Minor improvements in Makefile

[linpy.git] / doc / reference.rst

diff --git a/doc/reference.rst b/doc/reference.rst
index e6f291d..56986c5 100644 (file)
--- a/doc/reference.rst
+++ b/doc/reference.rst
@@ -164,6 +164,8 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl
     .. method:: __eq__(expr)
 
         Test whether two linear expressions are equal.
+        Unlike methods :meth:`LinExpr.__lt__`, :meth:`LinExpr.__le__`, :meth:`LinExpr.__ge__`, :meth:`LinExpr.__gt__`, the result is a boolean value, not a polyhedron.
+        To express that two linear expressions are equal or not equal, use functions :func:`Eq` and :func:`Ne` instead.
 
     As explained below, it is possible to create polyhedra from linear expressions using comparison methods.
 
@@ -241,6 +243,7 @@ Polyhedra
 
 A *convex polyhedron* (or simply "polyhedron") is the space defined by a system of linear equalities and inequalities.
 This space can be unbounded.
+A *Z-polyhedron* (simply called "polyhedron" in LinPy) is the set of integer points in a convex polyhedron.
 
 .. class:: Polyhedron(equalities, inequalities)
            Polyhedron(string)
@@ -250,27 +253,27 @@ This space can be unbounded.
     For example, the polyhedron ``0 <= x <= 2, 0 <= y <= 2`` can be constructed with:
 
     >>> x, y = symbols('x y')
-    >>> square = Polyhedron([], [x, 2 - x, y, 2 - y])
-    >>> square
+    >>> square1 = Polyhedron([], [x, 2 - x, y, 2 - y])
+    >>> square1
     And(0 <= x, x <= 2, 0 <= y, y <= 2)
 
     It may be easier to use comparison operators :meth:`LinExpr.__lt__`, :meth:`LinExpr.__le__`, :meth:`LinExpr.__ge__`, :meth:`LinExpr.__gt__`, or functions :func:`Lt`, :func:`Le`, :func:`Eq`, :func:`Ge` and :func:`Gt`, using one of the following instructions:
 
     >>> x, y = symbols('x y')
-    >>> square = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
-    >>> square = Le(0, x, 2) & Le(0, y, 2)
+    >>> square1 = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
+    >>> square1 = Le(0, x, 2) & Le(0, y, 2)
 
     It is also possible to build a polyhedron from a string.
 
-    >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+    >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
 
     Finally, a polyhedron can be constructed from a :class:`GeometricObject` instance, calling the :meth:`GeometricObject.aspolyedron` method.
     This way, it is possible to compute the polyhedral hull of a :class:`Domain` instance, i.e., the convex hull of two polyhedra:
 
-    >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
-    >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
-    >>> Polyhedron(square | square2)
-    And(x <= 4, 0 <= x, y <= 4, 0 <= y, x <= y + 2, y <= x + 2)
+    >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+    >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
+    >>> Polyhedron(square1 | square2)
+    And(0 <= x, 0 <= y, x <= y + 2, y <= x + 2, x <= 3, y <= 3)
 
     A polyhedron is a :class:`Domain` instance, and, therefore, inherits the functionalities of this class.
     It is also a :class:`GeometricObject` instance.
@@ -328,22 +331,20 @@ Unlike polyhedra, domains allow exact computation of union, subtraction and comp
 
     Return a domain from a sequence of polyhedra.
 
-    >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
-    >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
-    >>> dom = Domain(square, square2)
+    >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+    >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
+    >>> dom = Domain(square1, square2)
     >>> dom
-    Or(And(x <= 2, 0 <= x, y <= 2, 0 <= y), And(x <= 4, 2 <= x, y <= 4, 2 <= y))
+    Or(And(x <= 2, 0 <= x, y <= 2, 0 <= y), And(x <= 3, 1 <= x, y <= 3, 1 <= y))
 
-    It is also possible to build domains from polyhedra using arithmetic operators :meth:`Domain.__and__`, :meth:`Domain.__or__` or functions :func:`And` and :func:`Or`, using one of the following instructions:
+    It is also possible to build domains from polyhedra using arithmetic operators :meth:`Domain.__or__`, :meth:`Domain.__invert__` or functions :func:`Or` and :func:`Not`, using one of the following instructions:
 
-    >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
-    >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
-    >>> dom = square | square2
-    >>> dom = Or(square, square2)
+    >>> dom = square1 | square2
+    >>> dom = Or(square1, square2)
 
     Alternatively, a domain can be built from a string:
 
-    >>> dom = Domain('0 <= x <= 2, 0 <= y <= 2; 2 <= x <= 4, 2 <= y <= 4')
+    >>> dom = Domain('0 <= x <= 2, 0 <= y <= 2; 1 <= x <= 3, 1 <= y <= 3')
 
     Finally, a domain can be built from a :class:`GeometricObject` instance, calling the :meth:`GeometricObject.asdomain` method.