added functor Inverse + global function to AF (with respect to multiplication)

.gitattributes

@@ -68,7 +68,9 @@ Algebraic_foundations/doc_tex/Algebraic_foundations/interoperability.tex -text
 Algebraic_foundations/doc_tex/Algebraic_foundations/real_embeddable.tex -text
 Algebraic_foundations/doc_tex/Algebraic_foundations/real_number_types.tex -text
 Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_Divides.tex -text
+Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_Inverse.tex -text
 Algebraic_foundations/doc_tex/Algebraic_foundations_ref/Fraction.tex -text
+Algebraic_foundations/doc_tex/Algebraic_foundations_ref/inverse.tex -text
 Algebraic_foundations/examples/Algebraic_foundations/algebraic_structure_dispatch.cpp -text
 Algebraic_foundations/examples/Algebraic_foundations/fraction_traits.cpp -text
 Algebraic_foundations/examples/Algebraic_foundations/implicit_interoperable_dispatch.cpp -text

Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits.tex

@@ -132,6 +132,11 @@ A model of \ccc{AlgebraicStructureTraits::DivMod}.\\
 Required by the concept \ccc{EuclideanRing}.
 }
 
+\ccNestedType{Inverse}{ 
+A model of \ccc{AlgebraicStructureTraits::Inverse}.\\
+Required by the concept \ccc{Field}.
+}
+
 \ccNestedType{Sqrt}{ 
 A model of \ccc{AlgebraicStructureTraits::Sqrt}.\\
 Required by the concept \ccc{FieldWithSqrt}.

Algebraic_foundations/doc_tex/Algebraic_foundations_ref/AlgebraicStructureTraits_Inverse.tex

@@ -0,0 +1,32 @@
+\begin{ccRefFunctionObjectConcept}{AlgebraicStructureTraits::Inverse}
+
+\ccDefinition
+
+\ccc{AdaptableUnaryFunction} providing the inverse element with 
+respect to multiplication of a \ccc{Field}. 
+
+\ccRefines 
+
+\ccc{AdaptableUnaryFunction} 
+
+\ccTypes
+\ccNestedType{result_type} 
+        { Is \ccc{AlgebraicStructureTraits::Type}.}
+\ccNestedType{argument_type}
+        { Is \ccc{AlgebraicStructureTraits::Type}.}
+
+\ccOperations
+\ccThree{xxxxxxxxxxx}{xxxxxxxxxxx}{}
+\ccCreationVariable{inverse}
+\ccMethod{result_type operator()(argument_type  x) const;}
+        { returns  the inverse element of $x$ with respect to multiplication. 
+          \ccPrecond $x \neq 0$
+        }
+
+%\ccHasModels
+
+\ccSeeAlso
+
+\ccRefIdfierPage{AlgebraicStructureTraits}
+
+\end{ccRefFunctionObjectConcept} 

Algebraic_foundations/doc_tex/Algebraic_foundations_ref/Field.tex

@@ -12,8 +12,9 @@ operators / and /=.
 
 Moreover, \ccc{CGAL::Algebraic_structure_traits< Field >} is a model of 
 \ccc{AlgebraicStructureTraits} providing:\\
-- \ccc{CGAL::Algebraic_structure_traits< Field >::Algebraic_type} derived 
-from \ccc{Field_tag} \\
+- \ccc{CGAL::Algebraic_structure_traits< Field >::Algebraic_type} derived from \ccc{Field_tag} \\
+- \ccc{CGAL::Algebraic_structure_traits< FieldWithSqrt >::Inverse}\\
+
   
 \ccRefines
  \ccc{IntegralDomain}

Algebraic_foundations/doc_tex/Algebraic_foundations_ref/intro.tex

@@ -34,6 +34,7 @@
 \ccRefConceptPage{AlgebraicStructureTraits::DivMod}\\
 \ccRefConceptPage{AlgebraicStructureTraits::Div}\\
 \ccRefConceptPage{AlgebraicStructureTraits::Mod}\\
+\ccRefConceptPage{AlgebraicStructureTraits::Inverse}\\
 \ccRefConceptPage{AlgebraicStructureTraits::Sqrt}\\
 \ccRefConceptPage{AlgebraicStructureTraits::IsSquare}\\
 \ccRefConceptPage{AlgebraicStructureTraits::KthRoot}\\
@@ -61,6 +62,7 @@
 \ccRefIdfierPage{CGAL::div_mod}\\
 \ccRefIdfierPage{CGAL::div}\\
 \ccRefIdfierPage{CGAL::mod}\\
+\ccRefIdfierPage{CGAL::inverse}\\
 \ccRefIdfierPage{CGAL::sqrt}\\
 \ccRefIdfierPage{CGAL::kth_root}\\
 \ccRefIdfierPage{CGAL::root_of}\\

Algebraic_foundations/doc_tex/Algebraic_foundations_ref/inverse.tex

@@ -0,0 +1,22 @@
+\begin{ccRefFunction}{inverse}
+
+\ccDefinition
+
+The function \ccRefName\ returns the inverse element with respect to multiplication.
+
+The function is guaranteed to be well defined in case the argument type 
+is a model of the \ccc{Field} concept. 
+
+\ccInclude{CGAL/number_utils.h}
+
+\ccFunction{template <class NT> NT inverse(const NT& x);}
+{\ccPrecond $x \neq 0$}
+
+
+
+\ccSeeAlso
+
+\ccRefConceptPage{Field}\\
+\ccRefConceptPage{AlgebraicStructureTraits::Inverse}\\
+
+\end{ccRefFunction}

Algebraic_foundations/doc_tex/Algebraic_foundations_ref/main.tex

@@ -35,6 +35,7 @@
 \input{Algebraic_foundations_ref/AlgebraicStructureTraits_DivMod.tex}
 \input{Algebraic_foundations_ref/AlgebraicStructureTraits_Div.tex}
 \input{Algebraic_foundations_ref/AlgebraicStructureTraits_Mod.tex}
+\input{Algebraic_foundations_ref/AlgebraicStructureTraits_Inverse.tex}
 \input{Algebraic_foundations_ref/AlgebraicStructureTraits_Sqrt.tex}
 \input{Algebraic_foundations_ref/AlgebraicStructureTraits_KthRoot.tex}
 \input{Algebraic_foundations_ref/AlgebraicStructureTraits_RootOf.tex}
@@ -63,6 +64,7 @@
 \input{Algebraic_foundations_ref/div_mod.tex}
 \input{Algebraic_foundations_ref/div.tex}
 \input{Algebraic_foundations_ref/mod.tex}
+\input{Algebraic_foundations_ref/inverse.tex}
 \input{Algebraic_foundations_ref/sqrt.tex}
 \input{Algebraic_foundations_ref/kth_root.tex}
 \input{Algebraic_foundations_ref/root_of.tex}

Algebraic_foundations/include/CGAL/Algebraic_structure_traits.h

@@ -85,6 +85,7 @@ class Algebraic_structure_traits  {
     typedef Null_functor Kth_root;
     typedef Null_functor Root_of; 
     typedef Null_functor Divides; 
+    typedef Null_functor Inverse; 
 };
 
 // The algebraic structure traits base class
@@ -462,6 +463,17 @@ class Algebraic_structure_traits_base< Type_, Field_tag >
         }
       CGAL_IMPLICIT_INTEROPERABLE_BINARY_OPERATOR( Type )
     };
+  
+  // maps to \c 1/x by default.
+  class Inverse 
+    : public std::unary_function< Type, Type > { 
+  public:
+    Type operator()( const Type& x ) const { 
+      static const Type& one = Type(1);
+      return one/x;
+    }
+  };
+  
 
   // Default implementation of Divides functor for Field: 
   // returns always true

Algebraic_foundations/include/CGAL/Test/_test_algebraic_structure.h

@@ -358,6 +358,11 @@ void test_algebraic_structure_intern( const CGAL::Field_tag& ) {
     assert(!ie || tmp == AS(5)/AS(2));
     assert(CGAL_NTS divides(AS(5),AS(2),tmp));
     assert(!ie || tmp == AS(2)/AS(5));
+
+    typename AST::Inverse inverse;
+    assert(AS(1)/AS(2) == inverse(AS(2)));
+    assert(AS(1)/AS(2) == CGAL::inverse(AS(2)));
+    
     
 }
 

Algebraic_foundations/include/CGAL/number_utils.h

@@ -74,6 +74,16 @@ square( const AS& x ) {
     typename Algebraic_structure_traits< AS >::Square square;
     return square( x );
 }
+
+
+template< class AS >
+inline
+typename Algebraic_structure_traits< AS >::Inverse::result_type
+inverse( const AS& x ) {
+    typename Algebraic_structure_traits< AS >::Inverse inverse;
+    return inverse( x );
+}
+
 template< class AS >
 inline 
 typename Algebraic_structure_traits<AS>::Is_one::result_type