returns <c>true</c> if Set1 and Set2 + <p>Returns <c>true</c> if <anno>Set1</anno> + and <anno>Set2</anno> are <seealso marker="#disjoint">disjoint</seealso>, <c>false</c> otherwise.</p> </desc> </func> <func> - <name>is_empty_set(AnySet) -> Bool</name> + <name name="is_empty_set" arity="1"/> <fsummary>Test for an empty set.</fsummary> - <type> - <v>AnySet = anyset()</v> - <v>Bool = bool()</v> - </type> <desc> - <p>Returns <c>true</c> if Set is an empty unordered set, - <c>false</c> otherwise.</p> + <p>Returns <c>true</c> if <anno>AnySet</anno> is an empty + unordered set, <c>false</c> otherwise.</p> </desc> </func> <func> - <name>is_equal(AnySet1, AnySet2) -> Bool</name> + <name name="is_equal" arity="2"/> <fsummary>Test two sets for equality.</fsummary> - <type> - <v>AnySet1 = AnySet2 = anyset()</v> - <v>Bool = bool()</v> - </type> <desc> - <p>Returns <c>true</c> if the AnySet1 and AnySet2 + <p>Returns <c>true</c> if the <anno>AnySet1</anno> + and <anno>AnySet2</anno> are <seealso marker="#equal">equal</seealso>, <c>false</c> otherwise. This example shows that <c>==/2</c> is used when comparing sets for equality:</p> @@ -1119,67 +1064,49 @@ true</pre> </desc> </func> <func> - <name>is_set(AnySet) -> Bool</name> + <name name="is_set" arity="1"/> <fsummary>Test for an unordered set.</fsummary> - <type> - <v>AnySet = anyset()</v> - <v>Bool = bool()</v> - </type> <desc> - <p>Returns <c>true</c> if AnySet is + <p>Returns <c>true</c> if <anno>AnySet</anno> is an <seealso marker="#sets_definition">unordered set</seealso>, and - <c>false</c> if AnySet is an ordered set or an atomic set.</p> + <c>false</c> if <anno>AnySet</anno> is an ordered set or an + atomic set.</p> </desc> </func> <func> - <name>is_sofs_set(Term) -> Bool</name> + <name name="is_sofs_set" arity="1"/> <fsummary>Test for an unordered set.</fsummary> - <type> - <v>Bool = bool()</v> - <v>Term = term()</v> - </type> <desc> - <p>Returns <c>true</c> if Term is + <p>Returns <c>true</c> if <anno>Term</anno> is an <seealso marker="#sets_definition">unordered set</seealso>, an ordered set or an atomic set, <c>false</c> otherwise.</p> </desc> </func> <func> - <name>is_subset(Set1, Set2) -> Bool</name> + <name name="is_subset" arity="2"/> <fsummary>Test two sets for subset.</fsummary> - <type> - <v>Bool = bool()</v> - <v>Set1 = Set2 = set()</v> - </type> <desc> - <p>Returns <c>true</c> if Set1 is - a <seealso marker="#subset">subset</seealso> of Set2, <c>false</c> - otherwise.</p> + <p>Returns <c>true</c> if <anno>Set1</anno> is + a <seealso marker="#subset">subset</seealso> + of <anno>Set2</anno>, <c>false</c> otherwise.</p> </desc> </func> <func> - <name>is_type(Term) -> Bool</name> + <name name="is_type" arity="1"/> <fsummary>Test for a type.</fsummary> - <type> - <v>Bool = bool()</v> - <v>Term = term()</v> - </type> <desc> - <p>Returns <c>true</c> if the term Term is + <p>Returns <c>true</c> if the term <anno>Term</anno> is a <seealso marker="#type">type</seealso>.</p> </desc> </func> <func> - <name>join(Relation1, I, Relation2, J) -> Relation3</name> + <name name="join" arity="4"/> <fsummary>Return the join of two relations.</fsummary> - <type> - <v>Relation1 = Relation2 = Relation3 = relation()</v> - <v>I = J = integer() > 0</v> - </type> <desc> <p>Returns the <seealso marker="#natural_join">natural - join</seealso> of the relations Relation1 and Relation2 on - coordinates I and J.</p> + join</seealso> of the relations <anno>Relation1</anno> + and <anno>Relation2</anno> on coordinates <anno>I</anno> and + <anno>J</anno>.</p> <pre> 1> <input>R1 = sofs:relation([{a,x,1},{b,y,2}]),</input> <input>R2 = sofs:relation([{1,f,g},{1,h,i},{2,3,4}]),</input> @@ -1189,20 +1116,17 @@ true</pre> </desc> </func> <func> - <name>multiple_relative_product(TupleOfBinRels, BinRel1) -> BinRel2</name> + <name name="multiple_relative_product" arity="2"/> <fsummary>Return the multiple relative product of a tuple of binary relations and a relation.</fsummary> - <type> - <v>TupleOfBinRels = tuple-of(BinRel)</v> - <v>BinRel = BinRel1 = BinRel2 = binary_relation()</v> - </type> - <desc> - <p>If TupleOfBinRels is a non-empty tuple - {R[1], ..., R[n]} of binary relations and BinRel1 - is a binary relation, then BinRel2 is + <desc> + <p>If <anno>TupleOfBinRels</anno> is a non-empty tuple + {R[1], ..., R[n]} of binary relations + and <anno>BinRel1</anno> is a binary relation, + then <anno>BinRel2</anno> is the <seealso marker="#multiple_relative_product">multiple relative product</seealso> of the ordered set - (R[i], ..., R[n]) and BinRel1.</p> + (R[i], ..., R[n]) and <anno>BinRel1</anno>.</p> <pre> 1> <input>Ri = sofs:relation([{a,1},{b,2},{c,3}]),</input> <input>R = sofs:relation([{a,b},{b,c},{c,a}]),</input> @@ -1212,29 +1136,21 @@ true</pre> </desc> </func> <func> - <name>no_elements(ASet) -> NoElements</name> + <name name="no_elements" arity="1"/> <fsummary>Return the number of elements of a set.</fsummary> - <type> - <v>ASet = set() | ordset()</v> - <v>NoElements = integer() >= 0 </v> - </type> <desc> <p>Returns the number of elements of the ordered or unordered - set ASet.</p> + set <anno>ASet</anno>.</p> </desc> </func> <func> - <name>partition(SetOfSets) -> Partition</name> + <name name="partition" arity="1"/> <fsummary>Return the coarsest partition given a set of sets.</fsummary> - <type> - <v>SetOfSets = set_of_sets()</v> - <v>Partition = set()</v> - </type> <desc> <p>Returns the <seealso marker="#partition">partition</seealso> of - the union of the set of sets SetOfSets such that two + the union of the set of sets <anno>SetOfSets</anno> such that two elements are considered equal if they belong to the same - elements of SetOfSets.</p> + elements of <anno>SetOfSets</anno>.</p> <pre> 1> <input>Sets1 = sofs:from_term([[a,b,c],[d,e,f],[g,h,i]]),</input> <input>Sets2 = sofs:from_term([[b,c,d],[e,f,g],[h,i,j]]),</input> @@ -1244,17 +1160,12 @@ true</pre> </desc> </func> <func> - <name>partition(SetFun, Set) -> Partition</name> + <name name="partition" arity="2"/> <fsummary>Return a partition of a set.</fsummary> - <type> - <v>SetFun = set_fun()</v> - <v>Partition = set()</v> - <v>Set = set()</v> - </type> <desc> <p>Returns the <seealso marker="#partition">partition</seealso> of - Set such that two elements are considered equal if the - results of applying SetFun are equal.</p> + <anno>Set</anno> such that two elements are considered equal + if the results of applying <anno>SetFun</anno> are equal.</p> <pre> 1> <input>Ss = sofs:from_term([[a],[b],[c,d],[e,f]]),</input> <input>SetFun = fun(S) -> sofs:from_term(sofs:no_elements(S)) end,</input> @@ -1264,19 +1175,16 @@ true</pre> </desc> </func> <func> - <name>partition(SetFun, Set1, Set2) -> {Set3, Set4}</name> + <name name="partition" arity="3"/> <fsummary>Return a partition of a set.</fsummary> - <type> - <v>SetFun = set_fun()</v> - <v>Set1 = Set2 = Set3 = Set4 = set()</v> - </type> <desc> <p>Returns a pair of sets that, regarded as constituting a set, forms a <seealso marker="#partition">partition</seealso> 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.</p> + <anno>Set1</anno>. If the + result of applying <anno>SetFun</anno> to an element + of <anno>Set1</anno> yields an element in <anno>Set2</anno>, + the element belongs to <anno>Set3</anno>, otherwise the + element belongs to <anno>Set4</anno>.</p> <pre> 1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input> <input>S = sofs:set([2,4,6]),</input> @@ -1289,21 +1197,17 @@ true</pre> </desc> </func> <func> - <name>partition_family(SetFun, Set) -> Family</name> + <name name="partition_family" arity="2"/> <fsummary>Return a family indexing a partition.</fsummary> - <type> - <v>Family = family()</v> - <v>SetFun = set_fun()</v> - <v>Set = set()</v> - </type> <desc> <p>Returns the <seealso marker="#family">family</seealso> - Family where the indexed set is - a <seealso marker="#partition">partition</seealso> 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 <seealso marker="#equivalence_class">equivalence + <anno>Family</anno> where the indexed set is + a <seealso marker="#partition">partition</seealso> + of <anno>Set</anno> such that two elements are considered + equal if the results of applying <anno>SetFun</anno> are the + same value i. This i is the index that <anno>Family</anno> + maps onto + the <seealso marker="#equivalence_class">equivalence class</seealso>.</p> <pre> 1> <input>S = sofs:relation([{a,a,a,a},{a,a,b,b},{a,b,b,b}]),</input> @@ -1314,18 +1218,15 @@ true</pre> </desc> </func> <func> - <name>product(TupleOfSets) -> Relation</name> + <name name="product" arity="1"/> <fsummary>Return the Cartesian product of a tuple of sets.</fsummary> - <type> - <v>Relation = relation()</v> - <v>TupleOfSets = tuple-of(set())</v> - </type> <desc> <p>Returns the <seealso marker="#Cartesian_product_tuple">Cartesian product</seealso> 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.</p> + <anno>TupleOfSets</anno>. If (x[1], ..., x[n]) is + an element of the n-ary relation <anno>Relation</anno>, then + x[i] is drawn from element i + of <anno>TupleOfSets</anno>.</p> <pre> 1> <input>S1 = sofs:set([a,b]),</input> <input>S2 = sofs:set([1,2]),</input> @@ -1336,15 +1237,12 @@ true</pre> </desc> </func> <func> - <name>product(Set1, Set2) -> BinRel</name> + <name name="product" arity="2"/> <fsummary>Return the Cartesian product of two sets.</fsummary> - <type> - <v>BinRel = binary_relation()</v> - <v>Set1 = Set2 = set()</v> - </type> <desc> <p>Returns the <seealso marker="#Cartesian_product">Cartesian - product</seealso> of Set1 and Set2.</p> + product</seealso> of <anno>Set1</anno> + and <anno>Set2</anno>.</p> <pre> 1> <input>S1 = sofs:set([1,2]),</input> <input>S2 = sofs:set([a,b]),</input> @@ -1356,19 +1254,16 @@ true</pre> </desc> </func> <func> - <name>projection(SetFun, Set1) -> Set2</name> + <name name="projection" arity="2"/> <fsummary>Return a set of substituted elements.</fsummary> - <type> - <v>SetFun = set_fun()</v> - <v>Set1 = Set2 = set()</v> - </type> <desc> <p>Returns the set created by substituting each element of - Set1 by the result of applying SetFun to the element.</p> - <p>If SetFun is a number i >= 1 and Set1 is a - relation, then the returned set is - the <seealso marker="#projection">projection</seealso> of Set1 - onto coordinate i.</p> + <anno>Set1</anno> by the result of + applying <anno>SetFun</anno> to the element.</p> + <p>If <anno>SetFun</anno> is a number i >= 1 and + <anno>Set1</anno> is a relation, then the returned set is + the <seealso marker="#projection">projection</seealso> of + <anno>Set1</anno> onto coordinate i.</p> <pre> 1> <input>S1 = sofs:from_term([{1,a},{2,b},{3,a}]),</input> <input>S2 = sofs:projection(2, S1),</input> @@ -1377,15 +1272,11 @@ true</pre> </desc> </func> <func> - <name>range(BinRel) -> Set</name> + <name name="range" arity="1"/> <fsummary>Return the range of a binary relation.</fsummary> - <type> - <v>BinRel = binary_relation()</v> - <v>Set = set()</v> - </type> <desc> <p>Returns the <seealso marker="#range">range</seealso> of the - binary relation BinRel.</p> + binary relation <anno>BinRel</anno>.</p> <pre> 1> <input>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</input> <input>S = sofs:range(R),</input> @@ -1394,41 +1285,33 @@ true</pre> </desc> </func> <func> - <name>relation(Tuples [, Type]) -> Relation</name> + <name name="relation" arity="1"/> + <name name="relation" arity="2"/> <fsummary>Create a relation.</fsummary> - <type> - <v>N = integer()</v> - <v>Type = N | type()</v> - <v>Relation = relation()</v> - <v>Tuples = [tuple()]</v> - </type> <desc> <p>Creates a <seealso marker="#relation">relation</seealso>. <c>relation(R, T)</c> is equivalent to <c>from_term(R, T)</c>, if T is a <seealso marker="#type">type</seealso> and the result is a - relation. If Type is an integer N, then + relation. If <anno>Type</anno> is an integer N, then <c>[{atom, ..., atom}])</c>, 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 + explicitly given, the size of the first tuple of + <anno>Tuples</anno> is used if there is such a tuple. <c>relation([])</c> is equivalent to <c>relation([], 2)</c>.</p> </desc> </func> <func> - <name>relation_to_family(BinRel) -> Family</name> + <name name="relation_to_family" arity="1"/> <fsummary>Create a family from a binary relation.</fsummary> - <type> - <v>Family = family()</v> - <v>BinRel = binary_relation()</v> - </type> <desc> <p>Returns the <seealso marker="#family">family</seealso> - Family such that the index set is equal to + <anno>Family</anno> such that the index set is equal to the <seealso marker="#domain">domain</seealso> of the binary - relation BinRel, and Family[i] is + relation <anno>BinRel</anno>, and <anno>Family</anno>[i] is the <seealso marker="#image">image</seealso> of the set of i - under BinRel.</p> + under <anno>BinRel</anno>.</p> <pre> 1> <input>R = sofs:relation([{b,1},{c,2},{c,3}]),</input> <input>F = sofs:relation_to_family(R),</input> @@ -1437,63 +1320,57 @@ true</pre> </desc> </func> <func> - <name>relative_product(TupleOfBinRels [, BinRel1]) -> BinRel2</name> - <fsummary>Return the relative product of a tuple of binary relations + <name name="relative_product" arity="1"/> + <name name="relative_product" arity="2" clause_i="1"/> + <fsummary>Return the relative product of a list of binary relations and a binary relation.</fsummary> - <type> - <v>TupleOfBinRels = tuple-of(BinRel)</v> - <v>BinRel = BinRel1 = BinRel2 = binary_relation()</v> - </type> - <desc> - <p>If TupleOfBinRels is a non-empty tuple - {R[1], ..., R[n]} of binary relations and BinRel1 - is a binary relation, then BinRel2 is - the <seealso marker="#tuple_relative_product">relative - product</seealso> of the ordered set (R[i], ..., R[n]) - and BinRel1.</p> - <p>If BinRel1 is omitted, the relation of equality between the - elements of - the <seealso marker="#Cartesian_product_tuple">Cartesian - product</seealso> of the ranges of R[i], + <desc> + <p>If <anno>ListOfBinRels</anno> is a non-empty list + [R[1], ..., R[n]] of binary relations and + <anno>BinRel1</anno> + is a binary relation, then <anno>BinRel2</anno> is the <seealso + marker="#tuple_relative_product">relative product</seealso> + of the ordered set (R[i], ..., R[n]) and + <anno>BinRel1</anno>.</p> + <p>If <anno>BinRel1</anno> is omitted, the relation of equality + between the elements of + the <seealso marker="#Cartesian_product_tuple">Cartesian + product</seealso> of the ranges of R[i], range R[1] × ... × range R[n], is used instead (intuitively, nothing is "lost").</p> <pre> 1> <input>TR = sofs:relation([{1,a},{1,aa},{2,b}]),</input> <input>R1 = sofs:relation([{1,u},{2,v},{3,c}]),</input> -<input>R2 = sofs:relative_product({TR, R1}),</input> +<input>R2 = sofs:relative_product([TR, R1]),</input> <input>sofs:to_external(R2).</input> [{1,{a,u}},{1,{aa,u}},{2,{b,v}}]</pre> - <p>Note that <c>relative_product({R1}, R2)</c> is + <p>Note that <c>relative_product([R1], R2)</c> is different from <c>relative_product(R1, R2)</c>; the - tuple of one element is not identified with the element + list of one element is not identified with the element itself.</p> </desc> </func> <func> - <name>relative_product(BinRel1, BinRel2) -> BinRel3</name> + <name name="relative_product" arity="2" clause_i="2"/> <fsummary>Return the relative product of two binary relations.</fsummary> - <type> - <v>BinRel1 = BinRel2 = BinRel3 = binary_relation()</v> - </type> <desc> <p><marker id="relprod_impl"></marker>Returns the <seealso marker="#relative_product">relative - product</seealso> of the binary relations BinRel1 and BinRel2.</p> + product</seealso> of the binary relations <anno>BinRel1</anno> + and <anno>BinRel2</anno>.</p> </desc> </func> <func> - <name>relative_product1(BinRel1, BinRel2) -> BinRel3</name> + <name name="relative_product1" arity="2"/> <fsummary>Return the relative_product of two binary relations.</fsummary> - <type> - <v>BinRel1 = BinRel2 = BinRel3 = binary_relation()</v> - </type> <desc> <p>Returns the <seealso marker="#relative_product">relative product</seealso> of the <seealso marker="#converse">converse</seealso> of the - binary relation BinRel1 and the binary relation BinRel2.</p> + binary relation <anno>BinRel1</anno> and the binary + relation <anno>BinRel2</anno>.</p> <pre> 1> <input>R1 = sofs:relation([{1,a},{1,aa},{2,b}]),</input> <input>R2 = sofs:relation([{1,u},{2,v},{3,c}]),</input> @@ -1505,15 +1382,12 @@ true</pre> </desc> </func> <func> - <name>restriction(BinRel1, Set) -> BinRel2</name> + <name name="restriction" arity="2"/> <fsummary>Return a restriction of a binary relation.</fsummary> - <type> - <v>BinRel1 = BinRel2 = binary_relation()</v> - <v>Set = set()</v> - </type> <desc> <p>Returns the <seealso marker="#restriction">restriction</seealso> of - the binary relation BinRel1 to Set.</p> + the binary relation <anno>BinRel1</anno> + to <anno>Set</anno>.</p> <pre> 1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input> <input>S = sofs:set([1,2,4]),</input> @@ -1523,15 +1397,12 @@ true</pre> </desc> </func> <func> - <name>restriction(SetFun, Set1, Set2) -> Set3</name> + <name name="restriction" arity="3"/> <fsummary>Return a restriction of a set.</fsummary> - <type> - <v>SetFun = set_fun()</v> - <v>Set1 = Set2 = Set3 = set()</v> - </type> <desc> - <p>Returns a subset of Set1 containing those elements that - yield an element in Set2 as the result of applying SetFun.</p> + <p>Returns a subset of <anno>Set1</anno> containing those + elements that yield an element in <anno>Set2</anno> as the + result of applying <anno>SetFun</anno>.</p> <pre> 1> <input>S1 = sofs:relation([{1,a},{2,b},{3,c}]),</input> <input>S2 = sofs:set([b,c,d]),</input> @@ -1541,13 +1412,9 @@ true</pre> </desc> </func> <func> - <name>set(Terms [, Type]) -> Set</name> + <name name="set" arity="1"/> + <name name="set" arity="2"/> <fsummary>Create a set of atoms or any type of sets.</fsummary> - <type> - <v>Set = set()</v> - <v>Terms = [term()]</v> - <v>Type = type()</v> - </type> <desc> <p>Creates an <seealso marker="#sets_definition">unordered set</seealso>. <c>set(L, T)</c> is equivalent to @@ -1557,18 +1424,16 @@ true</pre> </desc> </func> <func> - <name>specification(Fun, Set1) -> Set2</name> + <name name="specification" arity="2"/> <fsummary>Select a subset using a predicate.</fsummary> - <type> - <v>Fun = spec_fun()</v> - <v>Set1 = Set2 = set()</v> - </type> <desc> - <p>Returns the set containing every element of Set1 for which - Fun returns <c>true</c>. If Fun is a tuple + <p>Returns the set containing every element + of <anno>Set1</anno> for which <anno>Fun</anno> + returns <c>true</c>. If <anno>Fun</anno> is a tuple <c>{external, Fun2}</c>, Fun2 is applied to the <seealso marker="#external_set">external set</seealso> of - each element, otherwise Fun is applied to each element.</p> + each element, otherwise <anno>Fun</anno> is applied to each + element.</p> <pre> 1> <input>R1 = sofs:relation([{a,1},{b,2}]),</input> <input>R2 = sofs:relation([{x,1},{x,2},{y,3}]),</input> @@ -1579,15 +1444,13 @@ true</pre> </desc> </func> <func> - <name>strict_relation(BinRel1) -> BinRel2</name> + <name name="strict_relation" arity="1"/> <fsummary>Return the strict relation corresponding to a given relation.</fsummary> - <type> - <v>BinRel1 = BinRel2 = binary_relation()</v> - </type> <desc> <p>Returns the <seealso marker="#strict_relation">strict - relation</seealso> corresponding to the binary relation BinRel1.</p> + relation</seealso> corresponding to the binary + relation <anno>BinRel1</anno>.</p> <pre> 1> <input>R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),</input> <input>R2 = sofs:strict_relation(R1),</input> @@ -1596,16 +1459,13 @@ true</pre> </desc> </func> <func> - <name>substitution(SetFun, Set1) -> Set2</name> + <name name="substitution" arity="2"/> <fsummary>Return a function with a given set as domain.</fsummary> - <type> - <v>SetFun = set_fun()</v> - <v>Set1 = Set2 = set()</v> - </type> - <desc> - <p>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.</p> + <desc> + <p>Returns a function, the domain of which + is <anno>Set1</anno>. The value of an element of the domain + is the result of applying <anno>SetFun</anno> to the + element.</p> <pre> 1> <input>L = [{a,1},{b,2}].</input> [{a,1},{b,2}] @@ -1647,14 +1507,12 @@ images2(SetOfSets, BinRel) -> </desc> </func> <func> - <name>symdiff(Set1, Set2) -> Set3</name> + <name name="symdiff" arity="2"/> <fsummary>Return the symmetric difference of two sets.</fsummary> - <type> - <v>Set1 = Set2 = Set3 = set()</v> - </type> <desc> <p>Returns the <seealso marker="#symmetric_difference">symmetric - difference</seealso> (or the Boolean sum) of Set1 and Set2.</p> + difference</seealso> (or the Boolean sum) + of <anno>Set1</anno> and <anno>Set2</anno>.</p> <pre> 1> <input>S1 = sofs:set([1,2,3]),</input> <input>S2 = sofs:set([2,3,4]),</input> @@ -1664,88 +1522,67 @@ images2(SetOfSets, BinRel) -> </desc> </func> <func> - <name>symmetric_partition(Set1, Set2) -> {Set3, Set4, Set5}</name> + <name name="symmetric_partition" arity="2"/> <fsummary>Return a partition of two sets.</fsummary> - <type> - <v>Set1 = Set2 = Set3 = Set4 = Set5 = set()</v> - </type> <desc> - <p>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.</p> + <p>Returns a triple of sets: <anno>Set3</anno> contains the + elements of <anno>Set1</anno> that do not belong + to <anno>Set2</anno>; <anno>Set4</anno> contains the + elements of <anno>Set1</anno> that belong + to <anno>Set2</anno>; <anno>Set5</anno> contains the + elements of <anno>Set2</anno> that do not belong + to <anno>Set1</anno>.</p> </desc> </func> <func> - <name>to_external(AnySet) -> ExternalSet</name> + <name name="to_external" arity="1"/> <fsummary>Return the elements of a set.</fsummary> - <type> - <v>ExternalSet = external_set()</v> - <v>AnySet = anyset()</v> - </type> <desc> <p>Returns the <seealso marker="#external_set">external set</seealso> of an atomic, ordered or unordered set.</p> </desc> </func> <func> - <name>to_sets(ASet) -> Sets</name> + <name name="to_sets" arity="1"/> <fsummary>Return a list or a tuple of the elements of set.</fsummary> - <type> - <v>ASet = set() | ordset()</v> - <v>Sets = tuple_of(AnySet) | [AnySet]</v> - </type> <desc> - <p>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.</p> + <p>Returns the elements of the ordered set <anno>ASet</anno> + as a tuple of sets, and the elements of the unordered set + <anno>ASet</anno> as a sorted list of sets without + duplicates.</p> </desc> </func> <func> - <name>type(AnySet) -> Type</name> + <name name="type" arity="1"/> <fsummary>Return the type of a set.</fsummary> - <type> - <v>AnySet = anyset()</v> - <v>Type = type()</v> - </type> <desc> <p>Returns the <seealso marker="#type">type</seealso> of an atomic, ordered or unordered set.</p> </desc> </func> <func> - <name>union(SetOfSets) -> Set</name> + <name name="union" arity="1"/> <fsummary>Return the union of a set of sets.</fsummary> - <type> - <v>Set = set()</v> - <v>SetOfSets = set_of_sets()</v> - </type> <desc> <p>Returns the <seealso marker="#union_n">union</seealso> of the - set of sets SetOfSets.</p> + set of sets <anno>SetOfSets</anno>.</p> </desc> </func> <func> - <name>union(Set1, Set2) -> Set3</name> + <name name="union" arity="2"/> <fsummary>Return the union of two sets.</fsummary> - <type> - <v>Set1 = Set2 = Set3 = set()</v> - </type> <desc> <p>Returns the <seealso marker="#union">union</seealso> of - Set1 and Set2.</p> + <anno>Set1</anno> and <anno>Set2</anno>.</p> </desc> </func> <func> - <name>union_of_family(Family) -> Set</name> + <name name="union_of_family" arity="1"/> <fsummary>Return the union of a family.</fsummary> - <type> - <v>Family = family()</v> - <v>Set = set()</v> - </type> <desc> <p>Returns the union of - the <seealso marker="#family">family</seealso> Family.</p> + the <seealso marker="#family">family</seealso> <anno>Family</anno>. + </p> <pre> 1> <input>F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),</input> <input>S = sofs:union_of_family(F),</input> @@ -1754,17 +1591,15 @@ images2(SetOfSets, BinRel) -> </desc> </func> <func> - <name>weak_relation(BinRel1) -> BinRel2</name> + <name name="weak_relation" arity="1"/> <fsummary>Return the weak relation corresponding to a given relation.</fsummary> - <type> - <v>BinRel1 = BinRel2 = binary_relation()</v> - </type> <desc> <p>Returns a subset S of the <seealso marker="#weak_relation">weak relation</seealso> W - corresponding to the binary relation BinRel1. Let F be the - <seealso marker="#field">field</seealso> of BinRel1. The + corresponding to the binary relation <anno>BinRel1</anno>. + Let F be the <seealso marker="#field">field</seealso> of + <anno>BinRel1</anno>. 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.</p> <pre> |