diff options
Diffstat (limited to 'lib/stdlib/doc/src/sofs.xml')
| -rw-r--r-- | lib/stdlib/doc/src/sofs.xml | 324 |
1 files changed, 162 insertions, 162 deletions
diff --git a/lib/stdlib/doc/src/sofs.xml b/lib/stdlib/doc/src/sofs.xml index c2c6675258..9b96c4d39a 100644 --- a/lib/stdlib/doc/src/sofs.xml +++ b/lib/stdlib/doc/src/sofs.xml @@ -273,7 +273,7 @@ <p>If S is an element (T, X) of Sets, then T is a <marker id="valid_type"></marker><em>valid type</em> of X, T is the type of S, and X is the external set of S. - <seealso marker="#from_term/2"><c>from_term/2</c></seealso> creates a + <seemfa marker="#from_term/2"><c>from_term/2</c></seemfa> creates a set from a type and an Erlang term turned into an external set.</p> <p>The sets represented by Sets are the elements of the range of function Set from Sets to Erlang terms and sets of Erlang terms:</p> @@ -288,7 +288,7 @@ </list> <p>When there is no risk of confusion, elements of Sets are identified with the sets they represent. For example, if U is the result of - calling <seealso marker="#union/2"><c>union/2</c></seealso> with S1 + calling <seemfa marker="#union/2"><c>union/2</c></seemfa> with S1 and S2 as arguments, then U is said to be the union of S1 and S2. A more precise formulation is that Set(U) is the union of Set(S1) and Set(S2).</p> @@ -300,8 +300,8 @@ 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, <seealso marker="#relative_product/2"> - <c>relative_product/2</c></seealso>, checks + relative product, <seemfa marker="#relative_product/2"> + <c>relative_product/2</c></seemfa>, 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 @@ -316,13 +316,13 @@ set.</p> <p>A few functions of this module - (<seealso marker="#drestriction/3"><c>drestriction/3</c></seealso>, - <seealso marker="#family_projection/2"><c>family_projection/2</c></seealso>, - <seealso marker="#partition/2"><c>partition/2</c></seealso>, - <seealso marker="#partition_family/2"><c>partition_family/2</c></seealso>, - <seealso marker="#projection/2"><c>projection/2</c></seealso>, - <seealso marker="#restriction/3"><c>restriction/3</c></seealso>, - <seealso marker="#substitution/2"><c>substitution/2</c></seealso>) + (<seemfa marker="#drestriction/3"><c>drestriction/3</c></seemfa>, + <seemfa marker="#family_projection/2"><c>family_projection/2</c></seemfa>, + <seemfa marker="#partition/2"><c>partition/2</c></seemfa>, + <seemfa marker="#partition_family/2"><c>partition_family/2</c></seemfa>, + <seemfa marker="#projection/2"><c>projection/2</c></seemfa>, + <seemfa marker="#restriction/3"><c>restriction/3</c></seemfa>, + <seemfa marker="#substitution/2"><c>substitution/2</c></seemfa>) accept an Erlang function as a means to modify each element of a given unordered set. <marker id="set_fun"></marker>Such a function, called @@ -377,12 +377,12 @@ fun(S) -> sofs:partition(1, S) end the execution time is in the worst case proportional to the sum of the sizes of the input arguments and the returned value. A few functions execute in constant time: - <seealso marker="#from_external/2"><c>from_external/2</c></seealso>, - <seealso marker="#is_empty_set/1"><c>is_empty_set/1</c></seealso>, - <seealso marker="#is_set/1"><c>is_set/1</c></seealso>, - <seealso marker="#is_sofs_set/1"><c>is_sofs_set/1</c></seealso>, - <seealso marker="#to_external/1"><c>to_external/1</c></seealso> - <seealso marker="#type/1"><c>type/1</c></seealso>.</p> + <seemfa marker="#from_external/2"><c>from_external/2</c></seemfa>, + <seemfa marker="#is_empty_set/1"><c>is_empty_set/1</c></seemfa>, + <seemfa marker="#is_set/1"><c>is_set/1</c></seemfa>, + <seemfa marker="#is_sofs_set/1"><c>is_sofs_set/1</c></seemfa>, + <seemfa marker="#to_external/1"><c>to_external/1</c></seemfa> + <seemfa marker="#type/1"><c>type/1</c></seemfa>.</p> <p>The functions of this module exit the process with a <c>badarg</c>, <c>bad_function</c>, or <c>type_mismatch</c> @@ -399,53 +399,53 @@ fun(S) -> sofs:partition(1, S) end </datatype> <datatype> <name name="binary_relation"></name> - <desc><p>A <seealso marker="#binary_relation">binary - relation</seealso>.</p></desc> + <desc><p>A <seeerl marker="#binary_relation">binary + relation</seeerl>.</p></desc> </datatype> <datatype> <name name="external_set"></name> - <desc><p>An <seealso marker="#external_set">external - set</seealso>.</p></desc> + <desc><p>An <seeerl marker="#external_set">external + set</seeerl>.</p></desc> </datatype> <datatype> <name name="family"></name> - <desc><p>A <seealso marker="#family">family</seealso> (of subsets).</p> + <desc><p>A <seeerl marker="#family">family</seeerl> (of subsets).</p> </desc> </datatype> <datatype> <name name="a_function"></name> - <desc><p>A <seealso marker="#function">function</seealso>.</p></desc> + <desc><p>A <seeerl marker="#function">function</seeerl>.</p></desc> </datatype> <datatype> <name name="ordset"></name> - <desc><p>An <seealso marker="#sets_definition">ordered - set</seealso>.</p></desc> + <desc><p>An <seeerl marker="#sets_definition">ordered + set</seeerl>.</p></desc> </datatype> <datatype> <name name="relation"></name> - <desc><p>An <seealso marker="#n_ary_relation">n-ary relation</seealso>. + <desc><p>An <seeerl marker="#n_ary_relation">n-ary relation</seeerl>. </p></desc> </datatype> <datatype> <name name="a_set"></name> - <desc><p>An <seealso marker="#sets_definition">unordered - set</seealso>.</p></desc> + <desc><p>An <seeerl marker="#sets_definition">unordered + set</seeerl>.</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> + <desc><p>An <seeerl marker="#sets_definition">unordered + set</seeerl> of unordered sets.</p></desc> </datatype> <datatype> <name name="set_fun"></name> - <desc><p>A <seealso marker="#set_fun">SetFun</seealso>.</p></desc> + <desc><p>A <seeerl marker="#set_fun">SetFun</seeerl>.</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> + <desc><p>A <seeerl marker="#type">type</seeerl>.</p></desc> </datatype> <datatype> <!-- Parameterized opaque types are NYI: --> @@ -460,10 +460,10 @@ fun(S) -> sofs:partition(1, S) end <name name="a_function" arity="2" since=""/> <fsummary>Create a function.</fsummary> <desc> - <p>Creates a <seealso marker="#function">function</seealso>. + <p>Creates a <seeerl marker="#function">function</seeerl>. <c>a_function(F, T)</c> is equivalent to <c>from_term(F, T)</c> if the result is a function. If - no <seealso marker="#type">type</seealso> is explicitly + no <seeerl marker="#type">type</seeerl> is explicitly specified, <c>[{atom, atom}]</c> is used as the function type.</p> </desc> @@ -476,10 +476,10 @@ fun(S) -> sofs:partition(1, S) end <p>Returns the binary relation containing the elements (E, Set) such that Set belongs to <c><anno>SetOfSets</anno></c> and E belongs to Set. If <c>SetOfSets</c> is - a <seealso marker="#partition">partition</seealso> of a set X and + a <seeerl marker="#partition">partition</seeerl> of a set X and R is the equivalence relation in X induced by <c>SetOfSets</c>, then the returned relation is - the <seealso marker="#canonical_map">canonical map</seealso> from + the <seeerl marker="#canonical_map">canonical map</seeerl> from X onto the equivalence classes with respect to R.</p> <pre> 1> <input>Ss = sofs:from_term([[a,b],[b,c]]),</input> @@ -493,7 +493,7 @@ fun(S) -> sofs:partition(1, S) end <name name="composite" arity="2" since=""/> <fsummary>Return the composite of two functions.</fsummary> <desc> - <p>Returns the <seealso marker="#composite">composite</seealso> of + <p>Returns the <seeerl marker="#composite">composite</seeerl> of the functions <c><anno>Function1</anno></c> and <c><anno>Function2</anno></c>.</p> <pre> @@ -510,7 +510,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Create the function that maps each element of a set onto another set.</fsummary> <desc> - <p>Creates the <seealso marker="#function">function</seealso> + <p>Creates the <seeerl marker="#function">function</seeerl> that maps each element of set <c>Set</c> onto <c>AnySet</c>.</p> <pre> 1> <input>S = sofs:set([a,b]),</input> @@ -525,7 +525,7 @@ fun(S) -> sofs:partition(1, S) end <name name="converse" arity="1" since=""/> <fsummary>Return the converse of a binary relation.</fsummary> <desc> - <p>Returns the <seealso marker="#converse">converse</seealso> + <p>Returns the <seeerl marker="#converse">converse</seeerl> of the binary relation <c><anno>BinRel1</anno></c>.</p> <pre> 1> <input>R1 = sofs:relation([{1,a},{2,b},{3,a}]),</input> @@ -539,7 +539,7 @@ fun(S) -> sofs:partition(1, S) end <name name="difference" arity="2" since=""/> <fsummary>Return the difference of two sets.</fsummary> <desc> - <p>Returns the <seealso marker="#difference">difference</seealso> of + <p>Returns the <seeerl marker="#difference">difference</seeerl> of the sets <c><anno>Set1</anno></c> and <c><anno>Set2</anno></c>.</p> </desc> </func> @@ -549,14 +549,14 @@ fun(S) -> sofs:partition(1, S) end <name name="digraph_to_family" arity="2" since=""/> <fsummary>Create a family from a directed graph.</fsummary> <desc> - <p>Creates a <seealso marker="#family">family</seealso> from + <p>Creates a <seeerl marker="#family">family</seeerl> from the directed graph <c><anno>Graph</anno></c>. Each vertex a of <c><anno>Graph</anno></c> is represented by a pair (a, {b[1], ..., b[n]}), where the b[i]:s are the out-neighbors of a. If no type is explicitly specified, [{atom, [atom]}] is used as type of the family. It is assumed that <c><anno>Type</anno></c> is - a <seealso marker="#valid_type">valid type</seealso> of the + a <seeerl marker="#valid_type">valid type</seeerl> of the external set of the family.</p> <p>If G is a directed graph, it holds that the vertices and edges of G are the same as the vertices and edges of @@ -568,7 +568,7 @@ fun(S) -> sofs:partition(1, S) end <name name="domain" arity="1" since=""/> <fsummary>Return the domain of a binary relation.</fsummary> <desc> - <p>Returns the <seealso marker="#domain">domain</seealso> of + <p>Returns the <seeerl marker="#domain">domain</seeerl> of the binary relation <c><anno>BinRel</anno></c>.</p> <pre> 1> <input>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</input> @@ -584,7 +584,7 @@ fun(S) -> sofs:partition(1, S) end <desc> <p>Returns the difference between the binary relation <c><anno>BinRel1</anno></c> - and the <seealso marker="#restriction">restriction</seealso> + and the <seeerl marker="#restriction">restriction</seeerl> of <c><anno>BinRel1</anno></c> to <c><anno>Set</anno></c>.</p> <pre> 1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input> @@ -621,8 +621,8 @@ fun(S) -> sofs:partition(1, S) end <name name="empty_set" arity="0" since=""/> <fsummary>Return the untyped empty set.</fsummary> <desc> - <p>Returns the <seealso marker="#sets_definition">untyped empty - set</seealso>. <c>empty_set()</c> is equivalent to + <p>Returns the <seeerl marker="#sets_definition">untyped empty + set</seeerl>. <c>empty_set()</c> is equivalent to <c>from_term([], ['_'])</c>.</p> </desc> </func> @@ -631,10 +631,10 @@ fun(S) -> sofs:partition(1, S) end <name name="extension" arity="3" since=""/> <fsummary>Extend the domain of a binary relation.</fsummary> <desc> - <p>Returns the <seealso marker="#extension">extension</seealso> of + <p>Returns the <seeerl marker="#extension">extension</seeerl> of <c><anno>BinRel1</anno></c> such that for each element E in <c><anno>Set</anno></c> that does not belong to the - <seealso marker="#domain">domain</seealso> of + <seeerl marker="#domain">domain</seeerl> of <c><anno>BinRel1</anno></c>, <c><anno>BinRel2</anno></c> contains the pair (E, <c>AnySet</c>).</p> <pre> @@ -652,10 +652,10 @@ fun(S) -> sofs:partition(1, S) end <name name="family" arity="2" since=""/> <fsummary>Create a family of subsets.</fsummary> <desc> - <p>Creates a <seealso marker="#family">family of subsets</seealso>. + <p>Creates a <seeerl marker="#family">family of subsets</seeerl>. <c>family(F, T)</c> is equivalent to <c>from_term(F, T)</c> if the result is a family. If - no <seealso marker="#type">type</seealso> is explicitly + no <seeerl marker="#type">type</seeerl> is explicitly specified, <c>[{atom, [atom]}]</c> is used as the family type.</p> </desc> @@ -666,7 +666,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return the difference of two families.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> and <c><anno>Family2</anno></c> - are <seealso marker="#family">families</seealso>, then + are <seeerl marker="#family">families</seeerl>, then <c><anno>Family3</anno></c> is the family such that the index set is equal to the index set of <c><anno>Family1</anno></c>, and <c><anno>Family3</anno></c>[i] is @@ -687,13 +687,13 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return a family of domains.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> is - a <seealso marker="#family">family</seealso> + a <seeerl marker="#family">family</seeerl> and <c><anno>Family1</anno></c>[i] is a binary relation for every i in the index set of <c><anno>Family1</anno></c>, then <c><anno>Family2</anno></c> is the family with the same index set as <c><anno>Family1</anno></c> such that <c><anno>Family2</anno></c>[i] is - the <seealso marker="#domain">domain</seealso> of + the <seeerl marker="#domain">domain</seeerl> of <c><anno>Family1</anno>[i]</c>.</p> <pre> 1> <input>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</input> @@ -708,13 +708,13 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return a family of fields.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> is - a <seealso marker="#family">family</seealso> + a <seeerl marker="#family">family</seeerl> and <c><anno>Family1</anno></c>[i] is a binary relation for every i in the index set of <c><anno>Family1</anno></c>, then <c><anno>Family2</anno></c> is the family with the same index set as <c><anno>Family1</anno></c> such that <c><anno>Family2</anno></c>[i] is - the <seealso marker="#field">field</seealso> of + the <seeerl marker="#field">field</seeerl> of <c><anno>Family1</anno></c>[i].</p> <pre> 1> <input>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</input> @@ -733,13 +733,13 @@ fun(S) -> sofs:partition(1, S) end of sets of sets.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> is - a <seealso marker="#family">family</seealso> + a <seeerl marker="#family">family</seeerl> and <c><anno>Family1</anno></c>[i] is a set of sets for every i in the index set of <c><anno>Family1</anno></c>, then <c><anno>Family2</anno></c> is the family with the same index set as <c><anno>Family1</anno></c> such that <c><anno>Family2</anno></c>[i] is - the <seealso marker="#intersection_n">intersection</seealso> + the <seeerl marker="#intersection_n">intersection</seeerl> of <c><anno>Family1</anno></c>[i].</p> <p>If <c><anno>Family1</anno></c>[i] is an empty set for some i, the process exits with a <c>badarg</c> message.</p> @@ -756,7 +756,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return the intersection of two families.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> and <c><anno>Family2</anno></c> - are <seealso marker="#family">families</seealso>, + are <seeerl marker="#family">families</seeerl>, then <c><anno>Family3</anno></c> is the family such that the index set is the intersection of <c><anno>Family1</anno></c>:s and <c><anno>Family2</anno></c>:s index sets, @@ -776,7 +776,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return a family of modified subsets.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> is - a <seealso marker="#family">family</seealso>, + a <seeerl marker="#family">family</seeerl>, then <c><anno>Family2</anno></c> is the family with the same index set as <c><anno>Family1</anno></c> such that <c><anno>Family2</anno></c>[i] is the result of @@ -795,13 +795,13 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return a family of ranges.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> is - a <seealso marker="#family">family</seealso> + a <seeerl marker="#family">family</seeerl> and <c><anno>Family1</anno></c>[i] is a binary relation for every i in the index set of <c><anno>Family1</anno></c>, then <c><anno>Family2</anno></c> is the family with the same index set as <c><anno>Family1</anno></c> such that <c><anno>Family2</anno></c>[i] is - the <seealso marker="#range">range</seealso> of + the <seeerl marker="#range">range</seeerl> of <c><anno>Family1</anno></c>[i].</p> <pre> 1> <input>FR = sofs:from_term([{a,[{1,a},{2,b},{3,c}]},{b,[]},{c,[{4,d},{5,e}]}]),</input> @@ -816,15 +816,15 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Select a subset of a family using a predicate.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> is - a <seealso marker="#family">family</seealso>, + a <seeerl marker="#family">family</seeerl>, then <c><anno>Family2</anno></c> is - the <seealso marker="#restriction">restriction</seealso> of + the <seeerl marker="#restriction">restriction</seeerl> of <c><anno>Family1</anno></c> to those elements i of the index set for which <c><anno>Fun</anno></c> applied to <c><anno>Family1</anno></c>[i] returns <c>true</c>. If <c><anno>Fun</anno></c> is a tuple <c>{external, Fun2}</c>, then <c>Fun2</c> is applied to - the <seealso marker="#external_set">external set</seealso> + the <seeerl marker="#external_set">external set</seeerl> of <c><anno>Family1</anno></c>[i], otherwise <c><anno>Fun</anno></c> is applied to <c><anno>Family1</anno></c>[i].</p> <pre> @@ -842,17 +842,17 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Create a directed graph from a family.</fsummary> <desc> <p>Creates a directed graph from - <seealso marker="#family">family</seealso> <c><anno>Family</anno></c>. + <seeerl marker="#family">family</seeerl> <c><anno>Family</anno></c>. For each pair (a, {b[1], ..., b[n]}) of <c><anno>Family</anno></c>, vertex a and the edges (a, b[i]) for 1 <= i <= n are added to a newly created directed graph.</p> - <p>If no graph type is specified, <seealso marker="digraph#new/0"> - <c>digraph:new/0</c></seealso> is used for + <p>If no graph type is specified, <seemfa marker="digraph#new/0"> + <c>digraph:new/0</c></seemfa> is used for creating the directed graph, otherwise argument <c><anno>GraphType</anno></c> is passed on as second argument to - <seealso marker="digraph#new/1"><c>digraph:new/1</c></seealso>.</p> + <seemfa marker="digraph#new/1"><c>digraph:new/1</c></seemfa>.</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 @@ -867,7 +867,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Create a binary relation from a family.</fsummary> <desc> <p>If <c><anno>Family</anno></c> is - a <seealso marker="#family">family</seealso>, + a <seeerl marker="#family">family</seeerl>, then <c><anno>BinRel</anno></c> is the binary relation containing all pairs (i, x) such that i belongs to the index set of <c><anno>Family</anno></c> and x belongs @@ -885,13 +885,13 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return the union of a family of sets of sets.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> is - a <seealso marker="#family">family</seealso> + a <seeerl marker="#family">family</seeerl> and <c><anno>Family1</anno></c>[i] is a set of sets for each i in the index set of <c><anno>Family1</anno></c>, then <c><anno>Family2</anno></c> is the family with the same index set as <c><anno>Family1</anno></c> such that <c><anno>Family2</anno></c>[i] is - the <seealso marker="#union_n">union</seealso> of + the <seeerl marker="#union_n">union</seeerl> of <c><anno>Family1</anno></c>[i].</p> <pre> 1> <input>F1 = sofs:from_term([{a,[[1,2],[2,3]]},{b,[[]]}]),</input> @@ -908,7 +908,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return the union of two families.</fsummary> <desc> <p>If <c><anno>Family1</anno></c> and <c><anno>Family2</anno></c> - are <seealso marker="#family">families</seealso>, + are <seeerl marker="#family">families</seeerl>, then <c><anno>Family3</anno></c> is the family such that the index set is the union of <c><anno>Family1</anno></c>:s and <c><anno>Family2</anno></c>:s index sets, @@ -929,7 +929,7 @@ fun(S) -> sofs:partition(1, S) end <name name="field" arity="1" since=""/> <fsummary>Return the field of a binary relation.</fsummary> <desc> - <p>Returns the <seealso marker="#field">field</seealso> of the + <p>Returns the <seeerl marker="#field">field</seeerl> of the binary relation <c><anno>BinRel</anno></c>.</p> <pre> 1> <input>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</input> @@ -945,12 +945,12 @@ fun(S) -> sofs:partition(1, S) end <name name="from_external" arity="2" since=""/> <fsummary>Create a set.</fsummary> <desc> - <p>Creates a set from the <seealso marker="#external_set">external - set</seealso> <c><anno>ExternalSet</anno></c> and - the <seealso marker="#type">type</seealso> <c><anno>Type</anno></c>. + <p>Creates a set from the <seeerl marker="#external_set">external + set</seeerl> <c><anno>ExternalSet</anno></c> and + the <seeerl marker="#type">type</seeerl> <c><anno>Type</anno></c>. It is assumed that <c><anno>Type</anno></c> is - a <seealso marker="#valid_type">valid - type</seealso> of <c><anno>ExternalSet</anno></c>.</p> + a <seeerl marker="#valid_type">valid + type</seeerl> of <c><anno>ExternalSet</anno></c>.</p> </desc> </func> @@ -958,8 +958,8 @@ fun(S) -> sofs:partition(1, S) end <name name="from_sets" arity="1" clause_i="1" since=""/> <fsummary>Create a set out of a list of sets.</fsummary> <desc> - <p>Returns the <seealso marker="#sets_definition">unordered - set</seealso> containing the sets of list + <p>Returns the <seeerl marker="#sets_definition">unordered + set</seeerl> containing the sets of list <c><anno>ListOfSets</anno></c>.</p> <pre> 1> <input>S1 = sofs:relation([{a,1},{b,2}]),</input> @@ -974,8 +974,8 @@ fun(S) -> sofs:partition(1, S) end <name name="from_sets" arity="1" clause_i="2" since=""/> <fsummary>Create an ordered set out of a tuple of sets.</fsummary> <desc> - <p>Returns the <seealso marker="#sets_definition">ordered - set</seealso> containing the sets of the non-empty tuple + <p>Returns the <seeerl marker="#sets_definition">ordered + set</seeerl> containing the sets of the non-empty tuple <c><anno>TupleOfSets</anno></c>.</p> </desc> </func> @@ -986,12 +986,12 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Create a set.</fsummary> <desc> <p><marker id="from_term"></marker>Creates an element - of <seealso marker="#sets_definition">Sets</seealso> by + of <seeerl marker="#sets_definition">Sets</seeerl> by traversing term <c><anno>Term</anno></c>, sorting lists, removing duplicates, and - deriving or verifying a <seealso marker="#valid_type">valid - type</seealso> for the so obtained external set. An - explicitly specified <seealso marker="#type">type</seealso> + deriving or verifying a <seeerl marker="#valid_type">valid + type</seeerl> for the so obtained external set. An + explicitly specified <seeerl marker="#type">type</seeerl> <c><anno>Type</anno></c> can be used to limit the depth of the traversal; an atomic type stops the traversal, as shown by the following example @@ -1019,14 +1019,14 @@ fun(S) -> sofs:partition(1, S) end <input>sofs:to_external(Ss).</input> [{a,[1,2,3]},{b,[4,5,6]}]</pre> <p>Other functions that create sets are - <seealso marker="#from_external/2"><c>from_external/2</c></seealso> - and <seealso marker="#from_sets/1"><c>from_sets/1</c></seealso>. + <seemfa marker="#from_external/2"><c>from_external/2</c></seemfa> + and <seemfa marker="#from_sets/1"><c>from_sets/1</c></seemfa>. Special cases of <c>from_term/2</c> are - <seealso marker="#a_function/1"><c>a_function/1,2</c></seealso>, - <seealso marker="#empty_set/0"><c>empty_set/0</c></seealso>, - <seealso marker="#family/1"><c>family/1,2</c></seealso>, - <seealso marker="#relation/1"><c>relation/1,2</c></seealso>, and - <seealso marker="#set/1"><c>set/1,2</c></seealso>.</p> + <seemfa marker="#a_function/1"><c>a_function/1,2</c></seemfa>, + <seemfa marker="#empty_set/0"><c>empty_set/0</c></seemfa>, + <seemfa marker="#family/1"><c>family/1,2</c></seemfa>, + <seemfa marker="#relation/1"><c>relation/1,2</c></seemfa>, and + <seemfa marker="#set/1"><c>set/1,2</c></seemfa>.</p> </desc> </func> @@ -1034,7 +1034,7 @@ fun(S) -> sofs:partition(1, S) end <name name="image" arity="2" since=""/> <fsummary>Return the image of a set under a binary relation.</fsummary> <desc> - <p>Returns the <seealso marker="#image">image</seealso> of + <p>Returns the <seeerl marker="#image">image</seeerl> of set <c><anno>Set1</anno></c> under the binary relation <c><anno>BinRel</anno></c>.</p> <pre> @@ -1051,7 +1051,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return the intersection of a set of sets.</fsummary> <desc> <p>Returns - the <seealso marker="#intersection_n">intersection</seealso> of + the <seeerl marker="#intersection_n">intersection</seeerl> of the set of sets <c><anno>SetOfSets</anno></c>.</p> <p>Intersecting an empty set of sets exits the process with a <c>badarg</c> message.</p> @@ -1063,7 +1063,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return the intersection of two sets.</fsummary> <desc> <p>Returns - the <seealso marker="#intersection">intersection</seealso> of + the <seeerl marker="#intersection">intersection</seeerl> of <c><anno>Set1</anno></c> and <c><anno>Set2</anno></c>.</p> </desc> </func> @@ -1073,7 +1073,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return the intersection of a family.</fsummary> <desc> <p>Returns the intersection of - <seealso marker="#family">family</seealso> <c><anno>Family</anno></c>. + <seeerl marker="#family">family</seeerl> <c><anno>Family</anno></c>. </p> <p>Intersecting an empty family exits the process with a <c>badarg</c> message.</p> @@ -1089,7 +1089,7 @@ fun(S) -> sofs:partition(1, S) end <name name="inverse" arity="1" since=""/> <fsummary>Return the inverse of a function.</fsummary> <desc> - <p>Returns the <seealso marker="#inverse">inverse</seealso> + <p>Returns the <seeerl marker="#inverse">inverse</seeerl> of function <c><anno>Function1</anno></c>.</p> <pre> 1> <input>R1 = sofs:relation([{1,a},{2,b},{3,c}]),</input> @@ -1104,8 +1104,8 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Return the inverse image of a set under a binary relation.</fsummary> <desc> - <p>Returns the <seealso marker="#inverse_image">inverse - image</seealso> of <c><anno>Set1</anno></c> under the binary + <p>Returns the <seeerl marker="#inverse_image">inverse + image</seeerl> of <c><anno>Set1</anno></c> under the binary relation <c><anno>BinRel</anno></c>.</p> <pre> 1> <input>R = sofs:relation([{1,a},{2,b},{2,c},{3,d}]),</input> @@ -1121,7 +1121,7 @@ fun(S) -> sofs:partition(1, S) end <fsummary>Test for a function.</fsummary> <desc> <p>Returns <c>true</c> if the binary relation <c><anno>BinRel</anno></c> - is a <seealso marker="#function">function</seealso> or the + is a <seeerl marker="#function">function</seeerl> or the untyped empty set, otherwise <c>false</c>.</p> </desc> </func> @@ -1132,7 +1132,7 @@ fun(S) -> sofs:partition(1, S) end <desc> <p>Returns <c>true</c> if <c><anno>Set1</anno></c> and <c><anno>Set2</anno></c> - are <seealso marker="#disjoint">disjoint</seealso>, otherwise + are <seeerl marker="#disjoint">disjoint</seeerl>, otherwise <c>false</c>.</p> </desc> </func> @@ -1152,7 +1152,7 @@ fun(S) -> sofs:partition(1, S) end <desc> <p>Returns <c>true</c> if <c><anno>AnySet1</anno></c> and <c><anno>AnySet2</anno></c> - are <seealso marker="#equal">equal</seealso>, otherwise + are <seeerl marker="#equal">equal</seeerl>, otherwise <c>false</c>. The following example shows that <c>==/2</c> is used when comparing sets for equality:</p> <pre> @@ -1168,7 +1168,7 @@ true</pre> <fsummary>Test for an unordered set.</fsummary> <desc> <p>Returns <c>true</c> if <c><anno>AnySet</anno></c> is - an <seealso marker="#sets_definition">unordered set</seealso>, and + an <seeerl marker="#sets_definition">unordered set</seeerl>, and <c>false</c> if <c><anno>AnySet</anno></c> is an ordered set or an atomic set.</p> </desc> @@ -1179,7 +1179,7 @@ true</pre> <fsummary>Test for an unordered set.</fsummary> <desc> <p>Returns <c>true</c> if <c><anno>Term</anno></c> is - an <seealso marker="#sets_definition">unordered set</seealso>, an + an <seeerl marker="#sets_definition">unordered set</seeerl>, an ordered set, or an atomic set, otherwise <c>false</c>.</p> </desc> </func> @@ -1189,7 +1189,7 @@ true</pre> <fsummary>Test two sets for subset.</fsummary> <desc> <p>Returns <c>true</c> if <c><anno>Set1</anno></c> is - a <seealso marker="#subset">subset</seealso> + a <seeerl marker="#subset">subset</seeerl> of <c><anno>Set2</anno></c>, otherwise <c>false</c>.</p> </desc> </func> @@ -1199,7 +1199,7 @@ true</pre> <fsummary>Test for a type.</fsummary> <desc> <p>Returns <c>true</c> if term <c><anno>Term</anno></c> is - a <seealso marker="#type">type</seealso>.</p> + a <seeerl marker="#type">type</seeerl>.</p> </desc> </func> @@ -1207,8 +1207,8 @@ true</pre> <name name="join" arity="4" since=""/> <fsummary>Return the join of two relations.</fsummary> <desc> - <p>Returns the <seealso marker="#natural_join">natural - join</seealso> of the relations <c><anno>Relation1</anno></c> + <p>Returns the <seeerl marker="#natural_join">natural + join</seeerl> of the relations <c><anno>Relation1</anno></c> and <c><anno>Relation2</anno></c> on coordinates <c><anno>I</anno></c> and <c><anno>J</anno></c>.</p> <pre> @@ -1229,8 +1229,8 @@ true</pre> {R[1], ..., R[n]} of binary relations and <c><anno>BinRel1</anno></c> is a binary relation, then <c><anno>BinRel2</anno></c> is - the <seealso marker="#multiple_relative_product">multiple relative - product</seealso> of the ordered set + the <seeerl marker="#multiple_relative_product">multiple relative + product</seeerl> of the ordered set (R[i], ..., R[n]) and <c><anno>BinRel1</anno></c>.</p> <pre> 1> <input>Ri = sofs:relation([{a,1},{b,2},{c,3}]),</input> @@ -1254,7 +1254,7 @@ true</pre> <name name="partition" arity="1" since=""/> <fsummary>Return the coarsest partition given a set of sets.</fsummary> <desc> - <p>Returns the <seealso marker="#partition">partition</seealso> of + <p>Returns the <seeerl marker="#partition">partition</seeerl> of the union of the set of sets <c><anno>SetOfSets</anno></c> such that two elements are considered equal if they belong to the same elements of <c><anno>SetOfSets</anno></c>.</p> @@ -1271,7 +1271,7 @@ true</pre> <name name="partition" arity="2" since=""/> <fsummary>Return a partition of a set.</fsummary> <desc> - <p>Returns the <seealso marker="#partition">partition</seealso> of + <p>Returns the <seeerl marker="#partition">partition</seeerl> of <c><anno>Set</anno></c> such that two elements are considered equal if the results of applying <c><anno>SetFun</anno></c> are equal.</p> <pre> @@ -1288,7 +1288,7 @@ true</pre> <fsummary>Return a partition of a set.</fsummary> <desc> <p>Returns a pair of sets that, regarded as constituting a - set, forms a <seealso marker="#partition">partition</seealso> of + set, forms a <seeerl marker="#partition">partition</seeerl> of <c><anno>Set1</anno></c>. If the result of applying <c><anno>SetFun</anno></c> to an element of <c><anno>Set1</anno></c> gives an element in <c><anno>Set2</anno></c>, @@ -1310,14 +1310,14 @@ true</pre> <name name="partition_family" arity="2" since=""/> <fsummary>Return a family indexing a partition.</fsummary> <desc> - <p>Returns <seealso marker="#family">family</seealso> + <p>Returns <seeerl marker="#family">family</seeerl> <c><anno>Family</anno></c> where the indexed set is - a <seealso marker="#partition">partition</seealso> + a <seeerl marker="#partition">partition</seeerl> of <c><anno>Set</anno></c> such that two elements are considered equal if the results of applying <c><anno>SetFun</anno></c> are the same value i. This i is the index that <c><anno>Family</anno></c> - maps onto the <seealso marker="#equivalence_class">equivalence - class</seealso>.</p> + maps onto the <seeerl marker="#equivalence_class">equivalence + class</seeerl>.</p> <pre> 1> <input>S = sofs:relation([{a,a,a,a},{a,a,b,b},{a,b,b,b}]),</input> <input>SetFun = {external, fun({A,_,C,_}) -> {A,C} end},</input> @@ -1331,8 +1331,8 @@ true</pre> <name name="product" arity="1" since=""/> <fsummary>Return the Cartesian product of a tuple of sets.</fsummary> <desc> - <p>Returns the <seealso marker="#Cartesian_product_tuple">Cartesian - product</seealso> of the non-empty tuple of sets + <p>Returns the <seeerl marker="#Cartesian_product_tuple">Cartesian + product</seeerl> of the non-empty tuple of sets <c><anno>TupleOfSets</anno></c>. If (x[1], ..., x[n]) is an element of the n-ary relation <c><anno>Relation</anno></c>, then x[i] is drawn from element i of <c><anno>TupleOfSets</anno></c>.</p> @@ -1350,8 +1350,8 @@ true</pre> <name name="product" arity="2" since=""/> <fsummary>Return the Cartesian product of two sets.</fsummary> <desc> - <p>Returns the <seealso marker="#Cartesian_product">Cartesian - product</seealso> of <c><anno>Set1</anno></c> + <p>Returns the <seeerl marker="#Cartesian_product">Cartesian + product</seeerl> of <c><anno>Set1</anno></c> and <c><anno>Set2</anno></c>.</p> <pre> 1> <input>S1 = sofs:set([1,2]),</input> @@ -1373,7 +1373,7 @@ true</pre> applying <c><anno>SetFun</anno></c> to the element.</p> <p>If <c><anno>SetFun</anno></c> is a number i >= 1 and <c><anno>Set1</anno></c> is a relation, then the returned set is - the <seealso marker="#projection">projection</seealso> of + the <seeerl marker="#projection">projection</seeerl> of <c><anno>Set1</anno></c> onto coordinate i.</p> <pre> 1> <input>S1 = sofs:from_term([{1,a},{2,b},{3,a}]),</input> @@ -1387,7 +1387,7 @@ true</pre> <name name="range" arity="1" since=""/> <fsummary>Return the range of a binary relation.</fsummary> <desc> - <p>Returns the <seealso marker="#range">range</seealso> of the + <p>Returns the <seeerl marker="#range">range</seeerl> of the binary relation <c><anno>BinRel</anno></c>.</p> <pre> 1> <input>R = sofs:relation([{1,a},{1,b},{2,b},{2,c}]),</input> @@ -1402,10 +1402,10 @@ true</pre> <name name="relation" arity="2" since=""/> <fsummary>Create a relation.</fsummary> <desc> - <p>Creates a <seealso marker="#relation">relation</seealso>. + <p>Creates a <seeerl marker="#relation">relation</seeerl>. <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 + a <seeerl marker="#type">type</seeerl> and the result is a relation. If <c><anno>Type</anno></c> is an integer N, then <c>[{atom, ..., atom}])</c>, where the tuple size is N, is used as type of the relation. If no type is @@ -1420,11 +1420,11 @@ true</pre> <name name="relation_to_family" arity="1" since=""/> <fsummary>Create a family from a binary relation.</fsummary> <desc> - <p>Returns <seealso marker="#family">family</seealso> + <p>Returns <seeerl marker="#family">family</seeerl> <c><anno>Family</anno></c> such that the index set is equal to - the <seealso marker="#domain">domain</seealso> of the binary + the <seeerl marker="#domain">domain</seeerl> of the binary relation <c><anno>BinRel</anno></c>, and <c><anno>Family</anno></c>[i] - is the <seealso marker="#image">image</seealso> of the set of i + is the <seeerl marker="#image">image</seeerl> of the set of i under <c><anno>BinRel</anno></c>.</p> <pre> 1> <input>R = sofs:relation([{b,1},{c,2},{c,3}]),</input> @@ -1444,13 +1444,13 @@ true</pre> [R[1], ..., R[n]] of binary relations and <c><anno>BinRel1</anno></c> is a binary relation, then <c><anno>BinRel2</anno></c> is the - <seealso marker="#tuple_relative_product">relative product</seealso> + <seeerl marker="#tuple_relative_product">relative product</seeerl> of the ordered set (R[i], ..., R[n]) and <c><anno>BinRel1</anno></c>.</p> <p>If <c><anno>BinRel1</anno></c> 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], + the <seeerl marker="#Cartesian_product_tuple">Cartesian + product</seeerl> of the ranges of R[i], range R[1] × ... × range R[n], is used instead (intuitively, nothing is "lost").</p> <pre> @@ -1470,8 +1470,8 @@ true</pre> <fsummary>Return the relative product of two binary relations.</fsummary> <desc> - <p>Returns the <seealso marker="#relative_product">relative - product</seealso> of the binary relations <c><anno>BinRel1</anno></c> + <p>Returns the <seeerl marker="#relative_product">relative + product</seeerl> of the binary relations <c><anno>BinRel1</anno></c> and <c><anno>BinRel2</anno></c>.</p> </desc> </func> @@ -1481,9 +1481,9 @@ true</pre> <fsummary>Return the relative_product of two binary relations.</fsummary> <desc> - <p>Returns the <seealso marker="#relative_product">relative - product</seealso> of - the <seealso marker="#converse">converse</seealso> of the + <p>Returns the <seeerl marker="#relative_product">relative + product</seeerl> of + the <seeerl marker="#converse">converse</seeerl> of the binary relation <c><anno>BinRel1</anno></c> and the binary relation <c><anno>BinRel2</anno></c>.</p> <pre> @@ -1501,7 +1501,7 @@ true</pre> <name name="restriction" arity="2" since=""/> <fsummary>Return a restriction of a binary relation.</fsummary> <desc> - <p>Returns the <seealso marker="#restriction">restriction</seealso> of + <p>Returns the <seeerl marker="#restriction">restriction</seeerl> of the binary relation <c><anno>BinRel1</anno></c> to <c><anno>Set</anno></c>.</p> <pre> @@ -1534,10 +1534,10 @@ true</pre> <name name="set" arity="2" since=""/> <fsummary>Create a set of atoms or any type of sets.</fsummary> <desc> - <p>Creates an <seealso marker="#sets_definition">unordered - set</seealso>. <c>set(L, T)</c> is equivalent to + <p>Creates an <seeerl marker="#sets_definition">unordered + set</seeerl>. <c>set(L, T)</c> is equivalent to <c>from_term(L, T)</c>, if the result is an unordered - set. If no <seealso marker="#type">type</seealso> is + set. If no <seeerl marker="#type">type</seeerl> is explicitly specified, <c>[atom]</c> is used as the set type.</p> </desc> </func> @@ -1550,7 +1550,7 @@ true</pre> of <c><anno>Set1</anno></c> for which <c><anno>Fun</anno></c> returns <c>true</c>. If <c><anno>Fun</anno></c> is a tuple <c>{external, Fun2}</c>, <c>Fun2</c> is applied to the - <seealso marker="#external_set">external set</seealso> of + <seeerl marker="#external_set">external set</seeerl> of each element, otherwise <c><anno>Fun</anno></c> is applied to each element.</p> <pre> @@ -1568,8 +1568,8 @@ true</pre> <fsummary>Return the strict relation corresponding to a given relation.</fsummary> <desc> - <p>Returns the <seealso marker="#strict_relation">strict - relation</seealso> corresponding to the binary + <p>Returns the <seeerl marker="#strict_relation">strict + relation</seeerl> corresponding to the binary relation <c><anno>BinRel1</anno></c>.</p> <pre> 1> <input>R1 = sofs:relation([{1,1},{1,2},{2,1},{2,2}]),</input> @@ -1604,7 +1604,7 @@ true</pre> [{a,a},{b,b},{c,c}]</pre> <p>Let <c>SetOfSets</c> be a set of sets and <c>BinRel</c> a binary relation. The function that maps each element <c>Set</c> of - <c>SetOfSets</c> onto the <seealso marker="#image">image</seealso> + <c>SetOfSets</c> onto the <seeerl marker="#image">image</seeerl> of <c>Set</c> under <c>BinRel</c> is returned by the following function:</p> <pre> @@ -1614,7 +1614,7 @@ images(SetOfSets, BinRel) -> <p>External unordered sets are represented as sorted lists. So, creating the image of a set under a relation R can traverse all elements of R (to that comes the sorting of results, the - image). In <seealso marker="#image/2"><c>image/2</c></seealso>, + image). In <seemfa marker="#image/2"><c>image/2</c></seemfa>, <c>BinRel</c> is traversed once for each element of <c>SetOfSets</c>, which can take too long. The following efficient function can be used instead under the @@ -1632,8 +1632,8 @@ images2(SetOfSets, BinRel) -> <name name="symdiff" arity="2" since=""/> <fsummary>Return the symmetric difference of two sets.</fsummary> <desc> - <p>Returns the <seealso marker="#symmetric_difference">symmetric - difference</seealso> (or the Boolean sum) + <p>Returns the <seeerl marker="#symmetric_difference">symmetric + difference</seeerl> (or the Boolean sum) of <c><anno>Set1</anno></c> and <c><anno>Set2</anno></c>.</p> <pre> 1> <input>S1 = sofs:set([1,2,3]),</input> @@ -1669,8 +1669,8 @@ images2(SetOfSets, BinRel) -> <name name="to_external" arity="1" since=""/> <fsummary>Return the elements of a set.</fsummary> <desc> - <p>Returns the <seealso marker="#external_set">external - set</seealso> of an atomic, ordered, or unordered set.</p> + <p>Returns the <seeerl marker="#external_set">external + set</seeerl> of an atomic, ordered, or unordered set.</p> </desc> </func> @@ -1689,7 +1689,7 @@ images2(SetOfSets, BinRel) -> <name name="type" arity="1" since=""/> <fsummary>Return the type of a set.</fsummary> <desc> - <p>Returns the <seealso marker="#type">type</seealso> of an + <p>Returns the <seeerl marker="#type">type</seeerl> of an atomic, ordered, or unordered set.</p> </desc> </func> @@ -1698,7 +1698,7 @@ images2(SetOfSets, BinRel) -> <name name="union" arity="1" since=""/> <fsummary>Return the union of a set of sets.</fsummary> <desc> - <p>Returns the <seealso marker="#union_n">union</seealso> of the + <p>Returns the <seeerl marker="#union_n">union</seeerl> of the set of sets <c><anno>SetOfSets</anno></c>.</p> </desc> </func> @@ -1707,7 +1707,7 @@ images2(SetOfSets, BinRel) -> <name name="union" arity="2" since=""/> <fsummary>Return the union of two sets.</fsummary> <desc> - <p>Returns the <seealso marker="#union">union</seealso> of + <p>Returns the <seeerl marker="#union">union</seeerl> of <c><anno>Set1</anno></c> and <c><anno>Set2</anno></c>.</p> </desc> </func> @@ -1716,7 +1716,7 @@ images2(SetOfSets, BinRel) -> <name name="union_of_family" arity="1" since=""/> <fsummary>Return the union of a family.</fsummary> <desc> - <p>Returns the union of <seealso marker="#family">family</seealso> + <p>Returns the union of <seeerl marker="#family">family</seeerl> <c><anno>Family</anno></c>.</p> <pre> 1> <input>F = sofs:family([{a,[0,2,4]},{b,[0,1,2]},{c,[2,3]}]),</input> @@ -1731,10 +1731,10 @@ images2(SetOfSets, BinRel) -> <fsummary>Return the weak relation corresponding to a given relation.</fsummary> <desc> - <p>Returns a subset S of the <seealso marker="#weak_relation">weak - relation</seealso> W + <p>Returns a subset S of the <seeerl marker="#weak_relation">weak + relation</seeerl> W corresponding to the binary relation <c><anno>BinRel1</anno></c>. - Let F be the <seealso marker="#field">field</seealso> of + Let F be the <seeerl marker="#field">field</seeerl> of <c><anno>BinRel1</anno></c>. 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> @@ -1749,11 +1749,11 @@ images2(SetOfSets, BinRel) -> <section> <title>See Also</title> - <p><seealso marker="dict"><c>dict(3)</c></seealso>, - <seealso marker="digraph"><c>digraph(3)</c></seealso>, - <seealso marker="orddict"><c>orddict(3)</c></seealso>, - <seealso marker="ordsets"><c>ordsets(3)</c></seealso>, - <seealso marker="sets"><c>sets(3)</c></seealso></p> + <p><seeerl marker="dict"><c>dict(3)</c></seeerl>, + <seeerl marker="digraph"><c>digraph(3)</c></seeerl>, + <seeerl marker="orddict"><c>orddict(3)</c></seeerl>, + <seeerl marker="ordsets"><c>ordsets(3)</c></seeerl>, + <seeerl marker="sets"><c>sets(3)</c></seeerl></p> </section> </erlref> |