diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Cartesian_base.h b/Cartesian_kernel/include/CGAL/Cartesian/Cartesian_base.h
index 69891b49bb6..c8a9a70b70e 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Cartesian_base.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Cartesian_base.h
@@ -81,11 +81,11 @@ struct Cartesian_base
     typedef CGAL::Object                                Object_2;
     typedef CGAL::Object                                Object_3;
 
-    // Bool_type had originally been Bool. It was renamed to avoid a conflict
+    // Boolean   had originally been Bool. It was renamed to avoid a conflict
     // between a macro defined in Xlib.h poorly chosen to have the same name,
     // that is 'Bool'.
     typedef typename Same_uncertainty_nt<bool, FT>::type
-                                                        Bool_type;
+                                                        Boolean;
     typedef typename Same_uncertainty_nt<CGAL::Sign, FT>::type
                                                         Sign;
     typedef typename Same_uncertainty_nt<CGAL::Comparison_result, FT>::type
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h b/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h
index 44b6047e42c..ee06cb765d7 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Direction_3.h
@@ -66,8 +66,8 @@ public:
   DirectionC3(const FT &x, const FT &y, const FT &z)
     : base(CGAL::make_array(x, y, z)) {}
 
-  typename R::Bool_type operator==(const DirectionC3 &d) const;
-  typename R::Bool_type operator!=(const DirectionC3 &d) const;
+  typename R::Boolean   operator==(const DirectionC3 &d) const;
+  typename R::Boolean   operator!=(const DirectionC3 &d) const;
 
   Vector_3       to_vector() const;
   Vector_3       vector() const { return to_vector(); }
@@ -105,7 +105,7 @@ public:
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 DirectionC3<R>::operator==(const DirectionC3<R> &d) const
 {
   if (CGAL::identical(base, d.base))
@@ -115,7 +115,7 @@ DirectionC3<R>::operator==(const DirectionC3<R> &d) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 DirectionC3<R>::operator!=(const DirectionC3<R> &d) const
 {
   return !(*this == d);
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Iso_cuboid_3.h b/Cartesian_kernel/include/CGAL/Cartesian/Iso_cuboid_3.h
index 898159fbb1c..f0a217cadf5 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Iso_cuboid_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Iso_cuboid_3.h
@@ -105,8 +105,8 @@ public:
                                     Construct_point_3()(max_hx/hw, max_hy/hw, max_hz/hw)));
   }
 
-  typename R::Bool_type operator==(const Iso_cuboidC3& s) const;
-  typename R::Bool_type operator!=(const Iso_cuboidC3& s) const;
+  typename R::Boolean   operator==(const Iso_cuboidC3& s) const;
+  typename R::Boolean   operator!=(const Iso_cuboidC3& s) const;
 
   const Point_3 & min BOOST_PREVENT_MACRO_SUBSTITUTION () const
   {
@@ -125,11 +125,11 @@ public:
   }
 
   Bounded_side bounded_side(const Point_3& p) const;
-  typename R::Bool_type         has_on(const Point_3& p) const;
-  typename R::Bool_type         has_on_boundary(const Point_3& p) const;
-  typename R::Bool_type         has_on_bounded_side(const Point_3& p) const;
-  typename R::Bool_type         has_on_unbounded_side(const Point_3& p) const;
-  typename R::Bool_type         is_degenerate() const;
+  typename R::Boolean           has_on(const Point_3& p) const;
+  typename R::Boolean           has_on_boundary(const Point_3& p) const;
+  typename R::Boolean           has_on_bounded_side(const Point_3& p) const;
+  typename R::Boolean           has_on_unbounded_side(const Point_3& p) const;
+  typename R::Boolean           is_degenerate() const;
   const FT &   xmin() const;
   const FT &   ymin() const;
   const FT &   zmin() const;
@@ -144,7 +144,7 @@ public:
 
 template < class R >
 CGAL_KERNEL_INLINE
-typename R::Bool_type
+typename R::Boolean
 Iso_cuboidC3<R>::operator==(const Iso_cuboidC3<R>& r) const
 { // FIXME : predicate
   if (CGAL::identical(base, r.base))
@@ -154,7 +154,7 @@ Iso_cuboidC3<R>::operator==(const Iso_cuboidC3<R>& r) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 Iso_cuboidC3<R>::operator!=(const Iso_cuboidC3<R>& r) const
 {
   return !(*this == r);
@@ -287,7 +287,7 @@ bounded_side(const typename Iso_cuboidC3<R>::Point_3& p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 Iso_cuboidC3<R>::
 has_on_boundary(const typename Iso_cuboidC3<R>::Point_3& p) const
 {
@@ -296,7 +296,7 @@ has_on_boundary(const typename Iso_cuboidC3<R>::Point_3& p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 Iso_cuboidC3<R>::
 has_on(const typename Iso_cuboidC3<R>::Point_3& p) const
 {
@@ -305,7 +305,7 @@ has_on(const typename Iso_cuboidC3<R>::Point_3& p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 Iso_cuboidC3<R>::
 has_on_bounded_side(const typename Iso_cuboidC3<R>::Point_3& p) const
 {
@@ -314,7 +314,7 @@ has_on_bounded_side(const typename Iso_cuboidC3<R>::Point_3& p) const
 
 template < class R >
 CGAL_KERNEL_INLINE
-typename R::Bool_type
+typename R::Boolean
 Iso_cuboidC3<R>::
 has_on_unbounded_side(const typename Iso_cuboidC3<R>::Point_3& p)
     const
@@ -324,7 +324,7 @@ has_on_unbounded_side(const typename Iso_cuboidC3<R>::Point_3& p)
 
 template < class R >
 CGAL_KERNEL_INLINE
-typename R::Bool_type
+typename R::Boolean
 Iso_cuboidC3<R>::is_degenerate() const
 { // FIXME : predicate
   return (this->min)().hx() == (this->max)().hx()
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Line_2.h b/Cartesian_kernel/include/CGAL/Cartesian/Line_2.h
index 03737e04818..9c8448197ca 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Line_2.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Line_2.h
@@ -55,8 +55,8 @@ public:
   LineC2(const FT &a, const FT &b, const FT &c)
     : base(CGAL::make_array(a, b, c)) {}
   
-  typename R_::Bool_type operator==(const LineC2 &l) const;
-  typename R_::Bool_type operator!=(const LineC2 &l) const;
+  typename R_::Boolean   operator==(const LineC2 &l) const;
+  typename R_::Boolean   operator!=(const LineC2 &l) const;
 
   const FT & a() const
   {
@@ -75,7 +75,7 @@ public:
 
 template < class R >
 CGAL_KERNEL_INLINE
-typename R::Bool_type
+typename R::Boolean
 LineC2<R>::operator==(const LineC2<R> &l) const
 {
   if (CGAL::identical(base, l.base))
@@ -85,7 +85,7 @@ LineC2<R>::operator==(const LineC2<R> &l) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 LineC2<R>::operator!=(const LineC2<R> &l) const
 {
   return ! (*this == l);
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Plane_3.h b/Cartesian_kernel/include/CGAL/Cartesian/Plane_3.h
index 3daf7aa30eb..b398d28dd22 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Plane_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Plane_3.h
@@ -79,8 +79,8 @@ public:
   PlaneC3(const Ray_3 &r, const Point_3 &p)
   { *this = plane_from_points(r.start(), r.second_point(), p); }
 
-  typename R::Bool_type operator==(const PlaneC3 &p) const;
-  typename R::Bool_type operator!=(const PlaneC3 &p) const;
+  typename R::Boolean   operator==(const PlaneC3 &p) const;
+  typename R::Boolean   operator!=(const PlaneC3 &p) const;
 
   const FT & a() const
   {
@@ -115,18 +115,18 @@ public:
   Point_3      to_3d(const Point_2 &p) const;
 
   typename R::Oriented_side     oriented_side(const Point_3 &p) const;
-  typename R::Bool_type         has_on_positive_side(const Point_3 &l) const;
-  typename R::Bool_type         has_on_negative_side(const Point_3 &l) const;
-  typename R::Bool_type         has_on(const Point_3 &p) const
+  typename R::Boolean           has_on_positive_side(const Point_3 &l) const;
+  typename R::Boolean           has_on_negative_side(const Point_3 &l) const;
+  typename R::Boolean           has_on(const Point_3 &p) const
   {
     return oriented_side(p) == ON_ORIENTED_BOUNDARY;
   }
-  typename R::Bool_type         has_on(const Line_3 &l) const
+  typename R::Boolean           has_on(const Line_3 &l) const
   {
     return has_on(l.point())
        &&  has_on(l.point() + l.direction().to_vector());
   }
-  typename R::Bool_type         has_on(const Circle_3 &circle) const
+  typename R::Boolean           has_on(const Circle_3 &circle) const
   {
     if(circle.squared_radius() != FT(0)) {
       const Plane_3& p = circle.supporting_plane();
@@ -145,12 +145,12 @@ public:
     } else return has_on(circle.center());
   }
 
-  typename R::Bool_type         is_degenerate() const;
+  typename R::Boolean           is_degenerate() const;
 };
 
 template < class R >
 CGAL_KERNEL_INLINE
-typename R::Bool_type
+typename R::Boolean
 PlaneC3<R>::operator==(const PlaneC3<R> &p) const
 {
   if (CGAL::identical(base, p.base))
@@ -160,7 +160,7 @@ PlaneC3<R>::operator==(const PlaneC3<R> &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 PlaneC3<R>::operator!=(const PlaneC3<R> &p) const
 {
   return !(*this == p);
@@ -277,7 +277,7 @@ oriented_side(const typename PlaneC3<R>::Point_3 &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 PlaneC3<R>::
 has_on_positive_side(const  typename PlaneC3<R>::Point_3 &p) const
 {
@@ -286,7 +286,7 @@ has_on_positive_side(const  typename PlaneC3<R>::Point_3 &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 PlaneC3<R>::
 has_on_negative_side(const  typename PlaneC3<R>::Point_3 &p) const
 {
@@ -295,7 +295,7 @@ has_on_negative_side(const  typename PlaneC3<R>::Point_3 &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 PlaneC3<R>::
 is_degenerate() const
 { // FIXME : predicate
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Point_2.h b/Cartesian_kernel/include/CGAL/Cartesian/Point_2.h
index db6ffa87c04..bd1a508de64 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Point_2.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Point_2.h
@@ -89,11 +89,11 @@ public:
     return base.cartesian_end(); 
   }
 
-  typename R_::Bool_type operator==(const PointC2 &p) const
+  typename R_::Boolean   operator==(const PointC2 &p) const
   {
       return base == p.base;
   }
-  typename R_::Bool_type operator!=(const PointC2 &p) const
+  typename R_::Boolean   operator!=(const PointC2 &p) const
   {
       return !(*this == p);
   }
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Ray_2.h b/Cartesian_kernel/include/CGAL/Cartesian/Ray_2.h
index 511189c0813..684326f08ca 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Ray_2.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Ray_2.h
@@ -63,7 +63,7 @@ public:
     return get(base)[1];
   }
 
-  typename R_::Bool_type is_degenerate() const
+  typename R_::Boolean   is_degenerate() const
   {
     return source() == second_point();
   }
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Ray_3.h b/Cartesian_kernel/include/CGAL/Cartesian/Ray_3.h
index 47c73e06ab0..34f2bed25d2 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Ray_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Ray_3.h
@@ -61,8 +61,8 @@ public:
   RayC3(const Point_3 &sp, const Line_3 &l)
     : base(CGAL::make_array(sp, sp + l.to_vector())) {}
 
-  typename R::Bool_type        operator==(const RayC3 &r) const;
-  typename R::Bool_type        operator!=(const RayC3 &r) const;
+  typename R::Boolean          operator==(const RayC3 &r) const;
+  typename R::Boolean          operator!=(const RayC3 &r) const;
 
   const Point_3 &   source() const
   {
@@ -79,14 +79,14 @@ public:
   Line_3      supporting_line() const;
   Ray_3       opposite() const;
 
-  typename R::Bool_type        is_degenerate() const;
-  typename R::Bool_type        has_on(const Point_3 &p) const;
-  typename R::Bool_type        collinear_has_on(const Point_3 &p) const;
+  typename R::Boolean          is_degenerate() const;
+  typename R::Boolean          has_on(const Point_3 &p) const;
+  typename R::Boolean          collinear_has_on(const Point_3 &p) const;
 };
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 RayC3<R>::operator==(const RayC3<R> &r) const
 {
     if (CGAL::identical(base, r.base))
@@ -96,7 +96,7 @@ RayC3<R>::operator==(const RayC3<R> &r) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 RayC3<R>::operator!=(const RayC3<R> &r) const
 {
   return !(*this == r);
@@ -146,7 +146,7 @@ RayC3<R>::opposite() const
 }
 
 template < class R >
-typename R::Bool_type
+typename R::Boolean
 RayC3<R>::
 has_on(const typename RayC3<R>::Point_3 &p) const
 {
@@ -157,7 +157,7 @@ has_on(const typename RayC3<R>::Point_3 &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 RayC3<R>::is_degenerate() const
 {
   return source() == second_point();
@@ -165,7 +165,7 @@ RayC3<R>::is_degenerate() const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 RayC3<R>::
 collinear_has_on(const typename RayC3<R>::Point_3 &p) const
 {
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h b/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h
index 3bdabf9f495..c730e276c4d 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Sphere_3.h
@@ -101,8 +101,8 @@ public:
     base = Rep(center, FT(0), o);
   }
 
-  typename R::Bool_type operator==(const SphereC3 &) const;
-  typename R::Bool_type operator!=(const SphereC3 &) const;
+  typename R::Boolean   operator==(const SphereC3 &) const;
+  typename R::Boolean   operator!=(const SphereC3 &) const;
 
   const Point_3 & center() const
   {
@@ -120,7 +120,7 @@ public:
   }
 
   // A circle is degenerate if its (squared) radius is null or negative
-  typename R::Bool_type is_degenerate() const;
+  typename R::Boolean   is_degenerate() const;
 
   // Returns a circle with opposite orientation
   Sphere_3 opposite() const;
@@ -129,21 +129,21 @@ public:
   //! precond: ! x.is_degenerate() (when available)
   // Returns R::ON_POSITIVE_SIDE, R::ON_ORIENTED_BOUNDARY or
   // R::ON_NEGATIVE_SIDE
-  typename R::Bool_type has_on(const Circle_3 &p) const;
-  typename R::Bool_type has_on_boundary(const Point_3 &p) const;
-  typename R::Bool_type has_on_positive_side(const Point_3 &p) const;
-  typename R::Bool_type has_on_negative_side(const Point_3 &p) const;
+  typename R::Boolean   has_on(const Circle_3 &p) const;
+  typename R::Boolean   has_on_boundary(const Point_3 &p) const;
+  typename R::Boolean   has_on_positive_side(const Point_3 &p) const;
+  typename R::Boolean   has_on_negative_side(const Point_3 &p) const;
 
   typename R_::Bounded_side bounded_side(const Point_3 &p) const;
   //! precond: ! x.is_degenerate() (when available)
   // Returns R::ON_BOUNDED_SIDE, R::ON_BOUNDARY or R::ON_UNBOUNDED_SIDE
-  typename R::Bool_type has_on_bounded_side(const Point_3 &p) const;
-  typename R::Bool_type has_on_unbounded_side(const Point_3 &p) const;
+  typename R::Boolean   has_on_bounded_side(const Point_3 &p) const;
+  typename R::Boolean   has_on_unbounded_side(const Point_3 &p) const;
 };
 
 template < class R >
 CGAL_KERNEL_INLINE
-typename R::Bool_type
+typename R::Boolean
 SphereC3<R>::operator==(const SphereC3<R> &t) const
 {
   if (CGAL::identical(base, t.base))
@@ -155,7 +155,7 @@ SphereC3<R>::operator==(const SphereC3<R> &t) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 SphereC3<R>::operator!=(const SphereC3<R> &t) const
 {
   return !(*this == t);
@@ -182,7 +182,7 @@ bounded_side(const typename SphereC3<R>::Point_3 &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 SphereC3<R>::
 has_on(const typename SphereC3<R>::Circle_3 &c) const
 {
@@ -198,7 +198,7 @@ has_on(const typename SphereC3<R>::Circle_3 &c) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 SphereC3<R>::
 has_on_boundary(const typename SphereC3<R>::Point_3 &p) const
 {
@@ -211,7 +211,7 @@ has_on_boundary(const typename SphereC3<R>::Point_3 &p) const
 
 template < class R >
 CGAL_KERNEL_INLINE
-typename R::Bool_type
+typename R::Boolean
 SphereC3<R>::
 has_on_negative_side(const typename SphereC3<R>::Point_3 &p) const
 {
@@ -224,7 +224,7 @@ has_on_negative_side(const typename SphereC3<R>::Point_3 &p) const
 
 template < class R >
 CGAL_KERNEL_INLINE
-typename R::Bool_type
+typename R::Boolean
 SphereC3<R>::
 has_on_positive_side(const typename SphereC3<R>::Point_3 &p) const
 {
@@ -237,7 +237,7 @@ has_on_positive_side(const typename SphereC3<R>::Point_3 &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 SphereC3<R>::
 has_on_bounded_side(const typename SphereC3<R>::Point_3 &p) const
 {
@@ -249,7 +249,7 @@ has_on_bounded_side(const typename SphereC3<R>::Point_3 &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 SphereC3<R>::
 has_on_unbounded_side(const typename SphereC3<R>::Point_3 &p) const
 {
@@ -261,7 +261,7 @@ has_on_unbounded_side(const typename SphereC3<R>::Point_3 &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 SphereC3<R>::
 is_degenerate() const
 {
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h b/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h
index c51310bcd81..561c34c62c4 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/Tetrahedron_3.h
@@ -56,24 +56,24 @@ public:
   const Point_3 &    vertex(int i) const;
   const Point_3 &    operator[](int i) const;
 
-  typename R::Bool_type       operator==(const TetrahedronC3 &t) const;
-  typename R::Bool_type       operator!=(const TetrahedronC3 &t) const;
+  typename R::Boolean         operator==(const TetrahedronC3 &t) const;
+  typename R::Boolean         operator!=(const TetrahedronC3 &t) const;
 
   typename R::Orientation    orientation() const;
   typename R::Oriented_side  oriented_side(const Point_3 &p) const;
   typename R::Bounded_side   bounded_side(const Point_3 &p) const;
 
-  typename R::Bool_type       has_on_boundary(const Point_3 &p) const;
-  typename R::Bool_type       has_on_positive_side(const Point_3 &p) const;
-  typename R::Bool_type       has_on_negative_side(const Point_3 &p) const;
-  typename R::Bool_type       has_on_bounded_side(const Point_3 &p) const;
-  typename R::Bool_type       has_on_unbounded_side(const Point_3 &p) const;
+  typename R::Boolean         has_on_boundary(const Point_3 &p) const;
+  typename R::Boolean         has_on_positive_side(const Point_3 &p) const;
+  typename R::Boolean         has_on_negative_side(const Point_3 &p) const;
+  typename R::Boolean         has_on_bounded_side(const Point_3 &p) const;
+  typename R::Boolean         has_on_unbounded_side(const Point_3 &p) const;
 
-  typename R::Bool_type       is_degenerate() const;
+  typename R::Boolean         is_degenerate() const;
 };
 
 template < class R >
-typename R::Bool_type
+typename R::Boolean
 TetrahedronC3<R>::
 operator==(const TetrahedronC3<R> &t) const
 {
@@ -101,7 +101,7 @@ operator==(const TetrahedronC3<R> &t) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 TetrahedronC3<R>::
 operator!=(const TetrahedronC3<R> &t) const
 {
@@ -166,7 +166,7 @@ bounded_side(const typename TetrahedronC3<R>::Point_3 &p) const
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 TetrahedronC3<R>::has_on_boundary
   (const typename TetrahedronC3<R>::Point_3 &p) const
 {
@@ -175,7 +175,7 @@ TetrahedronC3<R>::has_on_boundary
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 TetrahedronC3<R>::has_on_positive_side
   (const typename TetrahedronC3<R>::Point_3 &p) const
 {
@@ -184,7 +184,7 @@ TetrahedronC3<R>::has_on_positive_side
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 TetrahedronC3<R>::has_on_negative_side
   (const typename TetrahedronC3<R>::Point_3 &p) const
 {
@@ -193,7 +193,7 @@ TetrahedronC3<R>::has_on_negative_side
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 TetrahedronC3<R>::has_on_bounded_side
   (const typename TetrahedronC3<R>::Point_3 &p) const
 {
@@ -202,7 +202,7 @@ TetrahedronC3<R>::has_on_bounded_side
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 TetrahedronC3<R>::has_on_unbounded_side
   (const typename TetrahedronC3<R>::Point_3 &p) const
 {
@@ -211,7 +211,7 @@ TetrahedronC3<R>::has_on_unbounded_side
 
 template < class R >
 inline
-typename R::Bool_type
+typename R::Boolean
 TetrahedronC3<R>::is_degenerate() const
 {
   return orientation() == COPLANAR;
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/function_objects.h b/Cartesian_kernel/include/CGAL/Cartesian/function_objects.h
index d1bc3c9d1ab..7f2d7efcf01 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/function_objects.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/function_objects.h
@@ -73,7 +73,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Ray_2           Ray_2;
 
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()(const Line_2& l1, const Line_2& l2) const
@@ -105,7 +105,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Plane_3         Plane_3;
 
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()(const Line_3& l1, const Line_3& l2) const
@@ -251,7 +251,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_2         Point_2;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()(const Point_2& p, const Point_2& q, const Point_2& r) const
@@ -267,7 +267,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3         Point_3;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()(const Point_3& p, const Point_3& q, const Point_3& r) const
@@ -284,7 +284,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_2         Point_2;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()(const Point_2& p, const Point_2& q, const Point_2& r) const
@@ -300,7 +300,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3         Point_3;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()(const Point_3& p, const Point_3& q, const Point_3& r) const
@@ -319,7 +319,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Ray_2                 Ray_2;
     typedef typename K::Segment_2             Segment_2;
   public:
-    typedef typename K::Bool_type             result_type;
+    typedef typename K::Boolean               result_type;
 
     result_type
     operator()( const Ray_2& r, const Point_2& p) const
@@ -357,7 +357,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Orientation_2  Orientation_2;
     Orientation_2 o;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     Collinear_2() {}
     Collinear_2(const Orientation_2 o_) : o(o_) {}
@@ -372,7 +372,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3    Point_3;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()(const Point_3& p, const Point_3& q, const Point_3& r) const
@@ -3421,7 +3421,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3    Point_3;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3435,7 +3435,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_2    Point_2;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -3447,7 +3447,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3    Point_3;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3459,7 +3459,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_2    Point_2;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -3471,7 +3471,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3    Point_3;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3483,7 +3483,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3    Point_3;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3504,7 +3504,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Circle_3         Circle_3;
     typedef typename K::Sphere_3         Sphere_3;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Line_3& l, const Point_3& p) const
@@ -3564,7 +3564,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_2   Point_2;
   public:
-    typedef typename K::Bool_type result_type;
+    typedef typename K::Boolean   result_type;
 
     result_type
     operator()(const Point_2& p, const Point_2& q, const Point_2& r) const
@@ -3580,7 +3580,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3   Point_3;
   public:
-    typedef typename K::Bool_type result_type;
+    typedef typename K::Boolean   result_type;
 
     result_type
     operator()(const Point_3& p, const Point_3& q, const Point_3& r) const
@@ -3599,7 +3599,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Line_2    Line_2;
     typedef typename K::Equal_2   Equal_2;
   public:
-    typedef typename K::Bool_type result_type;
+    typedef typename K::Boolean   result_type;
 
     result_type
     operator()(const Point_2& a, const Point_2& b,
@@ -3629,7 +3629,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Plane_3       Plane_3;
     typedef typename K::Collinear_3   Collinear_3;
   public:
-    typedef typename K::Bool_type     result_type;
+    typedef typename K::Boolean       result_type;
 
     result_type
     operator()( const Plane_3& h, const Point_3& p, const Point_3& q) const
@@ -3660,7 +3660,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Compare_xyz_3   Compare_xyz_3;
     Compare_xyz_3 c;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     Less_xyz_3() {}
     Less_xyz_3(const Compare_xyz_3& c_) : c(c_) {}
@@ -3677,7 +3677,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Compare_xy_2   Compare_xy_2;
     Compare_xy_2 c;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     Less_xy_2() {}
     Less_xy_2(const Compare_xy_2& c_) : c(c_) {}
@@ -3694,7 +3694,7 @@ namespace CartesianKernelFunctors {
     typedef typename K::Compare_xy_3   Compare_xy_3;
     Compare_xy_3 c;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     Less_xy_3() {}
     Less_xy_3(const Compare_xy_3& c_) : c(c_) {}
@@ -3709,7 +3709,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_2        Point_2;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -3721,7 +3721,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3        Point_3;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3733,7 +3733,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_2        Point_2;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -3748,7 +3748,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_2        Point_2;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -3760,7 +3760,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3        Point_3;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3772,7 +3772,7 @@ namespace CartesianKernelFunctors {
   {
     typedef typename K::Point_3        Point_3;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/predicates_on_directions_2.h b/Cartesian_kernel/include/CGAL/Cartesian/predicates_on_directions_2.h
index 98e40e3ed75..b97b5cc7703 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/predicates_on_directions_2.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/predicates_on_directions_2.h
@@ -28,7 +28,7 @@ CGAL_BEGIN_NAMESPACE
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 equal_direction(const DirectionC2<K> &d1,
                 const DirectionC2<K> &d2)
 {
diff --git a/Cartesian_kernel/include/CGAL/Cartesian/predicates_on_planes_3.h b/Cartesian_kernel/include/CGAL/Cartesian/predicates_on_planes_3.h
index fc0c8683e42..7a96732547a 100644
--- a/Cartesian_kernel/include/CGAL/Cartesian/predicates_on_planes_3.h
+++ b/Cartesian_kernel/include/CGAL/Cartesian/predicates_on_planes_3.h
@@ -40,7 +40,7 @@ side_of_oriented_plane(const PlaneC3<K> &h,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 equal_plane(const PlaneC3<K> &h, const PlaneC3<K> &p)
 { 
   return equal_planeC3(h.a(), h.b(), h.c(), h.d(),
diff --git a/Circular_kernel_2/include/CGAL/Filtered_bbox_circular_kernel_2/bbox_filtered_predicates.h b/Circular_kernel_2/include/CGAL/Filtered_bbox_circular_kernel_2/bbox_filtered_predicates.h
index 0bc6da7a2e5..32d9fce139c 100644
--- a/Circular_kernel_2/include/CGAL/Filtered_bbox_circular_kernel_2/bbox_filtered_predicates.h
+++ b/Circular_kernel_2/include/CGAL/Filtered_bbox_circular_kernel_2/bbox_filtered_predicates.h
@@ -686,7 +686,7 @@ class Do_overlap_2
   class Do_intersect_2
   {
   public:
-    typedef typename BK::Bool_type  result_type;
+    typedef typename BK::Boolean    result_type;
     template <class T1, class T2>
     result_type
     operator()(const T1& t1, const T2& t2) const
diff --git a/Filtered_kernel/include/CGAL/Lazy_kernel.h b/Filtered_kernel/include/CGAL/Lazy_kernel.h
index 039d96f99fc..8f930e36051 100644
--- a/Filtered_kernel/include/CGAL/Lazy_kernel.h
+++ b/Filtered_kernel/include/CGAL/Lazy_kernel.h
@@ -81,7 +81,7 @@ public:
   typedef FT RT;
 
   typedef typename Same_uncertainty_nt<bool, FT>::type
-	                                                              Bool_type;
+	                                                              Boolean;
   typedef typename Same_uncertainty_nt<CGAL::Sign, FT>::type
 	                                                              Sign;
   typedef typename Same_uncertainty_nt<CGAL::Comparison_result, FT>::type
diff --git a/Homogeneous_kernel/include/CGAL/Homogeneous/Homogeneous_base.h b/Homogeneous_kernel/include/CGAL/Homogeneous/Homogeneous_base.h
index 32ce413be98..ff4b26c0635 100644
--- a/Homogeneous_kernel/include/CGAL/Homogeneous/Homogeneous_base.h
+++ b/Homogeneous_kernel/include/CGAL/Homogeneous/Homogeneous_base.h
@@ -89,7 +89,7 @@ struct Homogeneous_base
     typedef CGAL::Object                            Object_3;
 
     typedef typename Same_uncertainty_nt<bool, FT>::type
-                                                    Bool_type;
+                                                    Boolean;
     typedef typename Same_uncertainty_nt<CGAL::Sign, FT>::type
                                                     Sign;
     typedef typename Same_uncertainty_nt<CGAL::Comparison_result, FT>::type
diff --git a/Homogeneous_kernel/include/CGAL/Homogeneous/function_objects.h b/Homogeneous_kernel/include/CGAL/Homogeneous/function_objects.h
index 23580992e16..25b4b7422bf 100644
--- a/Homogeneous_kernel/include/CGAL/Homogeneous/function_objects.h
+++ b/Homogeneous_kernel/include/CGAL/Homogeneous/function_objects.h
@@ -241,7 +241,7 @@ namespace HomogeneousKernelFunctors {
     Collinear_2 c;
 #endif // CGAL_kernel_exactness_preconditions
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
 #ifdef CGAL_kernel_exactness_preconditions
     Collinear_are_ordered_along_line_2() {}
@@ -291,7 +291,7 @@ namespace HomogeneousKernelFunctors {
     Collinear_3 c;
 #endif // CGAL_kernel_exactness_preconditions
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
 #ifdef CGAL_kernel_exactness_preconditions
     Collinear_are_ordered_along_line_3() {}
@@ -372,7 +372,7 @@ namespace HomogeneousKernelFunctors {
     Collinear_2 c;
 #endif // CGAL_kernel_exactness_preconditions
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
 #ifdef CGAL_kernel_exactness_preconditions
     Collinear_are_strictly_ordered_along_line_2() {}
@@ -423,7 +423,7 @@ namespace HomogeneousKernelFunctors {
     Collinear_3 c;
 #endif // CGAL_kernel_exactness_preconditions
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
 #ifdef CGAL_kernel_exactness_preconditions
     Collinear_are_strictly_ordered_along_line_3() {}
@@ -457,7 +457,7 @@ namespace HomogeneousKernelFunctors {
     Construct_point_on_2 cp;
     Compare_xy_2 cxy;
   public:
-    typedef typename K::Bool_type             result_type;
+    typedef typename K::Boolean               result_type;
 
     Collinear_has_on_2() {}
     Collinear_has_on_2(const Construct_point_on_2& cp_,
@@ -486,7 +486,7 @@ namespace HomogeneousKernelFunctors {
     typedef typename K::Orientation_2  Orientation_2;
     Orientation_2 o;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     Collinear_2() {}
     Collinear_2(const Orientation_2 o_) : o(o_) {}
@@ -3636,7 +3636,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_3    Point_3;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3651,7 +3651,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_2    Point_2;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -3663,7 +3663,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_3    Point_3;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3675,7 +3675,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_2    Point_2;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -3687,7 +3687,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_3    Point_3;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3699,7 +3699,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_3    Point_3;
   public:
-    typedef typename K::Bool_type  result_type;
+    typedef typename K::Boolean    result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -3717,7 +3717,7 @@ namespace HomogeneousKernelFunctors {
     typedef typename K::Triangle_3       Triangle_3;
     typedef typename K::Tetrahedron_3    Tetrahedron_3;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Line_3& l, const Point_3& p) const
@@ -3784,7 +3784,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_2   Point_2;
   public:
-    typedef typename K::Bool_type result_type;
+    typedef typename K::Boolean   result_type;
 
     result_type
     operator()(const Point_2& p, const Point_2& q, const Point_2& r) const
@@ -3836,7 +3836,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_3   Point_3;
   public:
-    typedef typename K::Bool_type result_type;
+    typedef typename K::Boolean   result_type;
 
     result_type
     operator()(const Point_3& p, const Point_3& q, const Point_3& r) const
@@ -3874,7 +3874,7 @@ namespace HomogeneousKernelFunctors {
     typedef typename K::Point_2   Point_2;
     typedef typename K::Line_2    Line_2;
   public:
-    typedef typename K::Bool_type result_type;
+    typedef typename K::Boolean   result_type;
 
     result_type
     operator()(const Point_2& p, const Point_2& q,
@@ -3933,7 +3933,7 @@ namespace HomogeneousKernelFunctors {
     typedef typename K::Plane_3            Plane_3;
     typedef typename K::Construct_plane_3  Construct_plane_3;
   public:
-    typedef typename K::Bool_type          result_type;
+    typedef typename K::Boolean            result_type;
 
     result_type
     operator()( const Plane_3& pl, const Point_3& p, const Point_3& q) const
@@ -3974,7 +3974,7 @@ namespace HomogeneousKernelFunctors {
     typedef typename K::Compare_xyz_3  Compare_xyz_3;
     Compare_xyz_3 c;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     Less_xyz_3() {}
     Less_xyz_3(const Compare_xyz_3& c_) : c(c_) {}
@@ -3991,7 +3991,7 @@ namespace HomogeneousKernelFunctors {
     typedef typename K::Compare_xy_2  Compare_xy_2;
     Compare_xy_2 c;
   public:
-    typedef typename K::Bool_type     result_type;
+    typedef typename K::Boolean       result_type;
 
     Less_xy_2() {}
     Less_xy_2(const Compare_xy_2& c_) : c(c_) {}
@@ -4008,7 +4008,7 @@ namespace HomogeneousKernelFunctors {
     typedef typename K::Compare_xy_3  Compare_xy_3;
     Compare_xy_3 c;
   public:
-    typedef typename K::Bool_type     result_type;
+    typedef typename K::Boolean       result_type;
 
     Less_xy_3() {}
     Less_xy_3(const Compare_xy_3& c_) : c(c_) {}
@@ -4023,7 +4023,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_2         Point_2;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -4035,7 +4035,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_3         Point_3;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -4047,7 +4047,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_2       Point_2;
   public:
-    typedef typename K::Bool_type     result_type;
+    typedef typename K::Boolean       result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -4082,7 +4082,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_2         Point_2;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()( const Point_2& p, const Point_2& q) const
@@ -4094,7 +4094,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_3         Point_3;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
@@ -4106,7 +4106,7 @@ namespace HomogeneousKernelFunctors {
   {
     typedef typename K::Point_3         Point_3;
   public:
-    typedef typename K::Bool_type       result_type;
+    typedef typename K::Boolean         result_type;
 
     result_type
     operator()( const Point_3& p, const Point_3& q) const
diff --git a/Intersections_3/include/CGAL/Triangle_3_Tetrahedron_3_do_intersect.h b/Intersections_3/include/CGAL/Triangle_3_Tetrahedron_3_do_intersect.h
index f77a9677af9..79632a77d6b 100644
--- a/Intersections_3/include/CGAL/Triangle_3_Tetrahedron_3_do_intersect.h
+++ b/Intersections_3/include/CGAL/Triangle_3_Tetrahedron_3_do_intersect.h
@@ -32,7 +32,7 @@ namespace CGALi {
 
 // This code is not optimized:
 template <class K>
-typename K::Bool_type
+typename K::Boolean
 do_intersect(const typename K::Triangle_3 &tr,
              const typename K::Tetrahedron_3 &tet,
              const K & k)
@@ -59,7 +59,7 @@ do_intersect(const typename K::Triangle_3 &tr,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 do_intersect(const typename K::Tetrahedron_3 &tet,
 	     const typename K::Triangle_3 &tr,
 	     const K & k)
diff --git a/Intersections_3/include/CGAL/Triangle_3_Triangle_3_do_intersect.h b/Intersections_3/include/CGAL/Triangle_3_Triangle_3_do_intersect.h
index 56bda724722..8f8fac1424b 100644
--- a/Intersections_3/include/CGAL/Triangle_3_Triangle_3_do_intersect.h
+++ b/Intersections_3/include/CGAL/Triangle_3_Triangle_3_do_intersect.h
@@ -194,7 +194,7 @@ bool do_intersect_coplanar(const typename K::Triangle_3 &t1,
 
 
 template <class K>
-typename K::Bool_type
+typename K::Boolean
 do_intersect(const typename K::Triangle_3 &t1, 
 	     const typename K::Triangle_3 &t2,
 	     const K & k) 
@@ -446,7 +446,7 @@ do_intersect(const typename K::Triangle_3 &t1,
 
 
 template <class K>
-inline typename K::Bool_type
+inline typename K::Boolean
 do_intersect(const Triangle_3<K> &t1, 
 	     const Triangle_3<K> &t2) 
 {
diff --git a/Kernel_23/doc_tex/Kernel_23_ref/Kernel.tex b/Kernel_23/doc_tex/Kernel_23_ref/Kernel.tex
index 4aabf4aa8df..6350e376ad5 100644
--- a/Kernel_23/doc_tex/Kernel_23_ref/Kernel.tex
+++ b/Kernel_23/doc_tex/Kernel_23_ref/Kernel.tex
@@ -35,7 +35,7 @@ map to \ccc{bool} and CGAL kernel enum types, except when an interval arithmetic
 number type is used such as within the filtering kernels, in which case it is
 \ccc{Uncertain<bool>} or similar.
 
-\ccNestedType{Bool_type}{\ccc{bool} or \ccc{Uncertain<bool>}}
+\ccNestedType{Boolean}{\ccc{bool} or \ccc{Uncertain<bool>}}
 \ccGlue
 \ccNestedType{Sign}{\ccc{CGAL::Sign} or \ccc{Uncertain<CGAL::Sign>}}
 \ccGlue
diff --git a/Kernel_23/include/CGAL/Circle_2.h b/Kernel_23/include/CGAL/Circle_2.h
index 88c31c1a9ee..0f710ca239d 100644
--- a/Kernel_23/include/CGAL/Circle_2.h
+++ b/Kernel_23/include/CGAL/Circle_2.h
@@ -126,25 +126,25 @@ public:
     return R().oriented_side_2_object()(*this, p);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_boundary(const Point_2 &p) const
   {
     return bounded_side(p) == ON_BOUNDARY;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_bounded_side(const Point_2 &p) const
   {
     return bounded_side(p) == ON_BOUNDED_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_unbounded_side(const Point_2 &p) const
   {
     return bounded_side(p) == ON_UNBOUNDED_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_negative_side(const Point_2 &p) const
   {
     if (orientation() == COUNTERCLOCKWISE)
@@ -152,7 +152,7 @@ public:
     return has_on_bounded_side(p);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_positive_side(const Point_2 &p) const
   {
     if (orientation() == COUNTERCLOCKWISE)
@@ -160,7 +160,7 @@ public:
     return has_on_unbounded_side(p);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   is_degenerate() const
   {
     return CGAL_NTS is_zero(squared_radius());
@@ -181,13 +181,13 @@ public:
     return R().construct_bbox_2_object()(*this);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   operator==(const Circle_2 &c) const
   {
     return R().equal_2_object()(*this, c);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   operator!=(const Circle_2 &c) const
   {
     return !(*this == c);
diff --git a/Kernel_23/include/CGAL/Direction_2.h b/Kernel_23/include/CGAL/Direction_2.h
index c41a45eaa75..be601005d9e 100644
--- a/Kernel_23/include/CGAL/Direction_2.h
+++ b/Kernel_23/include/CGAL/Direction_2.h
@@ -85,7 +85,7 @@ public:
   Direction_2(const RT &x, const RT &y)
     :  RDirection_2(typename R::Construct_direction_2()(Return_base_tag(), x,y)) {}
 
-  typename R::Bool_type
+  typename R::Boolean
   counterclockwise_in_between(const Direction_2 &d1,
 			      const Direction_2 &d2) const
   {
@@ -116,28 +116,28 @@ public:
     return (i==0) ? dx() : dy();
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   operator<(const Direction_2 &d) const
   {
     return R().compare_angle_with_x_axis_2_object()(*this, d) == SMALLER;
   }
 
 
-  typename R::Bool_type
+  typename R::Boolean
   operator>(const Direction_2 &d) const
   {
     return d < *this;
   }
 
 
-  typename R::Bool_type
+  typename R::Boolean
   operator>=(const Direction_2 &d) const
   {
     return R().compare_angle_with_x_axis_2_object()(*this, d) != SMALLER;
   }
 
 
-  typename R::Bool_type
+  typename R::Boolean
   operator<=(const Direction_2 &d) const
   {
     return R().compare_angle_with_x_axis_2_object()(*this, d) != LARGER;
@@ -159,13 +159,13 @@ public:
     return this->vector();
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   operator==(const Direction_2& d) const
   {
     return R().equal_2_object()(*this, d);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   operator!=(const Direction_2& d) const
   {
     return !(*this == d);
diff --git a/Kernel_23/include/CGAL/Kernel/function_objects.h b/Kernel_23/include/CGAL/Kernel/function_objects.h
index 302c8dc98b9..22d6795053c 100644
--- a/Kernel_23/include/CGAL/Kernel/function_objects.h
+++ b/Kernel_23/include/CGAL/Kernel/function_objects.h
@@ -48,7 +48,7 @@ namespace CommonKernelFunctors {
     Collinear_2 c;
     Collinear_are_ordered_along_line_2 cao;
   public:
-    typedef typename K::Bool_type   result_type;
+    typedef typename K::Boolean     result_type;
 
     Are_ordered_along_line_2() {}
     Are_ordered_along_line_2(const Collinear_2& c_,
@@ -72,7 +72,7 @@ namespace CommonKernelFunctors {
     Collinear_3 c;
     Collinear_are_ordered_along_line_3 cao;
   public:
-    typedef typename K::Bool_type   result_type;
+    typedef typename K::Boolean     result_type;
 
     Are_ordered_along_line_3() {}
     Are_ordered_along_line_3(const Collinear_3& c_,
@@ -96,7 +96,7 @@ namespace CommonKernelFunctors {
     Collinear_2 c;
     Collinear_are_strictly_ordered_along_line_2 cao;
   public:
-    typedef typename K::Bool_type   result_type;
+    typedef typename K::Boolean     result_type;
 
     Are_strictly_ordered_along_line_2() {}
     Are_strictly_ordered_along_line_2(
@@ -121,7 +121,7 @@ namespace CommonKernelFunctors {
     Collinear_3 c;
     Collinear_are_strictly_ordered_along_line_3 cao;
   public:
-    typedef typename K::Bool_type   result_type;
+    typedef typename K::Boolean     result_type;
 
     Are_strictly_ordered_along_line_3() {}
     Are_strictly_ordered_along_line_3(
@@ -140,7 +140,7 @@ namespace CommonKernelFunctors {
   {
     typedef typename K::Object_2  Object_2;
   public:
-    //typedef typename K::Bool_type result_type;
+    //typedef typename K::Boolean   result_type;
     typedef bool                  result_type;
 
     template <class T>
@@ -154,7 +154,7 @@ namespace CommonKernelFunctors {
   {
     typedef typename K::Object_3        Object_3;
   public:
-    //typedef typename K::Bool_type       result_type;
+    //typedef typename K::Boolean         result_type;
     typedef bool                        result_type;
 
     template <class T>
@@ -1807,7 +1807,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Orientation_3 Orientation_3;
     Orientation_3 o;
   public:
-    typedef typename K::Bool_type     result_type;
+    typedef typename K::Boolean       result_type;
 
     Coplanar_3() {}
     Coplanar_3(const Orientation_3& o_) : o(o_) {}
@@ -1825,7 +1825,7 @@ namespace CommonKernelFunctors {
   {
     typedef typename K::Direction_2  Direction_2;
   public:
-    typedef typename K::Bool_type    result_type;
+    typedef typename K::Boolean      result_type;
 
     result_type
     operator()( const Direction_2& p, const Direction_2& q,
@@ -1842,7 +1842,7 @@ namespace CommonKernelFunctors {
   class Do_intersect_2
   {
   public:
-    typedef typename K::Bool_type   result_type;
+    typedef typename K::Boolean     result_type;
 
     // There are 36 combinaisons, so I use a template.
     template <class T1, class T2>
@@ -1855,7 +1855,7 @@ namespace CommonKernelFunctors {
   class Do_intersect_3
   {
   public:
-    typedef typename K::Bool_type   result_type;
+    typedef typename K::Boolean     result_type;
 
     // There are x combinaisons, so I use a template.
     template <class T1, class T2>
@@ -1878,7 +1878,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Circle_2      Circle_2;
 
   public:
-    typedef typename K::Bool_type     result_type;
+    typedef typename K::Boolean       result_type;
 
     result_type
     operator()(const Point_2 &p, const Point_2 &q) const
@@ -1966,7 +1966,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Circle_3      Circle_3;
 
   public:
-    typedef typename K::Bool_type     result_type;
+    typedef typename K::Boolean       result_type;
 
     // Point_3 is special case since the global operator== would recurse.
     result_type
@@ -2056,7 +2056,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Circle_2         Circle_2;
     typedef typename K::Triangle_2       Triangle_2;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Circle_2& c, const Point_2& p) const
@@ -2080,7 +2080,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Tetrahedron_3    Tetrahedron_3;
     typedef typename K::Plane_3          Plane_3;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Sphere_3& s, const Point_3& p) const
@@ -2104,7 +2104,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Circle_2         Circle_2;
     typedef typename K::Triangle_2       Triangle_2;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Circle_2& c, const Point_2& p) const
@@ -2128,7 +2128,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Tetrahedron_3    Tetrahedron_3;
     typedef typename K::Circle_3         Circle_3;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Sphere_3& s, const Point_3& p) const
@@ -2160,7 +2160,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Circle_2         Circle_2;
     typedef typename K::Triangle_2       Triangle_2;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Circle_2& c, const Point_2& p) const
@@ -2183,7 +2183,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Sphere_3         Sphere_3;
     typedef typename K::Tetrahedron_3    Tetrahedron_3;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Sphere_3& s, const Point_3& p) const
@@ -2206,7 +2206,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Circle_2         Circle_2;
     typedef typename K::Triangle_2       Triangle_2;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Circle_2& c, const Point_2& p) const
@@ -2229,7 +2229,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Sphere_3         Sphere_3;
     typedef typename K::Tetrahedron_3    Tetrahedron_3;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Sphere_3& s, const Point_3& p) const
@@ -2252,7 +2252,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Circle_2         Circle_2;
     typedef typename K::Triangle_2       Triangle_2;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Circle_2& c, const Point_2& p) const
@@ -2279,7 +2279,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Circle_3         Circle_3;
     typedef typename K::Tetrahedron_3    Tetrahedron_3;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Sphere_3& s, const Point_3& p) const
@@ -2311,7 +2311,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Ray_2            Ray_2;
     typedef typename K::Segment_2        Segment_2;
   public:
-    typedef typename K::Bool_type        result_type;
+    typedef typename K::Boolean          result_type;
 
     result_type
     operator()( const Line_2& l, const Point_2& p) const
@@ -2370,7 +2370,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Triangle_2        Triangle_2;
     typedef typename K::Circle_3          Circle_3;
   public:
-    typedef typename K::Bool_type         result_type;
+    typedef typename K::Boolean           result_type;
 
     result_type
     operator()( const Circle_2& c) const
@@ -2413,7 +2413,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Triangle_3        Triangle_3;
     typedef typename K::Tetrahedron_3     Tetrahedron_3;
   public:
-    typedef typename K::Bool_type         result_type;
+    typedef typename K::Boolean           result_type;
 
     result_type
     operator()( const Iso_cuboid_3& c) const
@@ -2455,7 +2455,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Segment_2 Segment_2;
     typedef typename K::Ray_2     Ray_2;
   public:
-    typedef typename K::Bool_type result_type;
+    typedef typename K::Boolean   result_type;
 
     result_type
     operator()( const Line_2& l) const
@@ -2477,7 +2477,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Segment_2 Segment_2;
     typedef typename K::Ray_2     Ray_2;
   public:
-    typedef typename K::Bool_type result_type;
+    typedef typename K::Boolean   result_type;
 
     result_type
     operator()( const Line_2& l) const
@@ -2499,7 +2499,7 @@ namespace CommonKernelFunctors {
     typedef typename K::Orientation_2  Orientation_2;
     Orientation_2 o;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     Left_turn_2() {}
     Left_turn_2(const Orientation_2& o_) : o(o_) {}
@@ -2519,7 +2519,7 @@ namespace CommonKernelFunctors {
     Orientation_2 o;
     Collinear_are_ordered_along_line_2 co;
   public:
-    typedef typename K::Bool_type      result_type;
+    typedef typename K::Boolean        result_type;
 
     Less_rotate_ccw_2() {}
     Less_rotate_ccw_2(const Orientation_2& o_,
diff --git a/Kernel_23/include/CGAL/Kernel/global_functions_2.h b/Kernel_23/include/CGAL/Kernel/global_functions_2.h
index 6751fd58052..7f075921761 100644
--- a/Kernel_23/include/CGAL/Kernel/global_functions_2.h
+++ b/Kernel_23/include/CGAL/Kernel/global_functions_2.h
@@ -36,14 +36,14 @@
 CGAL_BEGIN_NAMESPACE
 
 template < class K >
-typename K::Bool_type
+typename K::Boolean
 operator==(const Point_2<K> &p, const Origin& o)
 {
   return p == Point_2<K>(o);
 }
 
 template < class K >
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Point_2<K> &p, const Origin& o)
 {
   return p != Point_2<K>(o);
@@ -61,7 +61,7 @@ angle(const Point_2<K> &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_ordered_along_line(const Point_2<K> &p,
                        const Point_2<K> &q,
                        const Point_2<K> &r)
@@ -71,7 +71,7 @@ are_ordered_along_line(const Point_2<K> &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_strictly_ordered_along_line(const Point_2<K> &p,
                                 const Point_2<K> &q,
                                 const Point_2<K> &r)
@@ -221,7 +221,7 @@ circumcenter(const Triangle_2<K> &t)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear(const Point_2<K> &p, const Point_2<K> &q, const Point_2<K> &r)
 {
   return CGALi::collinear(p, q, r, K());
@@ -229,7 +229,7 @@ collinear(const Point_2<K> &p, const Point_2<K> &q, const Point_2<K> &r)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear_are_ordered_along_line(const Point_2<K> &p,
                                  const Point_2<K> &q,
                                  const Point_2<K> &r)
@@ -239,7 +239,7 @@ collinear_are_ordered_along_line(const Point_2<K> &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear_are_strictly_ordered_along_line(const Point_2<K> &p,
                                           const Point_2<K> &q,
                                           const Point_2<K> &r)
@@ -534,7 +534,7 @@ determinant(const Vector_2<K> &v0, const Vector_2<K> &v1)
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_distance_to_point(const Point_2<K>& p,
 			     const Point_2<K>& q,
 			     const Point_2<K>& r)
@@ -544,7 +544,7 @@ has_larger_distance_to_point(const Point_2<K>& p,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_distance_to_point(const Point_2<K>& p,
                               const Point_2<K>& q,
                               const Point_2<K>& r)
@@ -554,7 +554,7 @@ has_smaller_distance_to_point(const Point_2<K>& p,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_signed_distance_to_line(const Line_2<K>& l,
                                     const Point_2<K>& p,
                                     const Point_2<K>& q)
@@ -564,7 +564,7 @@ has_smaller_signed_distance_to_line(const Line_2<K>& l,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_signed_distance_to_line(const Line_2<K>& l,
 				   const Point_2<K>& p,
 				   const Point_2<K>& q)
@@ -574,7 +574,7 @@ has_larger_signed_distance_to_line(const Line_2<K>& l,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_signed_distance_to_line(const Point_2<K>& p,
 				   const Point_2<K>& q,
 				   const Point_2<K>& r,
@@ -585,7 +585,7 @@ has_larger_signed_distance_to_line(const Point_2<K>& p,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_signed_distance_to_line(const Point_2<K>& p,
                                     const Point_2<K>& q,
                                     const Point_2<K>& r,
@@ -596,7 +596,7 @@ has_smaller_signed_distance_to_line(const Point_2<K>& p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 left_turn(const Point_2<K> &p, const Point_2<K> &q, const Point_2<K> &r)
 {
   return CGALi::left_turn(p, q, r, K());
@@ -604,7 +604,7 @@ left_turn(const Point_2<K> &p, const Point_2<K> &q, const Point_2<K> &r)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_x(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::less_x(p, q, K());
@@ -612,7 +612,7 @@ less_x(const Point_2<K> &p, const Point_2<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_y(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::less_y(p, q, K());
@@ -620,7 +620,7 @@ less_y(const Point_2<K> &p, const Point_2<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xy_larger(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::lexicographically_xy_larger(p, q, K());
@@ -628,7 +628,7 @@ lexicographically_xy_larger(const Point_2<K> &p, const Point_2<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xy_larger_or_equal(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::lexicographically_xy_larger_or_equal(p, q, K());
@@ -636,7 +636,7 @@ lexicographically_xy_larger_or_equal(const Point_2<K> &p, const Point_2<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xy_smaller(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::lexicographically_xy_smaller(p, q, K());
@@ -644,7 +644,7 @@ lexicographically_xy_smaller(const Point_2<K> &p, const Point_2<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xy_smaller_or_equal(const Point_2<K> &p,
                                       const Point_2<K> &q)
 {
@@ -653,7 +653,7 @@ lexicographically_xy_smaller_or_equal(const Point_2<K> &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_yx_smaller(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::lexicographically_yx_smaller(p, q, K());
@@ -661,7 +661,7 @@ lexicographically_yx_smaller(const Point_2<K> &p, const Point_2<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_yx_smaller_or_equal(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::lexicographically_yx_smaller_or_equal(p, q, K());
@@ -670,7 +670,7 @@ lexicographically_yx_smaller_or_equal(const Point_2<K> &p, const Point_2<K> &q)
 // FIXME : Undocumented
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_yx_larger(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::lexicographically_yx_larger(p, q, K());
@@ -679,7 +679,7 @@ lexicographically_yx_larger(const Point_2<K> &p, const Point_2<K> &q)
 // FIXME : Undocumented
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_yx_larger_or_equal(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::lexicographically_yx_larger_or_equal(p, q, K());
@@ -713,73 +713,73 @@ min_vertex(const Iso_rectangle_2<K> &ir)
 // They have no counter part with the kernel argument...
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator<(const Direction_2<K>& d1, const Direction_2<K>& d2)
 { return compare_angle_with_x_axis(d1, d2) == SMALLER; }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator>(const Direction_2<K>& d1, const Direction_2<K>& d2)
 { return compare_angle_with_x_axis(d1, d2) == LARGER; }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator>=(const Direction_2<K>& d1, const Direction_2<K>& d2)
 { return compare_angle_with_x_axis(d1, d2) != SMALLER; }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator<=(const Direction_2<K>& d1, const Direction_2<K>& d2)
 { return compare_angle_with_x_axis(d1, d2) != LARGER; }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Point_2<K>& p, const Point_2<K>& q)
 { return K().equal_2_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Point_2<K>& p, const Point_2<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator<(const Point_2<K>& p, const Point_2<K>& q)
 { return K().less_xy_2_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator>(const Point_2<K>& p, const Point_2<K>& q)
 { return K().less_xy_2_object()(q, p); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator<=(const Point_2<K>& p, const Point_2<K>& q)
 { return ! K().less_xy_2_object()(q, p); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator>=(const Point_2<K>& p, const Point_2<K>& q)
 { return ! K().less_xy_2_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Vector_2<K>& v, const Vector_2<K>& w)
 { return K().equal_2_object()(v, w); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Vector_2<K>& v, const Vector_2<K>& w)
 { return ! (v == w); }
 
@@ -905,7 +905,7 @@ orientation(const Vector_2<K> &u, const Vector_2<K> &v)
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 right_turn(const Point_2<K> &p, const Point_2<K> &q, const Point_2<K> &r)
 {
   return CGALi::right_turn(p, q, r, K());
@@ -961,7 +961,7 @@ squared_radius(const Point_2<K>& p, const Point_2<K>& q, const Point_2<K>& r)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 x_equal(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::x_equal(p, q, K());
@@ -969,7 +969,7 @@ x_equal(const Point_2<K> &p, const Point_2<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 y_equal(const Point_2<K> &p, const Point_2<K> &q)
 {
   return CGALi::y_equal(p, q, K());
diff --git a/Kernel_23/include/CGAL/Kernel/global_functions_3.h b/Kernel_23/include/CGAL/Kernel/global_functions_3.h
index 4d1b8c51481..d799a57fe0c 100644
--- a/Kernel_23/include/CGAL/Kernel/global_functions_3.h
+++ b/Kernel_23/include/CGAL/Kernel/global_functions_3.h
@@ -43,7 +43,7 @@ angle(const Point_3<K> &p, const Point_3<K> &q, const Point_3<K> &r)
 
 template < typename K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_negative_oriented(const Point_3<K>& p, const Point_3<K>& q,
                       const Point_3<K>& r, const Point_3<K>& s)
 {
@@ -52,7 +52,7 @@ are_negative_oriented(const Point_3<K>& p, const Point_3<K>& q,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_ordered_along_line(const Point_3<K> &p,
                        const Point_3<K> &q,
                        const Point_3<K> &r)
@@ -62,7 +62,7 @@ are_ordered_along_line(const Point_3<K> &p,
 
 template < typename K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_positive_oriented(const Point_3<K>& p, const Point_3<K>& q,
                       const Point_3<K>& r, const Point_3<K>& s)
 {
@@ -71,7 +71,7 @@ are_positive_oriented(const Point_3<K>& p, const Point_3<K>& q,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_strictly_ordered_along_line(const Point_3<K> &p,
                                 const Point_3<K> &q,
                                 const Point_3<K> &r)
@@ -234,7 +234,7 @@ circumcenter(const Triangle_3<K> &t)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear(const Point_3<K> &p, const Point_3<K> &q, const Point_3<K> &r)
 {
   return CGALi::collinear(p, q, r, K());
@@ -242,7 +242,7 @@ collinear(const Point_3<K> &p, const Point_3<K> &q, const Point_3<K> &r)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear_are_ordered_along_line(const Point_3<K> &p,
                                  const Point_3<K> &q,
                                  const Point_3<K> &r)
@@ -252,7 +252,7 @@ collinear_are_ordered_along_line(const Point_3<K> &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear_are_strictly_ordered_along_line(const Point_3<K> &p,
                                           const Point_3<K> &q,
                                           const Point_3<K> &r)
@@ -378,7 +378,7 @@ compare_xyz(const Point_3<K> &p, const Point_3<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 coplanar(const Point_3<K> &p, const Point_3<K> &q,
          const Point_3<K> &r, const Point_3<K> &s)
 {
@@ -445,7 +445,7 @@ equidistant_line(const Point_3<K> &p, const Point_3<K> &q, const Point_3<K> &r)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_distance_to_point(const Point_3<K> &p,
 			     const Point_3<K> &q,
 			     const Point_3<K> &r)
@@ -455,7 +455,7 @@ has_larger_distance_to_point(const Point_3<K> &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_signed_distance_to_plane(const Point_3<K> &hp,
 				    const Point_3<K> &hq,
 				    const Point_3<K> &hr,
@@ -467,7 +467,7 @@ has_larger_signed_distance_to_plane(const Point_3<K> &hp,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_signed_distance_to_plane(const Plane_3<K> &h,
 				    const Point_3<K> &p,
 				    const Point_3<K> &q)
@@ -477,7 +477,7 @@ has_larger_signed_distance_to_plane(const Plane_3<K> &h,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_distance_to_point(const Point_3<K> &p,
                               const Point_3<K> &q,
                               const Point_3<K> &r)
@@ -487,7 +487,7 @@ has_smaller_distance_to_point(const Point_3<K> &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_signed_distance_to_plane(const Point_3<K> &hp,
                                      const Point_3<K> &hq,
                                      const Point_3<K> &hr,
@@ -499,7 +499,7 @@ has_smaller_signed_distance_to_plane(const Point_3<K> &hp,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_signed_distance_to_plane(const Plane_3<K> &h,
                                      const Point_3<K> &p,
                                      const Point_3<K> &q)
@@ -509,7 +509,7 @@ has_smaller_signed_distance_to_plane(const Plane_3<K> &h,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_x(const Point_3<K> &p, const Point_3<K> &q)
 { 
   return CGALi::less_x(p, q, K());
@@ -517,7 +517,7 @@ less_x(const Point_3<K> &p, const Point_3<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_y(const Point_3<K> &p, const Point_3<K> &q)
 { 
   return CGALi::less_y(p, q, K());
@@ -525,7 +525,7 @@ less_y(const Point_3<K> &p, const Point_3<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_z(const Point_3<K> &p, const Point_3<K> &q)
 { 
   return CGALi::less_z(p, q, K());
@@ -533,7 +533,7 @@ less_z(const Point_3<K> &p, const Point_3<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xyz_smaller(const Point_3<K> &p, const Point_3<K> &q)
 {
   return CGALi::lexicographically_xyz_smaller(p, q, K());
@@ -541,7 +541,7 @@ lexicographically_xyz_smaller(const Point_3<K> &p, const Point_3<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xyz_smaller_or_equal(const Point_3<K> &p,
                                        const Point_3<K> &q)
 {
@@ -583,182 +583,182 @@ normal(const Point_3<K> &p, const Point_3<K> &q, const Point_3<K> &r)
 // FIXME TODO : what to do with the operators ?
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Point_3<K>& p, const Point_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Point_3<K>& p, const Point_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Point_3<K>& p, const Origin& o)
 { return K().equal_3_object()(p, Point_3<K>(o)); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Point_3<K>& p, const Origin& o)
 { return ! (p == o); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Iso_cuboid_3<K>& p, const Iso_cuboid_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Iso_cuboid_3<K>& p, const Iso_cuboid_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Plane_3<K>& p, const Plane_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Plane_3<K>& p, const Plane_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Segment_3<K>& p, const Segment_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Segment_3<K>& p, const Segment_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Line_3<K>& p, const Line_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Line_3<K>& p, const Line_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Ray_3<K>& p, const Ray_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Ray_3<K>& p, const Ray_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Triangle_3<K>& p, const Triangle_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Triangle_3<K>& p, const Triangle_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Tetrahedron_3<K>& p, const Tetrahedron_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Tetrahedron_3<K>& p, const Tetrahedron_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Direction_3<K>& p, const Direction_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Direction_3<K>& p, const Direction_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Sphere_3<K>& p, const Sphere_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Sphere_3<K>& p, const Sphere_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Vector_3<K>& p, const Vector_3<K>& q)
 { return K().equal_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Vector_3<K>& p, const Vector_3<K>& q)
 { return ! (p == q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator==(const Vector_3<K>& p, const Null_vector& o)
 { return K().equal_3_object()(p, Vector_3<K>(o)); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator!=(const Vector_3<K>& p, const Null_vector& o)
 { return ! (p == o); }
 
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator<(const Point_3<K>& p, const Point_3<K>& q)
 { return K().less_xyz_3_object()(p, q); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator>(const Point_3<K>& p, const Point_3<K>& q)
 { return K().less_xyz_3_object()(q, p); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator<=(const Point_3<K>& p, const Point_3<K>& q)
 { return ! K().less_xyz_3_object()(q, p); }
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 operator>=(const Point_3<K>& p, const Point_3<K>& q)
 { return ! K().less_xyz_3_object()(p, q); }
 
@@ -1010,7 +1010,7 @@ volume(const Point_3<K> &p, const Point_3<K> &q,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 x_equal(const Point_3<K> &p, const Point_3<K> &q)
 {
   return CGALi::x_equal(p, q, K());
@@ -1018,7 +1018,7 @@ x_equal(const Point_3<K> &p, const Point_3<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 y_equal(const Point_3<K> &p, const Point_3<K> &q)
 {
   return CGALi::y_equal(p, q, K());
@@ -1026,7 +1026,7 @@ y_equal(const Point_3<K> &p, const Point_3<K> &q)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 z_equal(const Point_3<K> &p, const Point_3<K> &q)
 {
   return CGALi::z_equal(p, q, K());
diff --git a/Kernel_23/include/CGAL/Kernel/global_functions_internal_2.h b/Kernel_23/include/CGAL/Kernel/global_functions_internal_2.h
index cadae8f17e3..145e9d0ad92 100644
--- a/Kernel_23/include/CGAL/Kernel/global_functions_internal_2.h
+++ b/Kernel_23/include/CGAL/Kernel/global_functions_internal_2.h
@@ -47,7 +47,7 @@ angle(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_ordered_along_line(const typename K::Point_2 &p,
                        const typename K::Point_2 &q,
                        const typename K::Point_2 &r, const K& k)
@@ -57,7 +57,7 @@ are_ordered_along_line(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_strictly_ordered_along_line(const typename K::Point_2 &p,
                                 const typename K::Point_2 &q,
                                 const typename K::Point_2 &r,
@@ -213,7 +213,7 @@ circumcenter(const typename K::Triangle_2 &t, const K& k)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear(const typename K::Point_2 &p,
           const typename K::Point_2 &q,
           const typename K::Point_2 &r, const K& k)
@@ -223,7 +223,7 @@ collinear(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear_are_ordered_along_line(const typename K::Point_2 &p,
                                  const typename K::Point_2 &q,
                                  const typename K::Point_2 &r,
@@ -234,7 +234,7 @@ collinear_are_ordered_along_line(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear_are_strictly_ordered_along_line(
              const typename K::Point_2 &p,
              const typename K::Point_2 &q,
@@ -537,7 +537,7 @@ determinant(const typename K::Vector_2 &v0,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_distance_to_point(const typename K::Point_2 &p,
 			     const typename K::Point_2 &q,
 			     const typename K::Point_2 &r,
@@ -548,7 +548,7 @@ has_larger_distance_to_point(const typename K::Point_2 &p,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_distance_to_point(const typename K::Point_2 &p,
                               const typename K::Point_2 &q,
                               const typename K::Point_2 &r,
@@ -559,7 +559,7 @@ has_smaller_distance_to_point(const typename K::Point_2 &p,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_signed_distance_to_line(const typename K::Line_2& l,
 				    const typename K::Point_2& p,
 				    const typename K::Point_2& q,
@@ -570,7 +570,7 @@ has_smaller_signed_distance_to_line(const typename K::Line_2& l,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_signed_distance_to_line(const typename K::Line_2& l,
 				   const typename K::Point_2& p,
 				   const typename K::Point_2& q,
@@ -581,7 +581,7 @@ has_larger_signed_distance_to_line(const typename K::Line_2& l,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_signed_distance_to_line(const typename K::Point_2& p,
 				   const typename K::Point_2& q,
 				   const typename K::Point_2& r,
@@ -593,7 +593,7 @@ has_larger_signed_distance_to_line(const typename K::Point_2& p,
 
 template <class K>
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_signed_distance_to_line(const typename K::Point_2& p,
                                     const typename K::Point_2& q,
                                     const typename K::Point_2& r,
@@ -605,7 +605,7 @@ has_smaller_signed_distance_to_line(const typename K::Point_2& p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 left_turn(const typename K::Point_2 &p,
           const typename K::Point_2 &q,
           const typename K::Point_2 &r, const K& k)
@@ -615,7 +615,7 @@ left_turn(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_x(const typename K::Point_2 &p,
        const typename K::Point_2 &q, const K& k)
 {
@@ -624,7 +624,7 @@ less_x(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_y(const typename K::Point_2 &p,
        const typename K::Point_2 &q, const K& k)
 {
@@ -633,7 +633,7 @@ less_y(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xy_larger(const typename K::Point_2 &p,
                             const typename K::Point_2 &q,
                             const K& k)
@@ -643,7 +643,7 @@ lexicographically_xy_larger(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xy_larger_or_equal(const typename K::Point_2 &p,
                                      const typename K::Point_2 &q,
                                      const K& k)
@@ -653,7 +653,7 @@ lexicographically_xy_larger_or_equal(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xy_smaller(const typename K::Point_2 &p,
                              const typename K::Point_2 &q,
                              const K& k)
@@ -663,7 +663,7 @@ lexicographically_xy_smaller(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xy_smaller_or_equal(const typename K::Point_2 &p,
                                       const typename K::Point_2 &q,
                                       const K& k)
@@ -673,7 +673,7 @@ lexicographically_xy_smaller_or_equal(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_yx_smaller(const typename K::Point_2 &p,
                              const typename K::Point_2 &q,
                              const K& k)
@@ -683,7 +683,7 @@ lexicographically_yx_smaller(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_yx_smaller_or_equal(const typename K::Point_2 &p,
                                       const typename K::Point_2 &q,
                                       const K& k)
@@ -694,7 +694,7 @@ lexicographically_yx_smaller_or_equal(const typename K::Point_2 &p,
 // FIXME : Undocumented
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_yx_larger(const typename K::Point_2 &p,
                             const typename K::Point_2 &q,
                             const K& k)
@@ -705,7 +705,7 @@ lexicographically_yx_larger(const typename K::Point_2 &p,
 // FIXME : Undocumented
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_yx_larger_or_equal(const typename K::Point_2 &p,
                                      const typename K::Point_2 &q,
                                      const K& k)
@@ -759,7 +759,7 @@ orientation(const typename K::Vector_2 &u,
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 parallel(const typename K::Line_2 &l1,
          const typename K::Line_2 &l2, const K &k)
 {
@@ -768,7 +768,7 @@ parallel(const typename K::Line_2 &l1,
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 parallel(const typename K::Ray_2 &r1,
          const typename K::Ray_2 &r2, const K &k)
 {
@@ -777,7 +777,7 @@ parallel(const typename K::Ray_2 &r1,
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 parallel(const typename K::Segment_2 &s1,
          const typename K::Segment_2 &s2, const K &k)
 {
@@ -786,7 +786,7 @@ parallel(const typename K::Segment_2 &s1,
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 right_turn(const typename K::Point_2 &p,
            const typename K::Point_2 &q,
            const typename K::Point_2 &r, const K &k)
@@ -847,7 +847,7 @@ squared_radius(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 x_equal(const typename K::Point_2 &p,
         const typename K::Point_2 &q, const K &k)
 {
@@ -856,7 +856,7 @@ x_equal(const typename K::Point_2 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 y_equal(const typename K::Point_2 &p,
         const typename K::Point_2 &q, const K &k)
 {
diff --git a/Kernel_23/include/CGAL/Kernel/global_functions_internal_3.h b/Kernel_23/include/CGAL/Kernel/global_functions_internal_3.h
index 26ecca5d93b..5075aa038b1 100644
--- a/Kernel_23/include/CGAL/Kernel/global_functions_internal_3.h
+++ b/Kernel_23/include/CGAL/Kernel/global_functions_internal_3.h
@@ -43,7 +43,7 @@ angle(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_ordered_along_line(const typename K::Point_3 &p,
                        const typename K::Point_3 &q,
                        const typename K::Point_3 &r, const K& k)
@@ -53,7 +53,7 @@ are_ordered_along_line(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 are_strictly_ordered_along_line(const typename K::Point_3 &p,
                                 const typename K::Point_3 &q,
                                 const typename K::Point_3 &r,
@@ -225,7 +225,7 @@ circumcenter(const typename K::Triangle_3 &t, const K& k)
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear(const typename K::Point_3 &p,
           const typename K::Point_3 &q,
           const typename K::Point_3 &r,
@@ -236,7 +236,7 @@ collinear(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear_are_ordered_along_line(
           const typename K::Point_3 &p,
           const typename K::Point_3 &q,
@@ -248,7 +248,7 @@ collinear_are_ordered_along_line(
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 collinear_are_strictly_ordered_along_line(
           const typename K::Point_3 &p,
           const typename K::Point_3 &q,
@@ -398,7 +398,7 @@ compare_xyz(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 coplanar(const typename K::Point_3 &p,
          const typename K::Point_3 &q,
          const typename K::Point_3 &r,
@@ -471,7 +471,7 @@ equidistant_line(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_distance_to_point(const typename K::Point_3 &p,
                               const typename K::Point_3 &q,
                               const typename K::Point_3 &r,
@@ -482,7 +482,7 @@ has_smaller_distance_to_point(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_distance_to_point(const typename K::Point_3 &p,
 			     const typename K::Point_3 &q,
 			     const typename K::Point_3 &r,
@@ -493,7 +493,7 @@ has_larger_distance_to_point(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_signed_distance_to_plane(const typename K::Plane_3 &h,
 				    const typename K::Point_3 &p,
 				    const typename K::Point_3 &q,
@@ -504,7 +504,7 @@ has_larger_signed_distance_to_plane(const typename K::Plane_3 &h,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_larger_signed_distance_to_plane(const typename K::Point_3 &hp,
 				    const typename K::Point_3 &hq,
 				    const typename K::Point_3 &hr,
@@ -517,7 +517,7 @@ has_larger_signed_distance_to_plane(const typename K::Point_3 &hp,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_signed_distance_to_plane(const typename K::Plane_3 &h,
                                      const typename K::Point_3 &p,
                                      const typename K::Point_3 &q,
@@ -528,7 +528,7 @@ has_smaller_signed_distance_to_plane(const typename K::Plane_3 &h,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 has_smaller_signed_distance_to_plane(const typename K::Point_3 &hp,
                                      const typename K::Point_3 &hq,
                                      const typename K::Point_3 &hr,
@@ -541,7 +541,7 @@ has_smaller_signed_distance_to_plane(const typename K::Point_3 &hp,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_x(const typename K::Point_3 &p,
        const typename K::Point_3 &q,
        const K &k)
@@ -551,7 +551,7 @@ less_x(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_y(const typename K::Point_3 &p,
        const typename K::Point_3 &q,
        const K &k)
@@ -561,7 +561,7 @@ less_y(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 less_z(const typename K::Point_3 &p,
        const typename K::Point_3 &q,
        const K &k)
@@ -571,7 +571,7 @@ less_z(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xyz_smaller(const typename K::Point_3 &p,
                               const typename K::Point_3 &q,
                               const K &k)
@@ -660,7 +660,7 @@ orthogonal_vector(const typename K::Plane_3 &p, const K &k)
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 parallel(const typename K::Line_3 &l1,
          const typename K::Line_3 &l2, const K &k)
 {
@@ -669,7 +669,7 @@ parallel(const typename K::Line_3 &l1,
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 parallel(const typename K::Plane_3 &h1,
          const typename K::Plane_3 &h2, const K &k)
 {
@@ -678,7 +678,7 @@ parallel(const typename K::Plane_3 &h1,
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 parallel(const typename K::Ray_3 &r1,
          const typename K::Ray_3 &r2, const K &k)
 {
@@ -687,7 +687,7 @@ parallel(const typename K::Ray_3 &r1,
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 parallel(const typename K::Segment_3 &s1,
          const typename K::Segment_3 &s2, const K &k)
 {
@@ -792,7 +792,7 @@ volume(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 x_equal(const typename K::Point_3 &p,
         const typename K::Point_3 &q, const K &k)
 {
@@ -801,7 +801,7 @@ x_equal(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 y_equal(const typename K::Point_3 &p,
         const typename K::Point_3 &q, const K &k)
 {
@@ -810,7 +810,7 @@ y_equal(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 z_equal(const typename K::Point_3 &p,
         const typename K::Point_3 &q, const K &k)
 {
@@ -822,7 +822,7 @@ z_equal(const typename K::Point_3 &p,
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 are_negative_oriented(const typename K::Point_3 &p,
                       const typename K::Point_3 &q,
                       const typename K::Point_3 &r,
@@ -833,7 +833,7 @@ are_negative_oriented(const typename K::Point_3 &p,
 
 template <typename K>
 inline
-typename K::Bool_type
+typename K::Boolean
 are_positive_oriented(const typename K::Point_3 &p,
                       const typename K::Point_3 &q,
                       const typename K::Point_3 &r,
@@ -844,7 +844,7 @@ are_positive_oriented(const typename K::Point_3 &p,
 
 template < class K >
 inline
-typename K::Bool_type
+typename K::Boolean
 lexicographically_xyz_smaller_or_equal(const typename K::Point_3 &p,
                                        const typename K::Point_3 &q,
                                        const K&k)
diff --git a/Kernel_23/include/CGAL/Kernel_checker.h b/Kernel_23/include/CGAL/Kernel_checker.h
index b758df18438..22628d291e4 100644
--- a/Kernel_23/include/CGAL/Kernel_checker.h
+++ b/Kernel_23/include/CGAL/Kernel_checker.h
@@ -251,7 +251,7 @@ class Kernel_checker
 
 public:
 
-    typedef bool                      Bool_type;
+    typedef bool                      Boolean;
     typedef CGAL::Sign                Sign;
     typedef CGAL::Comparison_result   Comparison_result;
     typedef CGAL::Orientation         Orientation;
diff --git a/Kernel_23/include/CGAL/Line_2.h b/Kernel_23/include/CGAL/Line_2.h
index 78c2029aee2..5302f482ab7 100644
--- a/Kernel_23/include/CGAL/Line_2.h
+++ b/Kernel_23/include/CGAL/Line_2.h
@@ -143,19 +143,19 @@ public:
     return R().construct_projected_point_2_object()(*this,p);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   is_horizontal() const
   {
     return R().is_horizontal_2_object()(*this);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   is_vertical() const
   {
     return R().is_vertical_2_object()(*this);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   is_degenerate() const
   { return R().is_degenerate_2_object()(*this); }
 
@@ -165,25 +165,25 @@ public:
     return R().oriented_side_2_object()(*this,p);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_boundary(const Point_2 &p) const
   {
     return oriented_side(p) == ON_ORIENTED_BOUNDARY;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_positive_side(const Point_2 &p) const
   {
     return oriented_side(p) == ON_POSITIVE_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_negative_side(const Point_2 &p) const
   {
     return oriented_side(p) == ON_NEGATIVE_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on(const Point_2 &p) const
   {
     return has_on_boundary(p);
@@ -213,13 +213,13 @@ public:
     return R().construct_point_2_object()(*this,i);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   operator==(const Line_2 &l) const
   {
     return R().equal_2_object()(*this, l);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   operator!=(const Line_2 &l) const
   {
     return !(*this == l);
diff --git a/Kernel_23/include/CGAL/Sphere_3.h b/Kernel_23/include/CGAL/Sphere_3.h
index 98433b4637d..c77eb6a4675 100644
--- a/Kernel_23/include/CGAL/Sphere_3.h
+++ b/Kernel_23/include/CGAL/Sphere_3.h
@@ -125,26 +125,26 @@ public:
     return R().oriented_side_3_object()(*this, p);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_boundary(const Point_3 &p) const
   {
     return R().has_on_boundary_3_object()(*this, p);
     //return bounded_side(p) == ON_BOUNDARY;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_bounded_side(const Point_3 &p) const
   {
     return bounded_side(p) == ON_BOUNDED_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_unbounded_side(const Point_3 &p) const
   {
     return bounded_side(p) == ON_UNBOUNDED_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_negative_side(const Point_3 &p) const
   {
     if (orientation() == COUNTERCLOCKWISE)
@@ -152,7 +152,7 @@ public:
     return has_on_bounded_side(p);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_positive_side(const Point_3 &p) const
   {
     if (orientation() == COUNTERCLOCKWISE)
@@ -160,7 +160,7 @@ public:
     return has_on_unbounded_side(p);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   is_degenerate() const
   {
     return R().is_degenerate_3_object()(*this);
diff --git a/Kernel_23/include/CGAL/Triangle_2.h b/Kernel_23/include/CGAL/Triangle_2.h
index 83dc3b3f70e..30f14b956e0 100644
--- a/Kernel_23/include/CGAL/Triangle_2.h
+++ b/Kernel_23/include/CGAL/Triangle_2.h
@@ -94,13 +94,13 @@ public:
     return R().oriented_side_2_object()(*this,p);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   operator==(const Triangle_2 &t) const
   {
     return R().equal_2_object()(*this,t);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   operator!=(const Triangle_2 &t) const
   {
     return !(*this == t);
@@ -118,37 +118,37 @@ public:
     return vertex(i);
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_bounded_side(const Point_2 &p) const
   {
     return bounded_side(p) == ON_BOUNDED_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_unbounded_side(const Point_2 &p) const
   {
     return bounded_side(p) == ON_UNBOUNDED_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_boundary(const Point_2 &p) const
   {
     return bounded_side(p) == ON_BOUNDARY;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_negative_side(const Point_2 &p) const
   {
     return oriented_side(p) == ON_NEGATIVE_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   has_on_positive_side(const Point_2 &p) const
   {
     return oriented_side(p) == ON_POSITIVE_SIDE;
   }
 
-  typename R::Bool_type
+  typename R::Boolean
   is_degenerate() const
   {
     return R().collinear_2_object()(vertex(0), vertex(1), vertex(2));
diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Are_parallel_C2.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Are_parallel_C2.h
index 73f8f8e0c3a..2fba8913112 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Are_parallel_C2.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Are_parallel_C2.h
@@ -37,8 +37,8 @@ class Are_parallel_C2
 
 public:
   typedef typename K::Site_2       Site_2;
-  typedef typename K::Bool_type    Bool_type;
-  typedef Bool_type                result_type;
+  typedef typename K::Boolean      Boolean;
+  typedef Boolean                  result_type;
   typedef Site_2                   argument_type;
 
 private:
@@ -46,7 +46,7 @@ private:
   typedef typename K::FT           FT;
 
 private:
-  Bool_type predicate(const Site_2& p, const Site_2& q) const {
+  Boolean   predicate(const Site_2& p, const Site_2& q) const {
     CGAL_precondition( p.is_segment() && q.is_segment() );
     
     Segment_2 s1 = p.segment();
diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Are_same_points_C2.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Are_same_points_C2.h
index 09ca9156cad..8e55be33afb 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Are_same_points_C2.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Are_same_points_C2.h
@@ -36,20 +36,20 @@ private:
   typedef typename K::Site_2      Site_2;
   typedef typename K::Compare_x_2 Compare_x_2;
   typedef typename K::Compare_y_2 Compare_y_2;
-  typedef typename K::Bool_type   Bool_type;
+  typedef typename K::Boolean     Boolean;
 
   typedef typename K::Intersections_tag  ITag;
 
   Compare_x_2 compare_x_2;
   Compare_y_2 compare_y_2;
 
-  Bool_type are_same(const Point_2& p, const Point_2& q) const
+  Boolean   are_same(const Point_2& p, const Point_2& q) const
   {
     return
       compare_x_2(p, q) == EQUAL && compare_y_2(p, q) == EQUAL;
   }
 
-  Bool_type are_same(const Site_2& s, const Site_2& t) const
+  Boolean   are_same(const Site_2& s, const Site_2& t) const
   {
     return
       ( are_same(s.source(), t.source()) &&
@@ -58,12 +58,12 @@ private:
 	are_same(s.target(), t.source()) );
   }
 
-  Bool_type predicate(const Site_2& p, const Site_2& q, const Tag_false&) const
+  Boolean   predicate(const Site_2& p, const Site_2& q, const Tag_false&) const
   {
     return are_same(p.point(), q.point()); 
   }
 
-  Bool_type predicate(const Site_2& p, const Site_2& q, const Tag_true&) const
+  Boolean   predicate(const Site_2& p, const Site_2& q, const Tag_true&) const
   {
     if ( !p.is_input() && !q.is_input() ) {
       Site_2 s[2] = { p.supporting_site(0), p.supporting_site(1) };
@@ -79,10 +79,10 @@ private:
   }
 
 public:
-  typedef Bool_type      result_type;
+  typedef Boolean        result_type;
   typedef Site_2         argument_type;
 
-  Bool_type operator()(const Site_2& p, const Site_2& q) const
+  Boolean   operator()(const Site_2& p, const Site_2& q) const
   {
     CGAL_precondition( p.is_point() && q.is_point() );
 
diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Arrangement_type_C2.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Arrangement_type_C2.h
index 8787ee281ba..1576b8caec1 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Arrangement_type_C2.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Arrangement_type_C2.h
@@ -60,7 +60,7 @@ private:
   typedef typename Base::Comparison_result   Comparison_result;
   typedef typename Base::Oriented_side       Oriented_side;
   typedef typename Base::Sign                Sign;
-  typedef typename Base::Bool_type           Bool_type;
+  typedef typename Base::Boolean             Boolean;
 
   typedef typename K::Orientation_2          Orientation_2;
 
@@ -285,7 +285,7 @@ private:
   }
 
 
-  Bool_type inside_segment(const Site_2& s, const Site_2& p) const
+  Boolean   inside_segment(const Site_2& s, const Site_2& p) const
   {
     CGAL_precondition( s.is_segment() && p.is_point() );
 
@@ -300,7 +300,7 @@ private:
       // the point does not belong to the same line as the segment
       return false;
     if (! is_certain( os != ON_ORIENTED_BOUNDARY ) )
-      return indeterminate<Bool_type>();
+      return indeterminate<Boolean>();
 
     Line_2 lp1 = compute_perpendicular(l, s.segment().source());
 
diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Basic_predicates_C2.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Basic_predicates_C2.h
index 5098e7c54b2..7e24012a80b 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Basic_predicates_C2.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Basic_predicates_C2.h
@@ -52,7 +52,7 @@ public:
   typedef typename K::Comparison_result   Comparison_result;
   typedef typename K::Sign                Sign;
   typedef typename K::Orientation         Orientation;
-  typedef typename K::Bool_type           Bool_type;
+  typedef typename K::Boolean             Boolean;
 
   typedef CGAL::Sqrt_extension_1<RT>       Sqrt_1;
   typedef CGAL::Sqrt_extension_2<RT>       Sqrt_2;
diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Finite_edge_interior_conflict_C2.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Finite_edge_interior_conflict_C2.h
index c6e8ef9d7d4..d95b0390d6a 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Finite_edge_interior_conflict_C2.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Finite_edge_interior_conflict_C2.h
@@ -51,7 +51,7 @@ public:
   typedef typename Base::Sign                 Sign;
   typedef typename Base::Orientation          Orientation;
   typedef typename Base::Oriented_side        Oriented_side;
-  typedef typename Base::Bool_type            Bool_type;
+  typedef typename Base::Boolean              Boolean;
 
   typedef typename Base::Homogeneous_point_2  Homogeneous_point_2;
 
@@ -71,12 +71,12 @@ private:
   //--------------------------------------------------------------------
   //--------------------------------------------------------------------
 
-  Bool_type
+  Boolean
   is_interior_in_conflict_both(const Site_2& p, const Site_2& q,
 			       const Site_2& r, const Site_2& s,
 			       const Site_2& t, Method_tag tag) const
   {
-    Bool_type in_conflict(false);
+    Boolean   in_conflict(false);
 
     if ( p.is_point() && q.is_point() ) {
       in_conflict = is_interior_in_conflict_both_pp(p, q, r, s, t, tag);
@@ -172,7 +172,7 @@ private:
 
   //--------------------------------------------------------------------
 
-  Bool_type
+  Boolean
   is_interior_in_conflict_both_ps(const Site_2& p, const Site_2& q,
 				  const Site_2& r, const Site_2& s,
 				  const Site_2& t, Method_tag tag) const
@@ -192,7 +192,7 @@ private:
 
   //--------------------------------------------------------------------
 
-  Bool_type
+  Boolean
   is_interior_in_conflict_both_ps_p(const Site_2& p, const Site_2& q,
 				    const Site_2& r, const Site_2& s,
 				    const Site_2& t, Method_tag ) const
@@ -207,7 +207,7 @@ private:
 
     //if ( res != SMALLER ) { return true; }
     if (certainly( res != SMALLER ) ) { return true; }
-    if (! is_certain( res != SMALLER ) ) { return indeterminate<Bool_type>(); }
+    if (! is_certain( res != SMALLER ) ) { return indeterminate<Boolean>(); }
 
     Voronoi_vertex_2 vpqr(p, q, r);
     Voronoi_vertex_2 vqps(q, p, s);
@@ -232,7 +232,7 @@ private:
     return t1.is_input();
   }
 
-  Bool_type
+  Boolean
   is_interior_in_conflict_both_ps_s(const Site_2& sp, const Site_2& sq,
 				    const Site_2& r, const Site_2& s,
 				    const Site_2& st, Method_tag ) const
@@ -258,13 +258,13 @@ private:
       Oriented_side opqr = vpqr.oriented_side(lqperp);
       Oriented_side oqps = vqps.oriented_side(lqperp);
 
-      Bool_type on_different_parabola_arcs =
+      Boolean   on_different_parabola_arcs =
 	 ((opqr == ON_NEGATIVE_SIDE) & (oqps == ON_POSITIVE_SIDE)) |
 	 ((opqr == ON_POSITIVE_SIDE) & (oqps == ON_NEGATIVE_SIDE));
 
       //if ( !on_different_parabola_arcs ) { return true; }
       if (certainly( !on_different_parabola_arcs ) ) { return true; }
-      if (! is_certain( !on_different_parabola_arcs ) ) { return indeterminate<Bool_type>(); }
+      if (! is_certain( !on_different_parabola_arcs ) ) { return indeterminate<Boolean>(); }
 
       Site_2 t1;
       if ( same_points(sp, st.source_site()) ) {
@@ -336,7 +336,7 @@ private:
 	           (o_l_qps == ON_POSITIVE_SIDE) ) )
 	return true;
     if (! is_certain((o_l_pqr == -o_l_qps) & (o_l_pqr != ZERO)))
-        return indeterminate<Bool_type>();
+        return indeterminate<Boolean>();
 
     //>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
     //>>>>>>>>>> HERE I NEED TO CHECK THE BOUNDARY CASES <<<<<<
@@ -346,11 +346,11 @@ private:
     Oriented_side opqr = vpqr.oriented_side(lqperp);
     Oriented_side oqps = vqps.oriented_side(lqperp);
 
-    Bool_type on_different_parabola_arcs = (opqr == -oqps) & (opqr != ZERO);
+    Boolean   on_different_parabola_arcs = (opqr == -oqps) & (opqr != ZERO);
 
     // if ( !on_different_parabola_arcs ) { return true; }
     if (certainly( !on_different_parabola_arcs ) ) { return true; }
-    if (! is_certain( !on_different_parabola_arcs ) ) { return indeterminate<Bool_type>(); }
+    if (! is_certain( !on_different_parabola_arcs ) ) { return indeterminate<Boolean>(); }
       
     Homogeneous_point_2 pv = projection_on_line(lq, p);
     Homogeneous_point_2 hp(p);
@@ -374,7 +374,7 @@ private:
 
   //--------------------------------------------------------------------
 
-  Bool_type
+  Boolean
   is_interior_in_conflict_both_sp(const Site_2& p, const Site_2& q,
 				  const Site_2& r, const Site_2& s,
 				  const Site_2& t, Method_tag tag) const
@@ -640,10 +640,10 @@ private:
   //------------------------------------------------------------------------
 
 public:
-  typedef Bool_type         result_type;
+  typedef Boolean           result_type;
   typedef Site_2            argument_type;
 
-  Bool_type operator()(const Site_2& p, const Site_2& q, const Site_2& r,
+  Boolean   operator()(const Site_2& p, const Site_2& q, const Site_2& r,
 		       const Site_2& s, const Site_2& t, Sign sgn) const
   {
     if ( sgn == POSITIVE ) {
@@ -656,7 +656,7 @@ public:
   }
 
 
-  Bool_type operator()(const Site_2& p, const Site_2& q, const Site_2& ,
+  Boolean   operator()(const Site_2& p, const Site_2& q, const Site_2& ,
 		       const Site_2& t, Sign sgn) const
   {
     if ( t.is_point() ) {
@@ -679,7 +679,7 @@ public:
     return ( p_is_endpoint && q_is_endpoint );
   }
 
-  Bool_type operator()(const Site_2& p, const Site_2& q, const Site_2& t,
+  Boolean   operator()(const Site_2& p, const Site_2& q, const Site_2& t,
 		       Sign ) const
   {
     if ( p.is_segment() || q.is_segment()) {
diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Infinite_edge_interior_conflict_C2.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Infinite_edge_interior_conflict_C2.h
index b60d92fa73e..a375c5b5b0e 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Infinite_edge_interior_conflict_C2.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Infinite_edge_interior_conflict_C2.h
@@ -37,11 +37,11 @@ class Infinite_edge_interior_conflict_C2
 public:
   typedef typename K::Site_2           Site_2;
   typedef typename K::RT               RT;
-  typedef typename K::Bool_type        Bool_type;
+  typedef typename K::Boolean          Boolean;
   typedef Are_same_points_C2<K>        Are_same_points_2;
   typedef Are_same_segments_C2<K>      Are_same_segments_2;
 
-  typedef Bool_type                    result_type;
+  typedef Boolean                      result_type;
   struct argument_type {};
 
 private:
@@ -49,7 +49,7 @@ private:
   Are_same_segments_2  same_segments;
 
 public:
-  Bool_type operator()(const Site_2& q, const Site_2& s, const Site_2& r,
+  Boolean   operator()(const Site_2& q, const Site_2& s, const Site_2& r,
 		       const Site_2& t, Sign sgn) const
   {
     if ( t.is_segment() ) {
diff --git a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Is_degenerate_edge_C2.h b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Is_degenerate_edge_C2.h
index df6b6516e48..6d64f74ea48 100644
--- a/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Is_degenerate_edge_C2.h
+++ b/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2/Is_degenerate_edge_C2.h
@@ -34,7 +34,7 @@ class Is_degenerate_edge_C2
 {
 public:
   typedef typename K::Site_2      Site_2;
-  typedef typename K::Bool_type   Bool_type;
+  typedef typename K::Boolean     Boolean;
 
 private:
   typedef Voronoi_vertex_C2<K,Method_tag>  Voronoi_vertex_2;
@@ -50,10 +50,10 @@ private:
   }
 
 public:
-  typedef Bool_type     result_type;
+  typedef Boolean       result_type;
   typedef Site_2        argument_type;
 
-  Bool_type operator()(const Site_2& p, const Site_2& q,
+  Boolean   operator()(const Site_2& p, const Site_2& q,
 		       const Site_2& r, const Site_2& s) const
   {
     Voronoi_vertex_2 vpqr(p, q, r);