X-Git-Url: https://svn.cri.ensmp.fr/git/linpy.git/blobdiff_plain/af70cb24e2db153fa0afad42296cc1528b82b2a2..d1c9ff71cf2d59fa6c56665d1569d16d3f1428a6:/doc/reference.rst?ds=inline

diff --git a/doc/reference.rst b/doc/reference.rst
index 4e71658..56986c5 100644
--- a/doc/reference.rst
+++ b/doc/reference.rst
@@ -1,7 +1,12 @@
 
+.. _reference:
+
 Module Reference
 ================
 
+
+.. _reference_symbols:
+
 Symbols
 -------
 
@@ -67,6 +72,8 @@ This is achieved using ``Dummy('x')``.
     True
 
 
+.. _reference_linexprs:
+
 Linear Expressions
 ------------------
 
@@ -77,12 +84,12 @@ Linear expressions are generally built using overloaded operators.
 For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :class:`LinExpr`.
 
 .. class:: LinExpr(coefficients=None, constant=0)
-              LinExpr(string)
+           LinExpr(string)
 
     Return a linear expression from a dictionary or a sequence, that maps symbols to their coefficients, and a constant term.
     The coefficients and the constant term must be rational numbers.
 
-    For example, the linear expression ``x + 2y + 1`` can be constructed using one of the following instructions:
+    For example, the linear expression ``x + 2*y + 1`` can be constructed using one of the following instructions:
 
     >>> x, y = symbols('x y')
     >>> LinExpr({x: 1, y: 2}, 1)
@@ -95,7 +102,7 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl
 
     Alternatively, linear expressions can be constructed from a string:
 
-    >>> LinExpr('x + 2*y + 1')
+    >>> LinExpr('x + 2y + 1')
 
     :class:`LinExpr` instances are hashable, and should be treated as immutable.
 
@@ -157,6 +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.
 
@@ -170,8 +179,7 @@ For example, if ``x`` is a :class:`Symbol`, then ``x + 1`` is an instance of :cl
 
         >>> x, y = symbols('x y')
         >>> x < y
-        Le(x - y + 1, 0)
-
+        x + 1 <= y
 
     .. method:: scaleint()
 
@@ -214,7 +222,7 @@ Apart from :mod:`Symbol`, a particular case of linear expressions are rational v
 They are implemented by the :class:`Rational` class, that inherits from both :class:`LinExpr` and :class:`fractions.Fraction` classes.
 
 .. class:: Rational(numerator, denominator=1)
-              Rational(string)
+           Rational(string)
 
     The first version requires that the *numerator* and *denominator* are instances of :class:`numbers.Rational` and returns a new :class:`Rational` instance with the value ``numerator/denominator``.
     If the denominator is ``0``, it raises a :exc:`ZeroDivisionError`.
@@ -227,38 +235,45 @@ They are implemented by the :class:`Rational` class, that inherits from both :cl
 
     See the documentation of :class:`fractions.Fraction` for more information and examples.
 
+
+.. _reference_polyhedra:
+
 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)
-              Polyhedron(geometric object)
+           Polyhedron(string)
+           Polyhedron(geometric object)
 
     Return a polyhedron from two sequences of linear expressions: *equalities* is a list of expressions equal to ``0``, and *inequalities* is a list of expressions greater or equal to ``0``.
     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])
+    >>> 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)
+    >>> 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.
@@ -278,10 +293,20 @@ This space can be unbounded.
         The tuple of constraints, i.e., equalities and inequalities.
         This is semantically equivalent to: ``equalities + inequalities``.
 
+    .. method:: convex_union(polyhedron[, ...])
+
+        Return the convex union of two or more polyhedra.
+
+    .. method:: asinequalities()
+
+        Express the polyhedron using inequalities, given as a list of expressions greater or equal to 0.
+
     .. method:: widen(polyhedron)
 
         Compute the *standard widening* of two polyhedra, à la Halbwachs.
 
+        In its current implementation, this method is slow and should not be used on large polyhedra.
+
 
 .. data:: Empty
 
@@ -291,32 +316,35 @@ This space can be unbounded.
 
     The universe polyhedron, whose set of constraints is always satisfiable, i.e. is empty.
 
+
+.. _reference_domains:
+
 Domains
 -------
 
 A *domain* is a union of polyhedra.
-Unlike polyhedra, domains allow exact computation of union and complementary operations.
+Unlike polyhedra, domains allow exact computation of union, subtraction and complementary operations.
 
 .. class:: Domain(*polyhedra)
-              Domain(string)
-              Domain(geometric object)
+           Domain(string)
+           Domain(geometric object)
 
     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 <= 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.
 
@@ -473,6 +501,8 @@ Unlike polyhedra, domains allow exact computation of union and complementary ope
         Convert the domain to a sympy expression.
 
 
+.. _reference_operators:
+
 Comparison and Logic Operators
 ------------------------------
 
@@ -493,7 +523,7 @@ The following functions create :class:`Polyhedron` or :class:`Domain` instances
 .. function:: Ne(expr1, expr2[, expr3, ...])
 
     Create the domain such that ``expr1 != expr2 != expr3 ...``.
-    The result is a :class:`Domain`, not a :class:`Polyhedron`.
+    The result is a :class:`Domain` object, not a :class:`Polyhedron`.
 
 .. function:: Ge(expr1, expr2[, expr3, ...])
 
@@ -505,19 +535,21 @@ The following functions create :class:`Polyhedron` or :class:`Domain` instances
 
 The following functions combine :class:`Polyhedron` or :class:`Domain` instances using logic operators:
 
-.. function:: Or(domain1, domain2[, ...])
-
-    Create the union domain of the domains given in arguments.
-
 .. function:: And(domain1, domain2[, ...])
 
     Create the intersection domain of the domains given in arguments.
 
+.. function:: Or(domain1, domain2[, ...])
+
+    Create the union domain of the domains given in arguments.
+
 .. function:: Not(domain)
 
     Create the complementary domain of the domain given in argument.
 
 
+.. _reference_geometry:
+
 Geometric Objects
 -----------------