Make Expression.subs() always return a LinExpr instance

[linpy.git] / doc / tutorial.rst

diff --git a/doc/tutorial.rst b/doc/tutorial.rst
index 8598cbd..67aecd6 100644 (file)
--- a/doc/tutorial.rst
+++ b/doc/tutorial.rst
@@ -4,8 +4,14 @@
 Tutorial
 ========
 
 Tutorial
 ========
 

-Polyhedra
----------
+This section a short introduction to some of LinPy's features.
+For a comprehensive description of its functionalities, please consult the :ref:`reference`.
+
+
+.. _tutorial_polyhedra:
+
+Z-Polyhedra
+-----------

 
 The following example shows how we can manipulate polyhedra using LinPy.
 Let us define two square polyhedra, corresponding to the sets ``square1 = {(x, y) | 0 <= x <= 2, 0 <= y <= 2}`` and ``square2 = {(x, y) | 2 <= x <= 4, 2 <= y <= 4}``.
 
 The following example shows how we can manipulate polyhedra using LinPy.
 Let us define two square polyhedra, corresponding to the sets ``square1 = {(x, y) | 0 <= x <= 2, 0 <= y <= 2}`` and ``square2 = {(x, y) | 2 <= x <= 4, 2 <= y <= 4}``.


@@ -18,31 +24,32 @@ Then, we can build the :class:`Polyhedron` object ``square1`` from its constrain
 
 >>> square1 = Le(0, x, 2) & Le(0, y, 2)
 >>> square1
 
 >>> square1 = Le(0, x, 2) & Le(0, y, 2)
 >>> square1

-And(Ge(x, 0), Ge(-x + 2, 0), Ge(y, 0), Ge(-y + 2, 0))
+And(0 <= x, x <= 2, 0 <= y, y <= 2)

 
 LinPy provides comparison functions :func:`Lt`, :func:`Le`, :func:`Eq`, :func:`Ne`, :func:`Ge` and :func:`Gt` to build constraints, and logical operators :func:`And`, :func:`Or`, :func:`Not` to combine them.
 Alternatively, a polyhedron can be built from a string:
 
 >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
 >>> square2
 
 LinPy provides comparison functions :func:`Lt`, :func:`Le`, :func:`Eq`, :func:`Ne`, :func:`Ge` and :func:`Gt` to build constraints, and logical operators :func:`And`, :func:`Or`, :func:`Not` to combine them.
 Alternatively, a polyhedron can be built from a string:
 
 >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
 >>> square2

-And(Ge(x - 1, 0), Ge(-x + 3, 0), Ge(y - 1, 0), Ge(-y + 3, 0))
+And(1 <= x, x <= 3, 1 <= y, y <= 3)

 
 The usual polyhedral operations are available, including intersection:
 
 
 The usual polyhedral operations are available, including intersection:
 

->>> inter = square1.intersection(square2)
+>>> inter = square1.intersection(square2) # or square1 & square2

 >>> inter
 >>> inter

-And(Ge(x - 1, 0), Ge(-x + 2, 0), Ge(y - 1, 0), Ge(-y + 2, 0))
+And(1 <= x, x <= 2, 1 <= y, y <= 2)

 
 convex union:
 
 >>> hull = square1.convex_union(square2)
 >>> hull
 
 convex union:
 
 >>> hull = square1.convex_union(square2)
 >>> hull

-And(Ge(x, 0), Ge(y, 0), Ge(-x + y + 2, 0), Ge(x - y + 2, 0), Ge(-x + 3, 0), Ge(-y + 3, 0))
+And(0 <= x, 0 <= y, x <= y + 2, y <= x + 2, x <= 3, y <= 3)

 
 and projection:
 
 
 and projection:
 

->>> square1.project([y])
-And(Ge(x, 0), Ge(-x + 2, 0))
+>>> proj = square1.project([y])
+>>> proj
+And(0 <= x, x <= 2)

 
 Equality and inclusion tests are also provided.
 Special values :data:`Empty` and :data:`Universe` represent the empty and universe polyhedra.
 
 Equality and inclusion tests are also provided.
 Special values :data:`Empty` and :data:`Universe` represent the empty and universe polyhedra.


@@ -53,6 +60,8 @@ True
 False
 
 
 False
 
 

+.. _tutorial_domains:
+

 Domains
 -------
 
 Domains
 -------
 


@@ -60,20 +69,22 @@ LinPy is also able to manipulate polyhedral *domains*, that is, unions of polyhe
 An example of domain is the set union (as opposed to convex union) of polyhedra ``square1`` and ``square2``.
 The result is a :class:`Domain` object.
 
 An example of domain is the set union (as opposed to convex union) of polyhedra ``square1`` and ``square2``.
 The result is a :class:`Domain` object.
 

->>> union = square1 | square2
+>>> union = square1.union(square2) # or square1 | square2

 >>> union
 >>> union

-Or(And(Ge(-x + 2, 0), Ge(x, 0), Ge(-y + 2, 0), Ge(y, 0)), And(Ge(-x + 3, 0), Ge(x - 1, 0), Ge(-y + 3, 0), Ge(y - 1, 0)))
+Or(And(x <= 2, 0 <= x, y <= 2, 0 <= y), And(x <= 3, 1 <= x, y <= 3, 1 <= y))

 >>> union <= hull
 True
 
 Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.
 
 >>> union <= hull
 True
 
 Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.
 

->>> diff = square1 - square2
+>>> diff = square1.difference(square2) # or square1 - square2

 >>> diff
 >>> diff

-Or(And(Eq(x, 0), Ge(y, 0), Ge(-y + 2, 0)), And(Eq(y, 0), Ge(x - 1, 0), Ge(-x + 2, 0)))
+Or(And(x == 0, 0 <= y, y <= 2), And(y == 0, 1 <= x, x <= 2))

 >>> ~square1
 >>> ~square1

-Or(Ge(-x - 1, 0), Ge(x - 3, 0), And(Ge(x, 0), Ge(-x + 2, 0), Ge(-y - 1, 0)), And(Ge(x, 0), Ge(-x + 2, 0), Ge(y - 3, 0)))
+Or(x + 1 <= 0, 3 <= x, And(0 <= x, x <= 2, y + 1 <= 0), And(0 <= x, x <= 2, 3 <= y))
+

 
 

+.. _tutorial_plot:

 
 Plotting
 --------
 
 Plotting
 --------


@@ -95,7 +106,7 @@ Also, keyword arguments can be passed such as color and the degree of transparen
 .. figure:: images/union.jpg
     :align:  center
 
 .. figure:: images/union.jpg
     :align:  center
 

-3D plots are also supported.
+3D plots are also supported:

 
 >>> import matplotlib.pyplot as plt
 >>> from matplotlib import pylab
 
 >>> import matplotlib.pyplot as plt
 >>> from matplotlib import pylab