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# Cython wrapper for the Parma Polyhedra Library (PPL)

Cython wrapper for the Parma Polyhedra Library (PPL)

The Parma Polyhedra Library (PPL) is a library for polyhedral computations over \(\QQ\). This interface tries to reproduce the C++ API as faithfully as possible in Cython/Sage. For example, the following C++ excerpt:

    Variable x(0);
    Variable y(1);
    Constraint_System cs;
    cs.insert(x >= 0);
    cs.insert(x <= 3);
    cs.insert(y >= 0);
    cs.insert(y <= 3);
    C_Polyhedron poly_from_constraints(cs);

translates into:

    sage: from sage.libs.ppl import Variable, Constraint_System, C_Polyhedron
    sage: x = Variable(0)
    sage: y = Variable(1)
    sage: cs = Constraint_System()
    sage: cs.insert(x >= 0)
    sage: cs.insert(x <= 3)
    sage: cs.insert(y >= 0)
    sage: cs.insert(y <= 3)
    sage: poly_from_constraints = C_Polyhedron(cs)

The same polyhedron constructed from generators:

    sage: from sage.libs.ppl import Variable, Generator_System, C_Polyhedron, point
    sage: gs = Generator_System()
    sage: gs.insert(point(0*x + 0*y))
    sage: gs.insert(point(0*x + 3*y))
    sage: gs.insert(point(3*x + 0*y))
    sage: gs.insert(point(3*x + 3*y))
    sage: poly_from_generators = C_Polyhedron(gs)

Rich comparisons test equality/inequality and strict/non-strict containment:

    sage: poly_from_generators == poly_from_constraints
    True
    sage: poly_from_generators >= poly_from_constraints
    True
    sage: poly_from_generators <  poly_from_constraints
    False
    sage: poly_from_constraints.minimized_generators()
    Generator_System {point(0/1, 0/1), point(0/1, 3/1), point(3/1, 0/1), point(3/1, 3/1)}
    sage: poly_from_constraints.minimized_constraints()
    Constraint_System {-x0+3>=0, -x1+3>=0, x0>=0, x1>=0}

As we see above, the library is generally easy to use. There are a few pitfalls that are not entirely obvious without consulting the documentation, in particular:

- There are no vectors used to describe `Generator` (points, closure points, rays, lines) or `Constraint` (strict inequalities, non-strict inequalities, or equations). Coordinates are always specified via linear polynomials in `Variable`

- All coordinates of rays and lines as well as all coefficients of constraint relations are (arbitrary precision) integers. Only the generators `point()` and `closure_point()` allow one to specify an overall divisor of the otherwise integral coordinates. For example:

      sage: from sage.libs.ppl import Variable, point
      sage: x = Variable(0); y = Variable(1)
      sage: p = point( 2*x+3*y, 5 ); p
      point(2/5, 3/5)
      sage: p.coefficient(x)
      2
      sage: p.coefficient(y)
      3
      sage: p.divisor()
      5

- PPL supports (topologically) closed polyhedra (`C_Polyhedron`) as well as not neccesarily closed polyhedra (`NNC_Polyhedron`). Only the latter allows closure points (=points of the closure but not of the actual polyhedron) and strict inequalities (`>` and `<`)

The naming convention for the C++ classes is that they start with `PPL_`, for example, the original `Linear_Expression` becomes `PPL_Linear_Expression`. The Python wrapper has the same name as the original library class, that is, just `Linear_Expression`. In short:

- If you are using the Python wrapper (if in doubt: thats you), then you use the same names as the PPL C++ class library.
- If you are writing your own Cython code, you can access the underlying C++ classes by adding the prefix `PPL_`.

Finally, PPL is fast. For example, here is the permutahedron of 5 basis vectors:

    sage: from sage.libs.ppl import Variable, Generator_System, point, C_Polyhedron
    sage: basis = range(0,5)
    sage: x = [ Variable(i) for i in basis ]
    sage: gs = Generator_System();
    sage: for coeff in Permutations(basis):
    ....:    gs.insert(point( sum( (coeff[i]+1)*x[i] for i in basis ) ))
    sage: C_Polyhedron(gs)
    A 4-dimensional polyhedron in QQ^5 defined as the convex hull of 120 points

The above computation (using PPL) finishes without noticeable delay (timeit measures it to be 90 microseconds on sage.math). Below we do the same computation with cddlib, which needs more than 3 seconds on the same hardware:

    sage: basis = range(0,5)
    sage: gs = [ tuple(coeff) for coeff in Permutations(basis) ]
    sage: Polyhedron(vertices=gs, backend='cdd')  # long time (3s on sage.math, 2011)
    A 4-dimensional polyhedron in QQ^5 defined as the convex hull of 120 vertices

DIFFERENCES VS. C++

Since Python and C++ syntax are not always compatible, there are necessarily some differences. The main ones are:

- The `Linear_Expression` also accepts an iterable as input for the homogeneous cooefficients.
- `Polyhedron` and its subclasses as well as `Generator_System` and `Constraint_System` can be set immutable via a `set_immutable()` method. This is the analog of declaring a C++ instance `const`. All other classes are immutable by themselves.

AUTHORS:

- Volker Braun (2010-10-08): initial version.
- Risan (2012-02-19): extension for MIP_Problem class

class `sage.libs.ppl.``C_Polyhedron`

  Bases: `sage.libs.ppl.Polyhedron`

  Wrapper for PPL’s `C_Polyhedron` class.

  An object of the class `C_Polyhedron` represents a topologically closed convex polyhedron in the vector space. See `NNC_Polyhedron` for more general (not necessarily closed) polyhedra.

  When building a closed polyhedron starting from a system of constraints, an exception is thrown if the system contains a strict inequality constraint. Similarly, an exception is thrown when building a closed polyhedron starting from a system of generators containing a closure point.

  INPUT:

  - `arg` – the defining data of the polyhedron. Any one of the following is accepted:
    - A non-negative integer. Depending on `degenerate_element`, either the space-filling or the empty polytope in the given dimension `arg` is constructed.
    - A `Constraint_System`.
    - A `Generator_System`.
    - A single `Constraint`.
    - A single `Generator`.
    - A `C_Polyhedron`.
  - `degenerate_element` – string, either `'universe'` or `'empty'`. Only used if `arg` is an integer.

  OUTPUT:

  A `C_Polyhedron`.

  EXAMPLES:

      sage: from sage.libs.ppl import Constraint, Constraint_System, Generator, Generator_System, Variable, C_Polyhedron, point, ray
      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: C_Polyhedron( 5*x-2*y >=  x+y-1 )
      A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point, 1 ray, 1 line
      sage: cs = Constraint_System()
      sage: cs.insert( x >= 0 )
      sage: cs.insert( y >= 0 )
      sage: C_Polyhedron(cs)
      A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point, 2 rays
      sage: C_Polyhedron( point(x+y) )
      A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
      sage: gs = Generator_System()
      sage: gs.insert( point(-x-y) )
      sage: gs.insert( ray(x) )
      sage: C_Polyhedron(gs)
      A 1-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point, 1 ray

  The empty and universe polyhedra are constructed like this:

      sage: C_Polyhedron(3, 'empty')
      The empty polyhedron in QQ^3
      sage: C_Polyhedron(3, 'empty').constraints()
      Constraint_System {-1==0}
      sage: C_Polyhedron(3, 'universe')
      The space-filling polyhedron in QQ^3
      sage: C_Polyhedron(3, 'universe').constraints()
      Constraint_System {}

  Note that, by convention, the generator system of a polyhedron is either empty or contains at least one point. In particular, if you define a polyhedron via a non-empty `Generator_System` it must contain a point (at any position). If you start with a single generator, this generator must be a point:

      sage: C_Polyhedron( ray(x) )
      Traceback (most recent call last):
      ...
      ValueError: PPL::C_Polyhedron::C_Polyhedron(gs):
      *this is an empty polyhedron and
      the non-empty generator system gs contains no points.

class `sage.libs.ppl.``Constraint`

  Bases: `object`

  Wrapper for PPL’s `Constraint` class.

  An object of the class `Constraint` is either:

  - an equality \(\sum_{i=0}^{n-1} a_i x_i + b = 0\)
  - a non-strict inequality \(\sum_{i=0}^{n-1} a_i x_i + b \geq 0\)
  - a strict inequality \(\sum_{i=0}^{n-1} a_i x_i + b > 0\)

  where \(n\) is the dimension of the space, \(a_i\) is the integer coefficient of variable \(x_i\), and \(b_i\) is the integer inhomogeneous term.

  INPUT/OUTPUT:

  You construct constraints by writing inequalities in `Linear_Expression`. Do not attempt to manually construct constraints.

  EXAMPLES:

      sage: from sage.libs.ppl import Constraint, Variable, Linear_Expression
      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: 5*x-2*y >  x+y-1
      4*x0-3*x1+1>0
      sage: 5*x-2*y >= x+y-1
      4*x0-3*x1+1>=0
      sage: 5*x-2*y == x+y-1
      4*x0-3*x1+1==0
      sage: 5*x-2*y <= x+y-1
      -4*x0+3*x1-1>=0
      sage: 5*x-2*y <  x+y-1
      -4*x0+3*x1-1>0
      sage: x > 0
      x0>0

  Special care is needed if the left hand side is a constant:

      sage: 0 == 1    # watch out!
      False
      sage: Linear_Expression(0) == 1
      -1==0

  `OK`()

    Check if all the invariants are satisfied.

    EXAMPLES:

        sage: from sage.libs.ppl import Linear_Expression, Variable
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: ineq = (3*x+2*y+1>=0)
        sage: ineq.OK()
        True

  `ascii_dump`()

    Write an ASCII dump to stderr.

    EXAMPLES:

        sage: sage_cmd  = 'from sage.libs.ppl import Linear_Expression, Variable\n'
        sage: sage_cmd += 'x = Variable(0)\n'
        sage: sage_cmd += 'y = Variable(1)\n'
        sage: sage_cmd += 'e = (3*x+2*y+1 > 0)\n'
        sage: sage_cmd += 'e.ascii_dump()\n'
        sage: from sage.tests.cmdline import test_executable
        sage: (out, err, ret) = test_executable(['sage', '-c', sage_cmd], timeout=100)  # long time, indirect doctest
        sage: print err  # long time
        size 4 1 3 2 -1 > (NNC)

  `coefficient`(v)

    Return the coefficient of the variable `v`.

    INPUT:

    - `v` – a `Variable`.

    OUTPUT:

    An integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: ineq = (3*x+1 > 0)
        sage: ineq.coefficient(x)
        3

  `coefficients`()

    Return the coefficients of the constraint.

    See also `coefficient()`.

    OUTPUT:

    A tuple of integers of length `space_dimension()`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0);  y = Variable(1)
        sage: ineq = ( 3*x+5*y+1 ==  2);  ineq
        3*x0+5*x1-1==0
        sage: ineq.coefficients()
        (3, 5)

  `inhomogeneous_term`()

    Return the inhomogeneous term of the constraint.

    OUTPUT:

    Integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: y = Variable(1)
        sage: ineq = ( 10+y > 9 )
        sage: ineq
        x1+1>0
        sage: ineq.inhomogeneous_term()
        1

  `is_equality`()

    Test whether `self` is an equality.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` is an equality constraint.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: (x==0).is_equality()
        True
        sage: (x>=0).is_equality()
        False
        sage: (x>0).is_equality()
        False

  `is_equivalent_to`(c)

    Test whether `self` and `c` are equivalent.

    INPUT:

    - `c` – a `Constraint`.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` and `c` are equivalent constraints.

    Note that constraints having different space dimensions are not equivalent. However, constraints having different types may nonetheless be equivalent, if they both are tautologies or inconsistent.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Linear_Expression
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: ( x>0 ).is_equivalent_to( Linear_Expression(0)<x )
        True
        sage: ( x>0 ).is_equivalent_to( 0*y<x )
        False
        sage: ( 0*x>1 ).is_equivalent_to( 0*x==-2 )
        True

  `is_inconsistent`()

    Test whether `self` is an inconsistent constraint, that is, always false.

    An inconsistent constraint can have either one of the following forms:

    - an equality: \(\sum 0 x_i + b = 0\) with \(b\not=0\),
    - a non-strict inequality: \(\sum 0 x_i + b \geq 0\) with \(b< 0\), or
    - a strict inequality: \(\sum 0 x_i + b > 0\) with \(b\leq 0\).

    OUTPUT:

    Boolean. Returns `True` if and only if `self` is an inconsistent constraint.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: (x==1).is_inconsistent()
        False
        sage: (0*x>=1).is_inconsistent()
        True

  `is_inequality`()

    Test whether `self` is an inequality.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` is an inequality constraint, either strict or non-strict.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: (x==0).is_inequality()
        False
        sage: (x>=0).is_inequality()
        True
        sage: (x>0).is_inequality()
        True

  `is_nonstrict_inequality`()

    Test whether `self` is a non-strict inequality.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` is an non-strict inequality constraint.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: (x==0).is_nonstrict_inequality()
        False
        sage: (x>=0).is_nonstrict_inequality()
        True
        sage: (x>0).is_nonstrict_inequality()
        False

  `is_strict_inequality`()

    Test whether `self` is a strict inequality.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` is an strict inequality constraint.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: (x==0).is_strict_inequality()
        False
        sage: (x>=0).is_strict_inequality()
        False
        sage: (x>0).is_strict_inequality()
        True

  `is_tautological`()

    Test whether `self` is a tautological constraint.

    A tautology can have either one of the following forms:

    - an equality: \(\sum 0 x_i + 0 = 0\),
    - a non-strict inequality: \(\sum 0 x_i + b \geq 0\) with \(b\geq 0\), or
    - a strict inequality: \(\sum 0 x_i + b > 0\) with \(b> 0\).

    OUTPUT:

    Boolean. Returns `True` if and only if `self` is a tautological constraint.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: (x==0).is_tautological()
        False
        sage: (0*x>=0).is_tautological()
        True

  `space_dimension`()

    Return the dimension of the vector space enclosing `self`.

    OUTPUT:

    Integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: (x>=0).space_dimension()
        1
        sage: (y==1).space_dimension()
        2

  `type`()

    Return the constraint type of `self`.

    OUTPUT:

    String. One of `'equality'`, `'nonstrict_inequality'`, or `'strict_inequality'`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: (x==0).type()
        'equality'
        sage: (x>=0).type()
        'nonstrict_inequality'
        sage: (x>0).type()
        'strict_inequality'

class `sage.libs.ppl.``Constraint_System`

  Bases: `sage.libs.ppl._mutable_or_immutable`

  Wrapper for PPL’s `Constraint_System` class.

  An object of the class Constraint_System is a system of constraints, i.e., a multiset of objects of the class Constraint. When inserting constraints in a system, space dimensions are automatically adjusted so that all the constraints in the system are defined on the same vector space.

  EXAMPLES:

      sage: from sage.libs.ppl import Constraint_System, Variable
      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: cs = Constraint_System( 5*x-2*y > 0 )
      sage: cs.insert( 6*x<3*y )
      sage: cs.insert( x >= 2*x-7*y )
      sage: cs
      Constraint_System {5*x0-2*x1>0, -2*x0+x1>0, -x0+7*x1>=0}

  `OK`()

    Check if all the invariants are satisfied.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: cs = Constraint_System( 3*x+2*y+1 <= 10 )
        sage: cs.OK()
        True

  `ascii_dump`()

    Write an ASCII dump to stderr.

    EXAMPLES:

        sage: sage_cmd  = 'from sage.libs.ppl import Constraint_System, Variable\n'
        sage: sage_cmd += 'x = Variable(0)\n'
        sage: sage_cmd += 'y = Variable(1)\n'
        sage: sage_cmd += 'cs = Constraint_System( 3*x > 2*y+1 )\n'
        sage: sage_cmd += 'cs.ascii_dump()\n'
        sage: from sage.tests.cmdline import test_executable
        sage: (out, err, ret) = test_executable(['sage', '-c', sage_cmd], timeout=100)  # long time, indirect doctest
        sage: print err  # long time
        topology NOT_NECESSARILY_CLOSED
        1 x 2 SPARSE (sorted)
        index_first_pending 1
        size 4 -1 3 -2 -1 > (NNC)

  `clear`()

    Removes all constraints from the constraint system and sets its space dimension to 0.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System
        sage: x = Variable(0)
        sage: cs = Constraint_System(x>0)
        sage: cs
        Constraint_System {x0>0}
        sage: cs.clear()
        sage: cs
        Constraint_System {}

  `empty`()

    Return `True` if and only if `self` has no constraints.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, point
        sage: x = Variable(0)
        sage: cs = Constraint_System()
        sage: cs.empty()
        True
        sage: cs.insert( x>0 )
        sage: cs.empty()
        False

  `has_equalities`()

    Tests whether `self` contains one or more equality constraints.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System
        sage: x = Variable(0)
        sage: cs = Constraint_System()
        sage: cs.insert( x>0 )
        sage: cs.insert( x<0 )
        sage: cs.has_equalities()
        False
        sage: cs.insert( x==0 )
        sage: cs.has_equalities()
        True

  `has_strict_inequalities`()

    Tests whether `self` contains one or more strict inequality constraints.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System
        sage: x = Variable(0)
        sage: cs = Constraint_System()
        sage: cs.insert( x>=0 )
        sage: cs.insert( x==-1 )
        sage: cs.has_strict_inequalities()
        False
        sage: cs.insert( x>0 )
        sage: cs.has_strict_inequalities()
        True

  `insert`(c)

    Insert `c` into the constraint system.

    INPUT:

    - `c` – a `Constraint`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System
        sage: x = Variable(0)
        sage: cs = Constraint_System()
        sage: cs.insert( x>0 )
        sage: cs
        Constraint_System {x0>0}

  `space_dimension`()

    Return the dimension of the vector space enclosing `self`.

    OUTPUT:

    Integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System
        sage: x = Variable(0)
        sage: cs = Constraint_System( x>0 )
        sage: cs.space_dimension()
        1

class `sage.libs.ppl.``Constraint_System_iterator`

  Bases: `object`

  Wrapper for PPL’s `Constraint_System::const_iterator` class.

  EXAMPLES:

      sage: from sage.libs.ppl import Constraint_System, Variable, Constraint_System_iterator
      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: cs = Constraint_System( 5*x < 2*y )
      sage: cs.insert( 6*x-3*y==0 )
      sage: cs.insert( x >= 2*x-7*y )
      sage: Constraint_System_iterator(cs).next()
      -5*x0+2*x1>0
      sage: list(cs)
      [-5*x0+2*x1>0, 2*x0-x1==0, -x0+7*x1>=0]

  `next`()

    x.next() -> the next value, or raise StopIteration

class `sage.libs.ppl.``Generator`

  Bases: `object`

  Wrapper for PPL’s `Generator` class.

  An object of the class Generator is one of the following:

  > - a line \(\ell = (a_0, \dots, a_{n-1})^T\)
  > - a ray \(r = (a_0, \dots, a_{n-1})^T\)
  > - a point \(p = (\tfrac{a_0}{d}, \dots, \tfrac{a_{n-1}}{d})^T\)
  > - a closure point \(c = (\tfrac{a_0}{d}, \dots, \tfrac{a_{n-1}}{d})^T\)

  where \(n\) is the dimension of the space and, for points and closure points, \(d\) is the divisor.

  INPUT/OUTPUT:

  Use the helper functions `line()`, `ray()`, `point()`, and `closure_point()` to construct generators. Analogous class methods are also available, see `Generator.line()`, `Generator.ray()`, `Generator.point()`, `Generator.closure_point()`. Do not attempt to construct generators manually.

  Note

  The generators are constructed from linear expressions. The inhomogeneous term is always silently discarded.

  EXAMPLES:

      sage: from sage.libs.ppl import Generator, Variable
      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: Generator.line(5*x-2*y)
      line(5, -2)
      sage: Generator.ray(5*x-2*y)
      ray(5, -2)
      sage: Generator.point(5*x-2*y, 7)
      point(5/7, -2/7)
      sage: Generator.closure_point(5*x-2*y, 7)
      closure_point(5/7, -2/7)

  `OK`()

    Check if all the invariants are satisfied.

    EXAMPLES:

        sage: from sage.libs.ppl import Linear_Expression, Variable
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: e = 3*x+2*y+1
        sage: e.OK()
        True

  `ascii_dump`()

    Write an ASCII dump to stderr.

    EXAMPLES:

        sage: sage_cmd  = 'from sage.libs.ppl import Linear_Expression, Variable, point\n'
        sage: sage_cmd += 'x = Variable(0)\n'
        sage: sage_cmd += 'y = Variable(1)\n'
        sage: sage_cmd += 'p = point(3*x+2*y)\n'
        sage: sage_cmd += 'p.ascii_dump()\n'
        sage: from sage.tests.cmdline import test_executable
        sage: (out, err, ret) = test_executable(['sage', '-c', sage_cmd], timeout=100)  # long time, indirect doctest
        sage: print err  # long time
        size 3 1 3 2 P (C)

  static `closure_point`(expression=0, divisor=1)

    Construct a closure point.

    A closure point is a point of the topological closure of a polyhedron that is not a point of the polyhedron itself.

    INPUT:

    - `expression` – a `Linear_Expression` or something convertible to it (`Variable` or integer).
    - `divisor` – an integer.

    OUTPUT:

    A new `Generator` representing the point.

    Raises a `ValueError` if ``divisor==0`.

    EXAMPLES:

        sage: from sage.libs.ppl import Generator, Variable
        sage: y = Variable(1)
        sage: Generator.closure_point(2*y+7, 3)
        closure_point(0/3, 2/3)
        sage: Generator.closure_point(y+7, 3)
        closure_point(0/3, 1/3)
        sage: Generator.closure_point(7, 3)
        closure_point()
        sage: Generator.closure_point(0, 0)
        Traceback (most recent call last):
        ...
        ValueError: PPL::closure_point(e, d):
        d == 0.

  `coefficient`(v)

    Return the coefficient of the variable `v`.

    INPUT:

    - `v` – a `Variable`.

    OUTPUT:

    An integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, line
        sage: x = Variable(0)
        sage: line = line(3*x+1)
        sage: line
        line(1)
        sage: line.coefficient(x)
        1

  `coefficients`()

    Return the coefficients of the generator.

    See also `coefficient()`.

    OUTPUT:

    A tuple of integers of length `space_dimension()`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, point
        sage: x = Variable(0);  y = Variable(1)
        sage: p = point(3*x+5*y+1, 2); p
        point(3/2, 5/2)
        sage: p.coefficients()
        (3, 5)

  `divisor`()

    If `self` is either a point or a closure point, return its divisor.

    OUTPUT:

    An integer. If `self` is a ray or a line, raises `ValueError`.

    EXAMPLES:

        sage: from sage.libs.ppl import Generator, Variable
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: point = Generator.point(2*x-y+5)
        sage: point.divisor()
        1
        sage: line = Generator.line(2*x-y+5)
        sage: line.divisor()
        Traceback (most recent call last):
        ...
        ValueError: PPL::Generator::divisor():
        *this is neither a point nor a closure point.

  `is_closure_point`()

    Test whether `self` is a closure point.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, point, closure_point, ray, line
        sage: x = Variable(0)
        sage: line(x).is_closure_point()
        False
        sage: ray(x).is_closure_point()
        False
        sage: point(x,2).is_closure_point()
        False
        sage: closure_point(x,2).is_closure_point()
        True

  `is_equivalent_to`(g)

    Test whether `self` and `g` are equivalent.

    INPUT:

    - `g` – a `Generator`.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` and `g` are equivalent generators.

    Note that generators having different space dimensions are not equivalent.

    EXAMPLES:

        sage: from sage.libs.ppl import Generator, Variable, point, line
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: point(2*x    , 2).is_equivalent_to( point(x) )
        True
        sage: point(2*x+0*y, 2).is_equivalent_to( point(x) )
        False
        sage: line(4*x).is_equivalent_to(line(x))
        True

  `is_line`()

    Test whether `self` is a line.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, point, closure_point, ray, line
        sage: x = Variable(0)
        sage: line(x).is_line()
        True
        sage: ray(x).is_line()
        False
        sage: point(x,2).is_line()
        False
        sage: closure_point(x,2).is_line()
        False

  `is_line_or_ray`()

    Test whether `self` is a line or a ray.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, point, closure_point, ray, line
        sage: x = Variable(0)
        sage: line(x).is_line_or_ray()
        True
        sage: ray(x).is_line_or_ray()
        True
        sage: point(x,2).is_line_or_ray()
        False
        sage: closure_point(x,2).is_line_or_ray()
        False

  `is_point`()

    Test whether `self` is a point.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, point, closure_point, ray, line
        sage: x = Variable(0)
        sage: line(x).is_point()
        False
        sage: ray(x).is_point()
        False
        sage: point(x,2).is_point()
        True
        sage: closure_point(x,2).is_point()
        False

  `is_ray`()

    Test whether `self` is a ray.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, point, closure_point, ray, line
        sage: x = Variable(0)
        sage: line(x).is_ray()
        False
        sage: ray(x).is_ray()
        True
        sage: point(x,2).is_ray()
        False
        sage: closure_point(x,2).is_ray()
        False

  static `line`(expression)

    Construct a line.

    INPUT:

    - `expression` – a `Linear_Expression` or something convertible to it (`Variable` or integer).

    OUTPUT:

    A new `Generator` representing the line.

    Raises a `ValueError` if the homogeneous part of ``expression` represents the origin of the vector space.

    EXAMPLES:

        sage: from sage.libs.ppl import Generator, Variable
        sage: y = Variable(1)
        sage: Generator.line(2*y)
        line(0, 1)
        sage: Generator.line(y)
        line(0, 1)
        sage: Generator.line(1)
        Traceback (most recent call last):
        ...
        ValueError: PPL::line(e):
        e == 0, but the origin cannot be a line.

  static `point`(expression=0, divisor=1)

    Construct a point.

    INPUT:

    - `expression` – a `Linear_Expression` or something convertible to it (`Variable` or integer).
    - `divisor` – an integer.

    OUTPUT:

    A new `Generator` representing the point.

    Raises a `ValueError` if ``divisor==0`.

    EXAMPLES:

        sage: from sage.libs.ppl import Generator, Variable
        sage: y = Variable(1)
        sage: Generator.point(2*y+7, 3)
        point(0/3, 2/3)
        sage: Generator.point(y+7, 3)
        point(0/3, 1/3)
        sage: Generator.point(7, 3)
        point()
        sage: Generator.point(0, 0)
        Traceback (most recent call last):
        ...
        ValueError: PPL::point(e, d):
        d == 0.

  static `ray`(expression)

    Construct a ray.

    INPUT:

    - `expression` – a `Linear_Expression` or something convertible to it (`Variable` or integer).

    OUTPUT:

    A new `Generator` representing the ray.

    Raises a `ValueError` if the homogeneous part of ``expression` represents the origin of the vector space.

    EXAMPLES:

        sage: from sage.libs.ppl import Generator, Variable
        sage: y = Variable(1)
        sage: Generator.ray(2*y)
        ray(0, 1)
        sage: Generator.ray(y)
        ray(0, 1)
        sage: Generator.ray(1)
        Traceback (most recent call last):
        ...
        ValueError: PPL::ray(e):
        e == 0, but the origin cannot be a ray.

  `space_dimension`()

    Return the dimension of the vector space enclosing `self`.

    OUTPUT:

    Integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, point
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: point(x).space_dimension()
        1
        sage: point(y).space_dimension()
        2

  `type`()

    Return the generator type of `self`.

    OUTPUT:

    String. One of `'line'`, `'ray'`, `'point'`, or `'closure_point'`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, point, closure_point, ray, line
        sage: x = Variable(0)
        sage: line(x).type()
        'line'
        sage: ray(x).type()
        'ray'
        sage: point(x,2).type()
        'point'
        sage: closure_point(x,2).type()
        'closure_point'

class `sage.libs.ppl.``Generator_System`

  Bases: `sage.libs.ppl._mutable_or_immutable`

  Wrapper for PPL’s `Generator_System` class.

  An object of the class Generator_System is a system of generators, i.e., a multiset of objects of the class Generator (lines, rays, points and closure points). When inserting generators in a system, space dimensions are automatically adjusted so that all the generators in the system are defined on the same vector space. A system of generators which is meant to define a non-empty polyhedron must include at least one point: the reason is that lines, rays and closure points need a supporting point (lines and rays only specify directions while closure points only specify points in the topological closure of the NNC polyhedron).

  EXAMPLES:

      sage: from sage.libs.ppl import Generator_System, Variable, line, ray, point, closure_point
      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: gs = Generator_System( line(5*x-2*y) )
      sage: gs.insert( ray(6*x-3*y) )
      sage: gs.insert( point(2*x-7*y, 5) )
      sage: gs.insert( closure_point(9*x-1*y, 2) )
      sage: gs
      Generator_System {line(5, -2), ray(2, -1), point(2/5, -7/5), closure_point(9/2, -1/2)}

  `OK`()

    Check if all the invariants are satisfied.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Generator_System, point
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: gs = Generator_System( point(3*x+2*y+1) )
        sage: gs.OK()
        True

  `ascii_dump`()

    Write an ASCII dump to stderr.

    EXAMPLES:

        sage: sage_cmd  = 'from sage.libs.ppl import Generator_System, point, Variable\n'
        sage: sage_cmd += 'x = Variable(0)\n'
        sage: sage_cmd += 'y = Variable(1)\n'
        sage: sage_cmd += 'gs = Generator_System( point(3*x+2*y+1) )\n'
        sage: sage_cmd += 'gs.ascii_dump()\n'
        sage: from sage.tests.cmdline import test_executable
        sage: (out, err, ret) = test_executable(['sage', '-c', sage_cmd], timeout=100)  # long time, indirect doctest
        sage: print err  # long time
        topology NECESSARILY_CLOSED
        1 x 2 SPARSE (sorted)
        index_first_pending 1
        size 3 1 3 2 P (C)

  `clear`()

    Removes all generators from the generator system and sets its space dimension to 0.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Generator_System, point
        sage: x = Variable(0)
        sage: gs = Generator_System( point(3*x) ); gs
        Generator_System {point(3/1)}
        sage: gs.clear()
        sage: gs
        Generator_System {}

  `empty`()

    Return `True` if and only if `self` has no generators.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Generator_System, point
        sage: x = Variable(0)
        sage: gs = Generator_System()
        sage: gs.empty()
        True
        sage: gs.insert( point(-3*x) )
        sage: gs.empty()
        False

  `insert`(g)

    Insert `g` into the generator system.

    The number of space dimensions of `self` is increased, if needed.

    INPUT:

    - `g` – a `Generator`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Generator_System, point
        sage: x = Variable(0)
        sage: gs = Generator_System( point(3*x) )
        sage: gs.insert( point(-3*x) )
        sage: gs
        Generator_System {point(3/1), point(-3/1)}

  `space_dimension`()

    Return the dimension of the vector space enclosing `self`.

    OUTPUT:

    Integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Generator_System, point
        sage: x = Variable(0)
        sage: gs = Generator_System( point(3*x) )
        sage: gs.space_dimension()
        1

class `sage.libs.ppl.``Generator_System_iterator`

  Bases: `object`

  Wrapper for PPL’s `Generator_System::const_iterator` class.

  EXAMPLES:

      sage: from sage.libs.ppl import Generator_System, Variable, line, ray, point, closure_point, Generator_System_iterator
      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: gs = Generator_System( line(5*x-2*y) )
      sage: gs.insert( ray(6*x-3*y) )
      sage: gs.insert( point(2*x-7*y, 5) )
      sage: gs.insert( closure_point(9*x-1*y, 2) )
      sage: Generator_System_iterator(gs).next()
      line(5, -2)
      sage: list(gs)
      [line(5, -2), ray(2, -1), point(2/5, -7/5), closure_point(9/2, -1/2)]

  `next`()

    x.next() -> the next value, or raise StopIteration

class `sage.libs.ppl.``Linear_Expression`

  Bases: `object`

  Wrapper for PPL’s `PPL_Linear_Expression` class.

  INPUT:

  The constructor accepts zero, one, or two arguments.

  If there are two arguments `Linear_Expression(a,b)`, they are interpreted as

  - `a` – an iterable of integer coefficients, for example a list.
  - `b` – an integer. The inhomogeneous term.

  A single argument `Linear_Expression(arg)` is interpreted as

  - `arg` – something that determines a linear expression. Possibilities are:
    - a `Variable`: The linear expression given by that variable.
    - a `Linear_Expression`: The copy constructor.
    - an integer: Constructs the constant linear expression.

  No argument is the default constructor and returns the zero linear expression.

  OUTPUT:

  A `Linear_Expression`

  EXAMPLES:

      sage: from sage.libs.ppl import Variable, Linear_Expression
      sage: Linear_Expression([1,2,3,4],5)
      x0+2*x1+3*x2+4*x3+5
      sage: Linear_Expression(10)
      10
      sage: Linear_Expression()
      0
      sage: Linear_Expression(10).inhomogeneous_term()
      10
      sage: x = Variable(123)
      sage: expr = x+1; expr
      x123+1
      sage: expr.OK()
      True
      sage: expr.coefficient(x)
      1
      sage: expr.coefficient( Variable(124) )
      0

  `OK`()

    Check if all the invariants are satisfied.

    EXAMPLES:

        sage: from sage.libs.ppl import Linear_Expression, Variable
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: e = 3*x+2*y+1
        sage: e.OK()
        True

  `all_homogeneous_terms_are_zero`()

    Test if `self` is a constant linear expression.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Linear_Expression
        sage: Linear_Expression(10).all_homogeneous_terms_are_zero()
        True

  `ascii_dump`()

    Write an ASCII dump to stderr.

    EXAMPLES:

        sage: sage_cmd  = 'from sage.libs.ppl import Linear_Expression, Variable\n'
        sage: sage_cmd += 'x = Variable(0)\n'
        sage: sage_cmd += 'y = Variable(1)\n'
        sage: sage_cmd += 'e = 3*x+2*y+1\n'
        sage: sage_cmd += 'e.ascii_dump()\n'
        sage: from sage.tests.cmdline import test_executable
        sage: (out, err, ret) = test_executable(['sage', '-c', sage_cmd], timeout=100)  # long time, indirect doctest
        sage: print err  # long time
        size 3 1 3 2

  `coefficient`(v)

    Return the coefficient of the variable `v`.

    INPUT:

    - `v` – a `Variable`.

    OUTPUT:

    An integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: e = 3*x+1
        sage: e.coefficient(x)
        3

  `coefficients`()

    Return the coefficients of the linear expression.

    OUTPUT:

    A tuple of integers of length `space_dimension()`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0);  y = Variable(1)
        sage: e = 3*x+5*y+1
        sage: e.coefficients()
        (3, 5)

  `inhomogeneous_term`()

    Return the inhomogeneous term of the linear expression.

    OUTPUT:

    Integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Linear_Expression
        sage: Linear_Expression(10).inhomogeneous_term()
        10

  `is_zero`()

    Test if `self` is the zero linear expression.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Linear_Expression
        sage: Linear_Expression(0).is_zero()
        True
        sage: Linear_Expression(10).is_zero()
        False

  `space_dimension`()

    Return the dimension of the vector space necessary for the linear expression.

    OUTPUT:

    Integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: ( x+y+1 ).space_dimension()
        2
        sage: ( x+y   ).space_dimension()
        2
        sage: (   y+1 ).space_dimension()
        2
        sage: ( x  +1 ).space_dimension()
        1
        sage: ( y+1-y ).space_dimension()
        2

class `sage.libs.ppl.``MIP_Problem`

  Bases: `sage.libs.ppl._mutable_or_immutable`

  wrapper for PPL’s MIP_Problem class

  An object of the class MIP_Problem represents a Mixed Integer (Linear) Program problem.

  INPUT:

  - `dim` – integer
  - `args` – an array of the defining data of the MIP_Problem. For each element, any one of the following is accepted:
    - A `Constraint_System`.
    - A `Linear_Expression`.

  OUTPUT:

  A `MIP_Problem`.

  EXAMPLES:

      sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: cs = Constraint_System()
      sage: cs.insert( x >= 0)
      sage: cs.insert( y >= 0 )
      sage: cs.insert( 3 * x + 5 * y <= 10 )
      sage: m = MIP_Problem(2, cs, x + y)
      sage: m.optimal_value()
      10/3
      sage: m.optimizing_point()
      point(10/3, 0/3)

  `OK`()

    Check if all the invariants are satisfied.

    OUTPUT:

    `True` if and only if `self` satisfies all the invariants.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: m = MIP_Problem()
        sage: m.add_space_dimensions_and_embed(2)
        sage: m.add_constraint(x >= 0)
        sage: m.OK()
        True

  `add_constraint`(c)

    Adds a copy of constraint c to the MIP problem.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: m = MIP_Problem()
        sage: m.add_space_dimensions_and_embed(2)
        sage: m.add_constraint(x >= 0)
        sage: m.add_constraint(y >= 0)
        sage: m.add_constraint(3 * x + 5 * y <= 10)
        sage: m.set_objective_function(x + y)
        sage: m.optimal_value()
        10/3

    TESTS:

        sage: z = Variable(2)
        sage: m.add_constraint(z >= -3)
        Traceback (most recent call last):
        ...
        ValueError: PPL::MIP_Problem::add_constraint(c):
        c.space_dimension() == 3 exceeds this->space_dimension == 2.

  `add_constraints`(cs)

    Adds a copy of the constraints in cs to the MIP problem.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: cs = Constraint_System()
        sage: cs.insert( x >= 0)
        sage: cs.insert( y >= 0 )
        sage: cs.insert( 3 * x + 5 * y <= 10 )
        sage: m = MIP_Problem(2)
        sage: m.set_objective_function(x + y)
        sage: m.add_constraints(cs)
        sage: m.optimal_value()
        10/3

    TESTS:

        sage: p = Variable(9)
        sage: cs.insert(p >= -3)
        sage: m.add_constraints(cs)
        Traceback (most recent call last):
        ...
        ValueError: PPL::MIP_Problem::add_constraints(cs):
        cs.space_dimension() == 10 exceeds this->space_dimension() == 2.

  `add_space_dimensions_and_embed`(m)

    Adds m new space dimensions and embeds the old MIP problem in the new vector space.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: cs = Constraint_System()
        sage: cs.insert( x >= 0)
        sage: cs.insert( y >= 0 )
        sage: cs.insert( 3 * x + 5 * y <= 10 )
        sage: m = MIP_Problem(2, cs, x + y)
        sage: m.add_space_dimensions_and_embed(5)
        sage: m.space_dimension()
        7

  `clear`()

    Reset the MIP_Problem to be equal to the trivial MIP_Problem.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: cs = Constraint_System()
        sage: cs.insert( x >= 0)
        sage: cs.insert( y >= 0 )
        sage: cs.insert( 3 * x + 5 * y <= 10 )
        sage: m = MIP_Problem(2, cs, x + y)
        sage: m.objective_function()
        x0+x1
        sage: m.clear()
        sage: m.objective_function()
        0

  `evaluate_objective_function`(evaluating_point)

    Return the result of evaluating the objective function on evaluating_point. ValueError thrown if self and evaluating_point are dimension-incompatible or if the generator evaluating_point is not a point.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem, Generator
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: m = MIP_Problem()
        sage: m.add_space_dimensions_and_embed(2)
        sage: m.add_constraint(x >= 0)
        sage: m.add_constraint(y >= 0)
        sage: m.add_constraint(3 * x + 5 * y <= 10)
        sage: m.set_objective_function(x + y)
        sage: g = Generator.point(5 * x - 2 * y, 7)
        sage: m.evaluate_objective_function(g)
        3/7
        sage: z = Variable(2)
        sage: g = Generator.point(5 * x - 2 * z, 7)
        sage: m.evaluate_objective_function(g)
        Traceback (most recent call last):
        ...
        ValueError: PPL::MIP_Problem::evaluate_objective_function(p, n, d):
        *this and p are dimension incompatible.

  `is_satisfiable`()

    Check if the MIP_Problem is satisfiable

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: m = MIP_Problem()
        sage: m.add_space_dimensions_and_embed(2)
        sage: m.add_constraint(x >= 0)
        sage: m.add_constraint(y >= 0)
        sage: m.add_constraint(3 * x + 5 * y <= 10)
        sage: m.is_satisfiable()
        True

  `objective_function`()

    Return the optimal value of the MIP_Problem.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: cs = Constraint_System()
        sage: cs.insert( x >= 0)
        sage: cs.insert( y >= 0 )
        sage: cs.insert( 3 * x + 5 * y <= 10 )
        sage: m = MIP_Problem(2, cs, x + y)
        sage: m.objective_function()
        x0+x1

  `optimal_value`()

    Return the optimal value of the MIP_Problem. ValueError thrown if self does not have an optimizing point, i.e., if the MIP problem is unbounded or not satisfiable.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: cs = Constraint_System()
        sage: cs.insert( x >= 0 )
        sage: cs.insert( y >= 0 )
        sage: cs.insert( 3 * x + 5 * y <= 10 )
        sage: m = MIP_Problem(2, cs, x + y)
        sage: m.optimal_value()
        10/3
        sage: cs = Constraint_System()
        sage: cs.insert( x >= 0 )
        sage: m = MIP_Problem(1, cs, x + x )
        sage: m.optimal_value()
        Traceback (most recent call last):
        ...
        ValueError: PPL::MIP_Problem::optimizing_point():
        *this does not have an optimizing point.

  `optimization_mode`()

    Return the optimization mode used in the MIP_Problem.

    It will return “maximization” if the MIP_Problem was set to MAXIMIZATION mode, and “minimization” otherwise.

    EXAMPLES:

        sage: from sage.libs.ppl import MIP_Problem
        sage: m = MIP_Problem()
        sage: m.optimization_mode()
        'maximization'

  `optimizing_point`()

    Returns an optimal point for the MIP_Problem, if it exists.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: m = MIP_Problem()
        sage: m.add_space_dimensions_and_embed(2)
        sage: m.add_constraint(x >= 0)
        sage: m.add_constraint(y >= 0)
        sage: m.add_constraint(3 * x + 5 * y <= 10)
        sage: m.set_objective_function(x + y)
        sage: m.optimizing_point()
        point(10/3, 0/3)

  `set_objective_function`(obj)

    Sets the objective function to obj.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: m = MIP_Problem()
        sage: m.add_space_dimensions_and_embed(2)
        sage: m.add_constraint(x >= 0)
        sage: m.add_constraint(y >= 0)
        sage: m.add_constraint(3 * x + 5 * y <= 10)
        sage: m.set_objective_function(x + y)
        sage: m.optimal_value()
        10/3

    TESTS:

        sage: z = Variable(2)
        sage: m.set_objective_function(x + y + z)
        Traceback (most recent call last):
        ...
        ValueError: PPL::MIP_Problem::set_objective_function(obj):
        obj.space_dimension() == 3 exceeds this->space_dimension == 2.

  `set_optimization_mode`(mode)

    Sets the optimization mode to mode.

    EXAMPLES:

        sage: from sage.libs.ppl import MIP_Problem
        sage: m = MIP_Problem()
        sage: m.optimization_mode()
        'maximization'
        sage: m.set_optimization_mode('minimization')
        sage: m.optimization_mode()
        'minimization'

    TESTS:

        sage: m.set_optimization_mode('max')
        Traceback (most recent call last):
        ...
        ValueError: Unknown value: mode=max.

  `solve`()

    Optimizes the MIP_Problem

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: m = MIP_Problem()
        sage: m.add_space_dimensions_and_embed(2)
        sage: m.add_constraint(x >= 0)
        sage: m.add_constraint(y >= 0)
        sage: m.add_constraint(3 * x + 5 * y <= 10)
        sage: m.set_objective_function(x + y)
        sage: m.solve()
        {'status': 'optimized'}

  `space_dimension`()

    Return the space dimension of the MIP_Problem.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, Constraint_System, MIP_Problem
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: cs = Constraint_System()
        sage: cs.insert( x >= 0)
        sage: cs.insert( y >= 0 )
        sage: cs.insert( 3 * x + 5 * y <= 10 )
        sage: m = MIP_Problem(2, cs, x + y)
        sage: m.space_dimension()
        2

class `sage.libs.ppl.``NNC_Polyhedron`

  Bases: `sage.libs.ppl.Polyhedron`

  Wrapper for PPL’s `NNC_Polyhedron` class.

  An object of the class `NNC_Polyhedron` represents a not necessarily closed (NNC) convex polyhedron in the vector space.

  Note: Since NNC polyhedra are a generalization of closed polyhedra, any object of the class `C_Polyhedron` can be (explicitly) converted into an object of the class `NNC_Polyhedron`. The reason for defining two different classes is that objects of the class `C_Polyhedron` are characterized by a more efficient implementation, requiring less time and memory resources.

  INPUT:

  - `arg` – the defining data of the polyhedron. Any one of the following is accepted:
    - An non-negative integer. Depending on `degenerate_element`, either the space-filling or the empty polytope in the given dimension `arg` is constructed.
    - A `Constraint_System`.
    - A `Generator_System`.
    - A single `Constraint`.
    - A single `Generator`.
    - A `NNC_Polyhedron`.
    - A `C_Polyhedron`.
  - `degenerate_element` – string, either `'universe'` or `'empty'`. Only used if `arg` is an integer.

  OUTPUT:

  A `C_Polyhedron`.

  EXAMPLES:

      sage: from sage.libs.ppl import Constraint, Constraint_System, Generator, Generator_System, Variable, NNC_Polyhedron, point, ray, closure_point
      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: NNC_Polyhedron( 5*x-2*y >  x+y-1 )
      A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point, 1 closure_point, 1 ray, 1 line
      sage: cs = Constraint_System()
      sage: cs.insert( x > 0 )
      sage: cs.insert( y > 0 )
      sage: NNC_Polyhedron(cs)
      A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point, 1 closure_point, 2 rays
      sage: NNC_Polyhedron( point(x+y) )
      A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
      sage: gs = Generator_System()
      sage: gs.insert( point(-y) )
      sage: gs.insert( closure_point(-x-y) )
      sage: gs.insert( ray(x) )
      sage: p = NNC_Polyhedron(gs); p
      A 1-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point, 1 closure_point, 1 ray
      sage: p.minimized_constraints()
      Constraint_System {x1+1==0, x0+1>0}

  Note that, by convention, every polyhedron must contain a point:

      sage: NNC_Polyhedron( closure_point(x+y) )
      Traceback (most recent call last):
      ...
      ValueError: PPL::NNC_Polyhedron::NNC_Polyhedron(gs):
      *this is an empty polyhedron and
      the non-empty generator system gs contains no points.

class `sage.libs.ppl.``Poly_Con_Relation`

  Bases: `object`

  Wrapper for PPL’s `Poly_Con_Relation` class.

  INPUT/OUTPUT:

  You must not construct `Poly_Con_Relation` objects manually. You will usually get them from `relation_with()`. You can also get pre-defined relations from the class methods `nothing()`, `is_disjoint()`, `strictly_intersects()`, `is_included()`, and `saturates()`.

  EXAMPLES:

      sage: from sage.libs.ppl import Poly_Con_Relation
      sage: saturates     = Poly_Con_Relation.saturates();  saturates
      saturates
      sage: is_included   = Poly_Con_Relation.is_included(); is_included
      is_included
      sage: is_included.implies(saturates)
      False
      sage: saturates.implies(is_included)
      False
      sage: rels = []
      sage: rels.append( Poly_Con_Relation.nothing() )
      sage: rels.append( Poly_Con_Relation.is_disjoint() )
      sage: rels.append( Poly_Con_Relation.strictly_intersects() )
      sage: rels.append( Poly_Con_Relation.is_included() )
      sage: rels.append( Poly_Con_Relation.saturates() )
      sage: rels
      [nothing, is_disjoint, strictly_intersects, is_included, saturates]
      sage: from sage.matrix.constructor import matrix
      sage: m = matrix(5,5)
      sage: for i, rel_i in enumerate(rels):
      ...       for j, rel_j in enumerate(rels):
      ...           m[i,j] = rel_i.implies(rel_j)
      sage: m
      [1 0 0 0 0]
      [1 1 0 0 0]
      [1 0 1 0 0]
      [1 0 0 1 0]
      [1 0 0 0 1]

  `OK`(check_non_empty=False)

    Check if all the invariants are satisfied.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Con_Relation
        sage: Poly_Con_Relation.nothing().OK()
        True

  `ascii_dump`()

    Write an ASCII dump to stderr.

    EXAMPLES:

        sage: sage_cmd  = 'from sage.libs.ppl import Poly_Con_Relation\n'
        sage: sage_cmd += 'Poly_Con_Relation.nothing().ascii_dump()\n'
        sage: from sage.tests.cmdline import test_executable
        sage: (out, err, ret) = test_executable(['sage', '-c', sage_cmd], timeout=100)  # long time, indirect doctest
        sage: print err  # long time
        NOTHING

  `implies`(y)

    Test whether `self` implies `y`.

    INPUT:

    - `y` – a `Poly_Con_Relation`.

    OUTPUT:

    Boolean. `True` if and only if `self` implies `y`.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Con_Relation
        sage: nothing = Poly_Con_Relation.nothing()
        sage: nothing.implies( nothing )
        True

  static `is_disjoint`()

    Return the assertion “The polyhedron and the set of points satisfying the constraint are disjoint”.

    OUTPUT:

    A `Poly_Con_Relation`.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Con_Relation
        sage: Poly_Con_Relation.is_disjoint()
        is_disjoint

  static `is_included`()

    Return the assertion “The polyhedron is included in the set of points satisfying the constraint”.

    OUTPUT:

    A `Poly_Con_Relation`.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Con_Relation
        sage: Poly_Con_Relation.is_included()
        is_included

  static `nothing`()

    Return the assertion that says nothing.

    OUTPUT:

    A `Poly_Con_Relation`.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Con_Relation
        sage: Poly_Con_Relation.nothing()
        nothing

  static `saturates`()

    Return the assertion “”.

    OUTPUT:

    A `Poly_Con_Relation`.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Con_Relation
        sage: Poly_Con_Relation.saturates()
        saturates

  static `strictly_intersects`()

    Return the assertion “The polyhedron intersects the set of points satisfying the constraint, but it is not included in it”.

    OUTPUT:

    A `Poly_Con_Relation`.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Con_Relation
        sage: Poly_Con_Relation.strictly_intersects()
        strictly_intersects

class `sage.libs.ppl.``Poly_Gen_Relation`

  Bases: `object`

  Wrapper for PPL’s `Poly_Con_Relation` class.

  INPUT/OUTPUT:

  You must not construct `Poly_Gen_Relation` objects manually. You will usually get them from `relation_with()`. You can also get pre-defined relations from the class methods `nothing()` and `subsumes()`.

  EXAMPLES:

      sage: from sage.libs.ppl import Poly_Gen_Relation
      sage: nothing = Poly_Gen_Relation.nothing(); nothing
      nothing
      sage: subsumes = Poly_Gen_Relation.subsumes(); subsumes
      subsumes
      sage: nothing.implies( subsumes )
      False
      sage: subsumes.implies( nothing )
      True

  `OK`(check_non_empty=False)

    Check if all the invariants are satisfied.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Gen_Relation
        sage: Poly_Gen_Relation.nothing().OK()
        True

  `ascii_dump`()

    Write an ASCII dump to stderr.

    EXAMPLES:

        sage: sage_cmd  = 'from sage.libs.ppl import Poly_Gen_Relation\n'
        sage: sage_cmd += 'Poly_Gen_Relation.nothing().ascii_dump()\n'
        sage: from sage.tests.cmdline import test_executable
        sage: (out, err, ret) = test_executable(['sage', '-c', sage_cmd], timeout=100)  # long time, indirect doctest
        sage: print err  # long time
        NOTHING

  `implies`(y)

    Test whether `self` implies `y`.

    INPUT:

    - `y` – a `Poly_Gen_Relation`.

    OUTPUT:

    Boolean. `True` if and only if `self` implies `y`.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Gen_Relation
        sage: nothing = Poly_Gen_Relation.nothing()
        sage: nothing.implies( nothing )
        True

  static `nothing`()

    Return the assertion that says nothing.

    OUTPUT:

    A `Poly_Gen_Relation`.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Gen_Relation
        sage: Poly_Gen_Relation.nothing()
        nothing

  static `subsumes`()

    Return the assertion “Adding the generator would not change the polyhedron”.

    OUTPUT:

    A `Poly_Gen_Relation`.

    EXAMPLES:

        sage: from sage.libs.ppl import Poly_Gen_Relation
        sage: Poly_Gen_Relation.subsumes()
        subsumes

class `sage.libs.ppl.``Polyhedron`

  Bases: `sage.libs.ppl._mutable_or_immutable`

  Wrapper for PPL’s `Polyhedron` class.

  An object of the class Polyhedron represents a convex polyhedron in the vector space.

  A polyhedron can be specified as either a finite system of constraints or a finite system of generators (see Section Representations of Convex Polyhedra) and it is always possible to obtain either representation. That is, if we know the system of constraints, we can obtain from this the system of generators that define the same polyhedron and vice versa. These systems can contain redundant members: in this case we say that they are not in the minimal form.

  INPUT/OUTPUT:

  This is an abstract base for `C_Polyhedron` and `NNC_Polyhedron`. You cannot instantiate this class.

  `OK`(check_non_empty=False)

    Check if all the invariants are satisfied.

    The check is performed so as to intrude as little as possible. If the library has been compiled with run-time assertions enabled, error messages are written on std::cerr in case invariants are violated. This is useful for the purpose of debugging the library.

    INPUT:

    - `check_not_empty` – boolean. `True` if and only if, in addition to checking the invariants, `self` must be checked to be not empty.

    OUTPUT:

    `True` if and only if `self` satisfies all the invariants and either `check_not_empty` is `False` or `self` is not empty.

    EXAMPLES:

        sage: from sage.libs.ppl import Linear_Expression, Variable
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: e = 3*x+2*y+1
        sage: e.OK()
        True

  `add_constraint`(c)

    Add a constraint to the polyhedron.

    Adds a copy of constraint `c` to the system of constraints of `self`, without minimizing the result.

    See alse `add_constraints()`.

    INPUT:

    - `c` – the `Constraint` that will be added to the system of constraints of `self`.

    OUTPUT:

    This method modifies the polyhedron `self` and does not return anything.

    Raises a `ValueError` if `self` and the constraint `c` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

           sage: from sage.libs.ppl import Variable, C_Polyhedron
           sage: x = Variable(0)
           sage: y = Variable(1)
           sage: p = C_Polyhedron( y>=0 )
           sage: p.add_constraint( x>=0 )

        We just added a 1-d constraint to a 2-d polyhedron, this is
        fine. The other way is not::

           sage: p = C_Polyhedron( x>=0 )
           sage: p.add_constraint( y>=0 )
           Traceback (most recent call last):
           ...
           ValueError: PPL::C_Polyhedron::add_constraint(c):
           this->space_dimension() == 1, c.space_dimension() == 2.

        The constraint must also be topology-compatible, that is,
        :class:`C_Polyhedron` only allows non-strict inequalities::

           sage: p = C_Polyhedron( x>=0 )
           sage: p.add_constraint( x< 1 )
           Traceback (most recent call last):
           ...
           ValueError: PPL::C_Polyhedron::add_constraint(c):
           c is a strict inequality.

  `add_constraints`(cs)

    Add constraints to the polyhedron.

    Adds a copy of constraints in `cs` to the system of constraints of `self`, without minimizing the result.

    See alse `add_constraint()`.

    INPUT:

    - `cs` – the `Constraint_System` that will be added to the system of constraints of `self`.

    OUTPUT:

    This method modifies the polyhedron `self` and does not return anything.

    Raises a `ValueError` if `self` and the constraints in `cs` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

           sage: from sage.libs.ppl import Variable, C_Polyhedron, Constraint_System
           sage: x = Variable(0)
           sage: y = Variable(1)
           sage: cs = Constraint_System()
           sage: cs.insert(x>=0)
           sage: cs.insert(y>=0)
           sage: p = C_Polyhedron( y<=1 )
           sage: p.add_constraints(cs)

        We just added a 1-d constraint to a 2-d polyhedron, this is
        fine. The other way is not::

           sage: p = C_Polyhedron( x<=1 )
           sage: p.add_constraints(cs)
           Traceback (most recent call last):
           ...
           ValueError: PPL::C_Polyhedron::add_recycled_constraints(cs):
           this->space_dimension() == 1, cs.space_dimension() == 2.

        The constraints must also be topology-compatible, that is,
        :class:`C_Polyhedron` only allows non-strict inequalities::

           sage: p = C_Polyhedron( x>=0 )
           sage: p.add_constraints( Constraint_System(x<0) )
           Traceback (most recent call last):
           ...
           ValueError: PPL::C_Polyhedron::add_recycled_constraints(cs):
           cs contains strict inequalities.

  `add_generator`(g)

    Add a generator to the polyhedron.

    Adds a copy of constraint `c` to the system of generators of `self`, without minimizing the result.

    INPUT:

    - `g` – the `Generator` that will be added to the system of Generators of `self`.

    OUTPUT:

    This method modifies the polyhedron `self` and does not return anything.

    Raises a `ValueError` if `self` and the generator `g` are topology-incompatible or dimension-incompatible, or if `self` is an empty polyhedron and `g` is not a point.

    EXAMPLES:

           sage: from sage.libs.ppl import Variable, C_Polyhedron, point, closure_point, ray
           sage: x = Variable(0)
           sage: y = Variable(1)
           sage: p = C_Polyhedron(1, 'empty')
           sage: p.add_generator( point(0*x) )

        We just added a 1-d generator to a 2-d polyhedron, this is
        fine. The other way is not::

           sage: p = C_Polyhedron(1, 'empty')
           sage: p.add_generator(  point(0*y) )
           Traceback (most recent call last):
           ...
           ValueError: PPL::C_Polyhedron::add_generator(g):
           this->space_dimension() == 1, g.space_dimension() == 2.

        The constraint must also be topology-compatible, that is,
        :class:`C_Polyhedron` does not allow :func:`closure_point`
        generators::

           sage: p = C_Polyhedron( point(0*x+0*y) )
           sage: p.add_generator( closure_point(0*x) )
           Traceback (most recent call last):
           ...
           ValueError: PPL::C_Polyhedron::add_generator(g):
           g is a closure point.

    Finally, ever non-empty polyhedron must have at least one point generator:

        sage: p = C_Polyhedron(3, 'empty')
        sage: p.add_generator( ray(x) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::add_generator(g):
        *this is an empty polyhedron and g is not a point.

  `add_generators`(gs)

    Add generators to the polyhedron.

    Adds a copy of the generators in `gs` to the system of generators of `self`, without minimizing the result.

    See alse `add_generator()`.

    INPUT:

    - `gs` – the `Generator_System` that will be added to the system of constraints of `self`.

    OUTPUT:

    This method modifies the polyhedron `self` and does not return anything.

    Raises a `ValueError` if `self` and one of the generators in `gs` are topology-incompatible or dimension-incompatible, or if `self` is an empty polyhedron and `gs` does not contain a point.

    EXAMPLES:

           sage: from sage.libs.ppl import Variable, C_Polyhedron, Generator_System, point, ray, closure_point
           sage: x = Variable(0)
           sage: y = Variable(1)
           sage: gs = Generator_System()
           sage: gs.insert(point(0*x+0*y))
           sage: gs.insert(point(1*x+1*y))
           sage: p = C_Polyhedron(2, 'empty')
           sage: p.add_generators(gs)

        We just added a 1-d constraint to a 2-d polyhedron, this is
        fine. The other way is not::

           sage: p = C_Polyhedron(1, 'empty')
           sage: p.add_generators(gs)
           Traceback (most recent call last):
           ...
           ValueError: PPL::C_Polyhedron::add_recycled_generators(gs):
           this->space_dimension() == 1, gs.space_dimension() == 2.

        The constraints must also be topology-compatible, that is,
        :class:`C_Polyhedron` does not allow :func:`closure_point`
        generators::

           sage: p = C_Polyhedron( point(0*x+0*y) )
           sage: p.add_generators( Generator_System(closure_point(x) ))
           Traceback (most recent call last):
           ...
           ValueError: PPL::C_Polyhedron::add_recycled_generators(gs):
           gs contains closure points.

  `add_space_dimensions_and_embed`(m)

    Add `m` new space dimensions and embed `self` in the new vector space.

    The new space dimensions will be those having the highest indexes in the new polyhedron, which is characterized by a system of constraints in which the variables running through the new dimensions are not constrained. For instance, when starting from the polyhedron \(P\) and adding a third space dimension, the result will be the polyhedron

    \[\Big\{ (x,y,z)^T \in \RR^3 \Big| (x,y)^T \in P \Big\}\]

    INPUT:

    - `m` – integer.

    OUTPUT:

    This method assigns the embedded polyhedron to `self` and does not return anything.

    Raises a `ValueError` if adding `m` new space dimensions would cause the vector space to exceed dimension `self.max_space_dimension()`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point
        sage: x = Variable(0)
        sage: p = C_Polyhedron( point(3*x) )
        sage: p.add_space_dimensions_and_embed(1)
        sage: p.minimized_generators()
        Generator_System {line(0, 1), point(3/1, 0/1)}
        sage: p.add_space_dimensions_and_embed( p.max_space_dimension() )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::add_space_dimensions_and_embed(m):
        adding m new space dimensions exceeds the maximum allowed space dimension.

  `add_space_dimensions_and_project`(m)

    Add `m` new space dimensions and embed `self` in the new vector space.

    The new space dimensions will be those having the highest indexes in the new polyhedron, which is characterized by a system of constraints in which the variables running through the new dimensions are all constrained to be equal to \(0\). For instance, when starting from the polyhedron \(P\) and adding a third space dimension, the result will be the polyhedron

    \[\Big\{ (x,y,0)^T \in \RR^3 \Big| (x,y)^T \in P \Big\}\]

    INPUT:

    - `m` – integer.

    OUTPUT:

    This method assigns the projected polyhedron to `self` and does not return anything.

    Raises a `ValueError` if adding `m` new space dimensions would cause the vector space to exceed dimension `self.max_space_dimension()`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point
        sage: x = Variable(0)
        sage: p = C_Polyhedron( point(3*x) )
        sage: p.add_space_dimensions_and_project(1)
        sage: p.minimized_generators()
        Generator_System {point(3/1, 0/1)}
        sage: p.add_space_dimensions_and_project( p.max_space_dimension() )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::add_space_dimensions_and_project(m):
        adding m new space dimensions exceeds the maximum allowed space dimension.

  `affine_dimension`()

    Return the affine dimension of `self`.

    OUTPUT:

    An integer. Returns 0 if `self` is empty. Otherwise, returns the affine dimension of `self`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron( 5*x-2*y ==  x+y-1 )
        sage: p.affine_dimension()
        1

  `ascii_dump`()

    Write an ASCII dump to stderr.

    EXAMPLES:

        sage: sage_cmd  = 'from sage.libs.ppl import C_Polyhedron, Variable\n'
        sage: sage_cmd += 'x = Variable(0)\n'
        sage: sage_cmd += 'y = Variable(1)\n'
        sage: sage_cmd += 'p = C_Polyhedron(3*x+2*y==1)\n'
        sage: sage_cmd += 'p.minimized_generators()\n'
        sage: sage_cmd += 'p.ascii_dump()\n'
        sage: from sage.tests.cmdline import test_executable
        sage: (out, err, ret) = test_executable(['sage', '-c', sage_cmd], timeout=100)  # long time, indirect doctest
        sage: print err  # long time
        space_dim 2
        -ZE -EM  +CM +GM  +CS +GS  -CP -GP  -SC +SG
        con_sys (up-to-date)
        topology NECESSARILY_CLOSED
        2 x 2 SPARSE (sorted)
        index_first_pending 2
        size 3 -1 3 2 = (C)
        size 3 1 0 0 >= (C)
        <BLANKLINE>
        gen_sys (up-to-date)
        topology NECESSARILY_CLOSED
        2 x 2 DENSE (not_sorted)
        index_first_pending 2
        size 3 0 2 -3 L (C)
        size 3 2 0 1 P (C)
        <BLANKLINE>
        sat_c
        0 x 0
        <BLANKLINE>
        sat_g
        2 x 2
        0 0
        0 1

  `bounds_from_above`(expr)

    Test whether the `expr` is bounded from above.

    INPUT:

    - `expr` – a `Linear_Expression`

    OUTPUT:

    Boolean. Returns `True` if and only if `expr` is bounded from above in `self`.

    Raises a `ValueError` if `expr` and `this` are dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, Linear_Expression
        sage: x = Variable(0);  y = Variable(1)
        sage: p = C_Polyhedron(y<=0)
        sage: p.bounds_from_above(x+1)
        False
        sage: p.bounds_from_above(Linear_Expression(y))
        True
        sage: p = C_Polyhedron(x<=0)
        sage: p.bounds_from_above(y+1)
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::bounds_from_above(e):
        this->space_dimension() == 1, e.space_dimension() == 2.

  `bounds_from_below`(expr)

    Test whether the `expr` is bounded from above.

    INPUT:

    - `expr` – a `Linear_Expression`

    OUTPUT:

    Boolean. Returns `True` if and only if `expr` is bounded from above in `self`.

    Raises a `ValueError` if `expr` and `this` are dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, Linear_Expression
        sage: x = Variable(0);  y = Variable(1)
        sage: p = C_Polyhedron(y>=0)
        sage: p.bounds_from_below(x+1)
        False
        sage: p.bounds_from_below(Linear_Expression(y))
        True
        sage: p = C_Polyhedron(x<=0)
        sage: p.bounds_from_below(y+1)
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::bounds_from_below(e):
        this->space_dimension() == 1, e.space_dimension() == 2.

  `concatenate_assign`(y)

    Assign to `self` the concatenation of `self` and `y`.

    This functions returns the Cartiesian product of `self` and `y`.

    Viewing a polyhedron as a set of tuples (its points), it is sometimes useful to consider the set of tuples obtained by concatenating an ordered pair of polyhedra. Formally, the concatenation of the polyhedra \(P\) and \(Q\) (taken in this order) is the polyhedron such that

    \[R = \Big\{ (x_0,\dots,x_{n-1},y_0,\dots,y_{m-1})^T \in \RR^{n+m} \Big| (x_0,\dots,x_{n-1})^T \in P ,~ (y_0,\dots,y_{m-1})^T \in Q \Big\}\]

    Another way of seeing it is as follows: first embed polyhedron \(P\) into a vector space of dimension \(n+m\) and then add a suitably renamed-apart version of the constraints defining \(Q\).

    INPUT:

    - `m` – integer.

    OUTPUT:

    This method assigns the concatenated polyhedron to `self` and does not return anything.

    Raises a `ValueError` if `self` and `y` are topology-incompatible or if adding `y.space_dimension()` new space dimensions would cause the vector space to exceed dimension `self.max_space_dimension()`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, NNC_Polyhedron, point
        sage: x = Variable(0)
        sage: p1 = C_Polyhedron( point(1*x) )
        sage: p2 = C_Polyhedron( point(2*x) )
        sage: p1.concatenate_assign(p2)
        sage: p1.minimized_generators()
        Generator_System {point(1/1, 2/1)}

    The polyhedra must be topology-compatible and not exceed the maximum space dimension:

        sage: p1.concatenate_assign( NNC_Polyhedron(1, 'universe') )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::concatenate_assign(y):
        y is a NNC_Polyhedron.
        sage: p1.concatenate_assign( C_Polyhedron(p1.max_space_dimension(), 'empty') )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::concatenate_assign(y):
        concatenation exceeds the maximum allowed space dimension.

  `constrains`(var)

    Test whether `var` is constrained in `self`.

    INPUT:

    - `var` – a `Variable`.

    OUTPUT:

    Boolean. Returns `True` if and only if `var` is constrained in `self`.

    Raises a `ValueError` if `var` is not a space dimension of `self`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron
        sage: x = Variable(0)
        sage: p = C_Polyhedron(1, 'universe')
        sage: p.constrains(x)
        False
        sage: p = C_Polyhedron(x>=0)
        sage: p.constrains(x)
        True
        sage: y = Variable(1)
        sage: p.constrains(y)
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::constrains(v):
        this->space_dimension() == 1, v.space_dimension() == 2.

  `constraints`()

    Returns the system of constraints.

    See also `minimized_constraints()`.

    OUTPUT:

    A `Constraint_System`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron( y>=0 )
        sage: p.add_constraint( x>=0 )
        sage: p.add_constraint( x+y>=0 )
        sage: p.constraints()
        Constraint_System {x1>=0, x0>=0, x0+x1>=0}
        sage: p.minimized_constraints()
        Constraint_System {x1>=0, x0>=0}

  `contains`(y)

    Test whether `self` contains `y`.

    INPUT:

    - `y` – a `Polyhedron`.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` contains `y`.

    Raises a `ValueError` if `self` and `y` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, NNC_Polyhedron
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p0 = C_Polyhedron( x>=0 )
        sage: p1 = C_Polyhedron( x>=1 )
        sage: p0.contains(p1)
        True
        sage: p1.contains(p0)
        False

    Errors are raised if the dimension or topology is not compatible:

        sage: p0.contains(C_Polyhedron(y>=0))
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::contains(y):
        this->space_dimension() == 1, y.space_dimension() == 2.
        sage: p0.contains(NNC_Polyhedron(x>0))
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::contains(y):
        y is a NNC_Polyhedron.

  `contains_integer_point`()

    Test whether `self` contains an integer point.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` contains an integer point.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, NNC_Polyhedron
        sage: x = Variable(0)
        sage: p = NNC_Polyhedron(x>0)
        sage: p.add_constraint(x<1)
        sage: p.contains_integer_point()
        False
        sage: p.topological_closure_assign()
        sage: p.contains_integer_point()
        True

  `difference_assign`(y)

    Assign to `self` the poly-difference of `self` and `y`.

    For any pair of NNC polyhedra \(P_1\) and \(P_2\) the convex polyhedral difference (or poly-difference) of \(P_1\) and \(P_2\) is defined as the smallest convex polyhedron containing the set-theoretic difference \(P_1\setminus P_2\) of \(P_1\) and \(P_2\).

    In general, even if \(P_1\) and \(P_2\) are topologically closed polyhedra, their poly-difference may be a convex polyhedron that is not topologically closed. For this reason, when computing the poly-difference of two `C_Polyhedron`, the library will enforce the topological closure of the result.

    INPUT:

    - `y` – a `Polyhedron`

    OUTPUT:

    This method assigns the poly-difference to `self` and does not return anything.

    Raises a `ValueError` if `self` and and `y` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point, closure_point, NNC_Polyhedron
        sage: x = Variable(0)
        sage: p = NNC_Polyhedron( point(0*x) )
        sage: p.add_generator( point(1*x) )
        sage: p.poly_difference_assign(NNC_Polyhedron( point(0*x) ))
        sage: p.minimized_constraints()
        Constraint_System {-x0+1>=0, x0>0}

    The poly-difference of `C_polyhedron` is really its closure:

        sage: p = C_Polyhedron( point(0*x) )
        sage: p.add_generator( point(1*x) )
        sage: p.poly_difference_assign(C_Polyhedron( point(0*x) ))
        sage: p.minimized_constraints()
        Constraint_System {x0>=0, -x0+1>=0}

    `self` and `y` must be dimension- and topology-compatible, or an exception is raised:

        sage: y = Variable(1)
        sage: p.poly_difference_assign( C_Polyhedron(y>=0) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::poly_difference_assign(y):
        this->space_dimension() == 1, y.space_dimension() == 2.
        sage: p.poly_difference_assign( NNC_Polyhedron(x+y<1) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::poly_difference_assign(y):
        y is a NNC_Polyhedron.

  `drop_some_non_integer_points`()

    Possibly tighten `self` by dropping some points with non-integer coordinates.

    The modified polyhedron satisfies:

    - it is (not necessarily strictly) contained in the original polyhedron.
    - integral vertices (generating points with integer coordinates) of the original polyhedron are not removed.

    Note

    The modified polyhedron is not neccessarily a lattice polyhedron; Some vertices will, in general, still be rational. Lattice points interior to the polyhedron may be lost in the process.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, NNC_Polyhedron, Constraint_System
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: cs = Constraint_System()
        sage: cs.insert( x>=0 )
        sage: cs.insert( y>=0 )
        sage: cs.insert( 3*x+2*y<5 )
        sage: p = NNC_Polyhedron(cs)
        sage: p.minimized_generators()
        Generator_System {point(0/1, 0/1), closure_point(0/2, 5/2), closure_point(5/3, 0/3)}
        sage: p.drop_some_non_integer_points()
        sage: p.minimized_generators()
        Generator_System {point(0/1, 0/1), point(0/1, 2/1), point(4/3, 0/3)}

  `generators`()

    Returns the system of generators.

    See also `minimized_generators()`.

    OUTPUT:

    A `Generator_System`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron(3,'empty')
        sage: p.add_generator( point(-x-y) )
        sage: p.add_generator( point(0) )
        sage: p.add_generator( point(+x+y) )
        sage: p.generators()
        Generator_System {point(-1/1, -1/1, 0/1), point(0/1, 0/1, 0/1), point(1/1, 1/1, 0/1)}
        sage: p.minimized_generators()
        Generator_System {point(-1/1, -1/1, 0/1), point(1/1, 1/1, 0/1)}

  `intersection_assign`(y)

    Assign to `self` the intersection of `self` and `y`.

    INPUT:

    - `y` – a `Polyhedron`

    OUTPUT:

    This method assigns the intersection to `self` and does not return anything.

    Raises a `ValueError` if `self` and and `y` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, NNC_Polyhedron
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron( 1*x+0*y >= 0 )
        sage: p.intersection_assign( C_Polyhedron(y>=0) )
        sage: p.constraints()
        Constraint_System {x0>=0, x1>=0}
        sage: z = Variable(2)
        sage: p.intersection_assign( C_Polyhedron(z>=0) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::intersection_assign(y):
        this->space_dimension() == 2, y.space_dimension() == 3.
        sage: p.intersection_assign( NNC_Polyhedron(x+y<1) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::intersection_assign(y):
        y is a NNC_Polyhedron.

  `is_bounded`()

    Test whether `self` is bounded.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` is a bounded polyhedron.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, NNC_Polyhedron, point, closure_point, ray
        sage: x = Variable(0)
        sage: p = NNC_Polyhedron( point(0*x) )
        sage: p.add_generator( closure_point(1*x) )
        sage: p.is_bounded()
        True
        sage: p.add_generator( ray(1*x) )
        sage: p.is_bounded()
        False

  `is_discrete`()

    Test whether `self` is discrete.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` is discrete.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point, ray
        sage: x = Variable(0);  y = Variable(1)
        sage: p = C_Polyhedron( point(1*x+2*y) )
        sage: p.is_discrete()
        True
        sage: p.add_generator( point(x) )
        sage: p.is_discrete()
        False

  `is_disjoint_from`(y)

    Tests whether `self` and `y` are disjoint.

    INPUT:

    - `y` – a `Polyhedron`.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` and `y` are disjoint.

    Rayises a `ValueError` if `self` and `y` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, NNC_Polyhedron
        sage: x = Variable(0);  y = Variable(1)
        sage: C_Polyhedron(x<=0).is_disjoint_from( C_Polyhedron(x>=1) )
        True

    This is not allowed:

        sage: x = Variable(0);  y = Variable(1)
        sage: poly_1d = C_Polyhedron(x<=0)
        sage: poly_2d = C_Polyhedron(x+0*y>=1)
        sage: poly_1d.is_disjoint_from(poly_2d)
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::intersection_assign(y):
        this->space_dimension() == 1, y.space_dimension() == 2.

    Nor is this:

        sage: x = Variable(0);  y = Variable(1)
        sage: c_poly   =   C_Polyhedron( x<=0 )
        sage: nnc_poly = NNC_Polyhedron( x >0 )
        sage: c_poly.is_disjoint_from(nnc_poly)
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::intersection_assign(y):
        y is a NNC_Polyhedron.
        sage: NNC_Polyhedron(c_poly).is_disjoint_from(nnc_poly)
        True

  `is_empty`()

    Test if `self` is an empty polyhedron.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import C_Polyhedron
        sage: C_Polyhedron(3, 'empty').is_empty()
        True
        sage: C_Polyhedron(3, 'universe').is_empty()
        False

  `is_topologically_closed`()

    Tests if `self` is topologically closed.

    OUTPUT:

    Returns `True` if and only if `self` is a topologically closed subset of the ambient vector space.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, NNC_Polyhedron
        sage: x = Variable(0);  y = Variable(1)
        sage: C_Polyhedron(3, 'universe').is_topologically_closed()
        True
        sage: C_Polyhedron( x>=1 ).is_topologically_closed()
        True
        sage: NNC_Polyhedron( x>1 ).is_topologically_closed()
        False

  `is_universe`()

    Test if `self` is a universe (space-filling) polyhedron.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import C_Polyhedron
        sage: C_Polyhedron(3, 'empty').is_universe()
        False
        sage: C_Polyhedron(3, 'universe').is_universe()
        True

  `max_space_dimension`()

    Return the maximum space dimension all kinds of Polyhedron can handle.

    OUTPUT:

    Integer.

    EXAMPLES:

        sage: from sage.libs.ppl import C_Polyhedron
        sage: C_Polyhedron(1, 'empty').max_space_dimension()   # random output
        1152921504606846974
        sage: C_Polyhedron(1, 'empty').max_space_dimension()
        357913940            # 32-bit
        1152921504606846974  # 64-bit

  `maximize`(expr)

    Maximize `expr`.

    INPUT:

    - `expr` – a `Linear_Expression`.

    OUTPUT:

    A dictionary with the following keyword:value pair:

    - `'bounded'`: Boolean. Whether the linear expression `expr` is bounded from above on `self`.

    If `expr` is bounded from above, the following additional keyword:value pairs are set to provide information about the supremum:

    - `'sup_n'`: Integer. The numerator of the supremum value.
    - `'sup_d'`: Non-zero integer. The denominator of the supremum value.
    - `'maximum'`: Boolean. `True` if and only if the supremum is also the maximum value.
    - `'generator'`: a `Generator`. A point or closure point where expr reaches its supremum value.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, NNC_Polyhedron, Constraint_System, Linear_Expression
        sage: x = Variable(0);  y = Variable(1)
        sage: cs = Constraint_System()
        sage: cs.insert( x>=0 )
        sage: cs.insert( y>=0 )
        sage: cs.insert( 3*x+5*y<=10 )
        sage: p = C_Polyhedron(cs)
        sage: p.maximize( x+y )
        {'bounded': True,
         'generator': point(10/3, 0/3),
         'maximum': True,
         'sup_d': 3,
         'sup_n': 10}

    Unbounded case:

        sage: cs = Constraint_System()
        sage: cs.insert( x>0 )
        sage: p = NNC_Polyhedron(cs)
        sage: p.maximize( +x )
        {'bounded': False}
        sage: p.maximize( -x )
        {'bounded': True,
         'generator': closure_point(0/1),
         'maximum': False,
         'sup_d': 1,
         'sup_n': 0}

  `minimize`(expr)

    Minimize `expr`.

    INPUT:

    - `expr` – a `Linear_Expression`.

    OUTPUT:

    A dictionary with the following keyword:value pair:

    - `'bounded'`: Boolean. Whether the linear expression `expr` is bounded from below on `self`.

    If `expr` is bounded from below, the following additional keyword:value pairs are set to provide information about the infimum:

    - `'inf_n'`: Integer. The numerator of the infimum value.
    - `'inf_d'`: Non-zero integer. The denominator of the infimum value.
    - `'minimum'`: Boolean. `True` if and only if the infimum is also the minimum value.
    - `'generator'`: a `Generator`. A point or closure point where expr reaches its infimum value.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, NNC_Polyhedron, Constraint_System, Linear_Expression
        sage: x = Variable(0);  y = Variable(1)
        sage: cs = Constraint_System()
        sage: cs.insert( x>=0 )
        sage: cs.insert( y>=0 )
        sage: cs.insert( 3*x+5*y<=10 )
        sage: p = C_Polyhedron(cs)
        sage: p.minimize( x+y )
        {'bounded': True,
         'generator': point(0/1, 0/1),
         'inf_d': 1,
         'inf_n': 0,
         'minimum': True}

    Unbounded case:

        sage: cs = Constraint_System()
        sage: cs.insert( x>0 )
        sage: p = NNC_Polyhedron(cs)
        sage: p.minimize( +x )
        {'bounded': True,
         'generator': closure_point(0/1),
         'inf_d': 1,
         'inf_n': 0,
         'minimum': False}
        sage: p.minimize( -x )
        {'bounded': False}

  `minimized_constraints`()

    Returns the minimized system of constraints.

    See also `constraints()`.

    OUTPUT:

    A `Constraint_System`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron( y>=0 )
        sage: p.add_constraint( x>=0 )
        sage: p.add_constraint( x+y>=0 )
        sage: p.constraints()
        Constraint_System {x1>=0, x0>=0, x0+x1>=0}
        sage: p.minimized_constraints()
        Constraint_System {x1>=0, x0>=0}

  `minimized_generators`()

    Returns the minimized system of generators.

    See also `generators()`.

    OUTPUT:

    A `Generator_System`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron(3,'empty')
        sage: p.add_generator( point(-x-y) )
        sage: p.add_generator( point(0) )
        sage: p.add_generator( point(+x+y) )
        sage: p.generators()
        Generator_System {point(-1/1, -1/1, 0/1), point(0/1, 0/1, 0/1), point(1/1, 1/1, 0/1)}
        sage: p.minimized_generators()
        Generator_System {point(-1/1, -1/1, 0/1), point(1/1, 1/1, 0/1)}

  `poly_difference_assign`(y)

    Assign to `self` the poly-difference of `self` and `y`.

    For any pair of NNC polyhedra \(P_1\) and \(P_2\) the convex polyhedral difference (or poly-difference) of \(P_1\) and \(P_2\) is defined as the smallest convex polyhedron containing the set-theoretic difference \(P_1\setminus P_2\) of \(P_1\) and \(P_2\).

    In general, even if \(P_1\) and \(P_2\) are topologically closed polyhedra, their poly-difference may be a convex polyhedron that is not topologically closed. For this reason, when computing the poly-difference of two `C_Polyhedron`, the library will enforce the topological closure of the result.

    INPUT:

    - `y` – a `Polyhedron`

    OUTPUT:

    This method assigns the poly-difference to `self` and does not return anything.

    Raises a `ValueError` if `self` and and `y` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point, closure_point, NNC_Polyhedron
        sage: x = Variable(0)
        sage: p = NNC_Polyhedron( point(0*x) )
        sage: p.add_generator( point(1*x) )
        sage: p.poly_difference_assign(NNC_Polyhedron( point(0*x) ))
        sage: p.minimized_constraints()
        Constraint_System {-x0+1>=0, x0>0}

    The poly-difference of `C_polyhedron` is really its closure:

        sage: p = C_Polyhedron( point(0*x) )
        sage: p.add_generator( point(1*x) )
        sage: p.poly_difference_assign(C_Polyhedron( point(0*x) ))
        sage: p.minimized_constraints()
        Constraint_System {x0>=0, -x0+1>=0}

    `self` and `y` must be dimension- and topology-compatible, or an exception is raised:

        sage: y = Variable(1)
        sage: p.poly_difference_assign( C_Polyhedron(y>=0) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::poly_difference_assign(y):
        this->space_dimension() == 1, y.space_dimension() == 2.
        sage: p.poly_difference_assign( NNC_Polyhedron(x+y<1) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::poly_difference_assign(y):
        y is a NNC_Polyhedron.

  `poly_hull_assign`(y)

    Assign to `self` the poly-hull of `self` and `y`.

    For any pair of NNC polyhedra \(P_1\) and \(P_2\), the convex polyhedral hull (or poly-hull) of is the smallest NNC polyhedron that includes both \(P_1\) and \(P_2\). The poly-hull of any pair of closed polyhedra in is also closed.

    INPUT:

    - `y` – a `Polyhedron`

    OUTPUT:

    This method assigns the poly-hull to `self` and does not return anything.

    Raises a `ValueError` if `self` and and `y` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point, NNC_Polyhedron
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron( point(1*x+0*y) )
        sage: p.poly_hull_assign(C_Polyhedron( point(0*x+1*y) ))
        sage: p.generators()
        Generator_System {point(0/1, 1/1), point(1/1, 0/1)}

    `self` and `y` must be dimension- and topology-compatible, or an exception is raised:

        sage: z = Variable(2)
        sage: p.poly_hull_assign( C_Polyhedron(z>=0) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::poly_hull_assign(y):
        this->space_dimension() == 2, y.space_dimension() == 3.
        sage: p.poly_hull_assign( NNC_Polyhedron(x+y<1) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::poly_hull_assign(y):
        y is a NNC_Polyhedron.

  `relation_with`(arg)

    Return the relations holding between the polyhedron `self` and the generator or constraint `arg`.

    INPUT:

    - `arg` – a `Generator` or a `Constraint`.

    OUTPUT:

    A `Poly_Gen_Relation` or a `Poly_Con_Relation` according to the type of the input.

    Raises `ValueError` if `self` and the generator/constraint `arg` are dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point, ray, Poly_Con_Relation
        sage: x = Variable(0);  y = Variable(1)
        sage: p = C_Polyhedron(2, 'empty')
        sage: p.add_generator( point(1*x+0*y) )
        sage: p.add_generator( point(0*x+1*y) )
        sage: p.minimized_constraints()
        Constraint_System {x0+x1-1==0, -x1+1>=0, x1>=0}
        sage: p.relation_with( point(1*x+1*y) )
        nothing
        sage: p.relation_with( point(1*x+1*y, 2) )
        subsumes
        sage: p.relation_with( x+y==-1 )
        is_disjoint
        sage: p.relation_with( x==y )
        strictly_intersects
        sage: p.relation_with( x+y<=1 )
        is_included, saturates
        sage: p.relation_with( x+y<1 )
        is_disjoint, saturates

    In a Sage program you will usually use `relation_with()` together with `implies()` or `implies()`, for example:

        sage: p.relation_with( x+y<1 ).implies(Poly_Con_Relation.saturates())
        True

    You can only get relations with dimension-compatible generators or constraints:

        sage: z = Variable(2)
        sage: p.relation_with( point(x+y+z) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::relation_with(g):
        this->space_dimension() == 2, g.space_dimension() == 3.
        sage: p.relation_with( z>0 )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::relation_with(c):
        this->space_dimension() == 2, c.space_dimension() == 3.

  `remove_higher_space_dimensions`(new_dimension)

    Remove the higher dimensions of the vector space so that the resulting space will have dimension `new_dimension`.

    OUTPUT:

    This method modifies `self` and does not return anything.

    Raises a `ValueError` if `new_dimensions` is greater than the space dimension of `self`.

    EXAMPLES:

        sage: from sage.libs.ppl import C_Polyhedron, Variable
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron(3*x+0*y==2)
        sage: p.remove_higher_space_dimensions(1)
        sage: p.minimized_constraints()
        Constraint_System {3*x0-2==0}
        sage: p.remove_higher_space_dimensions(2)
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::remove_higher_space_dimensions(nd):
        this->space_dimension() == 1, required space dimension == 2.

  `space_dimension`()

    Return the dimension of the vector space enclosing `self`.

    OUTPUT:

    Integer.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron( 5*x-2*y >=  x+y-1 )
        sage: p.space_dimension()
        2

  `strictly_contains`(y)

    Test whether `self` strictly contains `y`.

    INPUT:

    - `y` – a `Polyhedron`.

    OUTPUT:

    Boolean. Returns `True` if and only if `self` contains `y` and `self` does not equal `y`.

    Raises a `ValueError` if `self` and `y` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, NNC_Polyhedron
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p0 = C_Polyhedron( x>=0 )
        sage: p1 = C_Polyhedron( x>=1 )
        sage: p0.strictly_contains(p1)
        True
        sage: p1.strictly_contains(p0)
        False

    Errors are raised if the dimension or topology is not compatible:

        sage: p0.strictly_contains(C_Polyhedron(y>=0))
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::contains(y):
        this->space_dimension() == 1, y.space_dimension() == 2.
        sage: p0.strictly_contains(NNC_Polyhedron(x>0))
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::contains(y):
        y is a NNC_Polyhedron.

  `topological_closure_assign`()

    Assign to `self` its topological closure.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, NNC_Polyhedron
        sage: x = Variable(0)
        sage: p = NNC_Polyhedron(x>0)
        sage: p.is_topologically_closed()
        False
        sage: p.topological_closure_assign()
        sage: p.is_topologically_closed()
        True
        sage: p.minimized_constraints()
        Constraint_System {x0>=0}

  `unconstrain`(var)

    Compute the cylindrification of `self` with respect to space dimension `var`.

    INPUT:

    - `var` – a `Variable`. The space dimension that will be unconstrained. Exceptions:

    OUTPUT:

    This method assigns the cylindrification to `self` and does not return anything.

    Raises a `ValueError` if `var` is not a space dimension of `self`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron( point(x+y) ); p
        A 0-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point
        sage: p.unconstrain(x); p
        A 1-dimensional polyhedron in QQ^2 defined as the convex hull of 1 point, 1 line
        sage: z = Variable(2)
        sage: p.unconstrain(z)
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::unconstrain(var):
        this->space_dimension() == 2, required space dimension == 3.

  `upper_bound_assign`(y)

    Assign to `self` the poly-hull of `self` and `y`.

    For any pair of NNC polyhedra \(P_1\) and \(P_2\), the convex polyhedral hull (or poly-hull) of is the smallest NNC polyhedron that includes both \(P_1\) and \(P_2\). The poly-hull of any pair of closed polyhedra in is also closed.

    INPUT:

    - `y` – a `Polyhedron`

    OUTPUT:

    This method assigns the poly-hull to `self` and does not return anything.

    Raises a `ValueError` if `self` and and `y` are topology-incompatible or dimension-incompatible.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable, C_Polyhedron, point, NNC_Polyhedron
        sage: x = Variable(0)
        sage: y = Variable(1)
        sage: p = C_Polyhedron( point(1*x+0*y) )
        sage: p.poly_hull_assign(C_Polyhedron( point(0*x+1*y) ))
        sage: p.generators()
        Generator_System {point(0/1, 1/1), point(1/1, 0/1)}

    `self` and `y` must be dimension- and topology-compatible, or an exception is raised:

        sage: z = Variable(2)
        sage: p.poly_hull_assign( C_Polyhedron(z>=0) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::poly_hull_assign(y):
        this->space_dimension() == 2, y.space_dimension() == 3.
        sage: p.poly_hull_assign( NNC_Polyhedron(x+y<1) )
        Traceback (most recent call last):
        ...
        ValueError: PPL::C_Polyhedron::poly_hull_assign(y):
        y is a NNC_Polyhedron.

class `sage.libs.ppl.``Variable`

  Bases: `object`

  Wrapper for PPL’s `Variable` class.

  A dimension of the vector space.

  An object of the class Variable represents a dimension of the space, that is one of the Cartesian axes. Variables are used as basic blocks in order to build more complex linear expressions. Each variable is identified by a non-negative integer, representing the index of the corresponding Cartesian axis (the first axis has index 0). The space dimension of a variable is the dimension of the vector space made by all the Cartesian axes having an index less than or equal to that of the considered variable; thus, if a variable has index \(i\), its space dimension is \(i+1\).

  INPUT:

  - `i` – integer. The index of the axis.

  OUTPUT:

  A `Variable`

  EXAMPLES:

      sage: from sage.libs.ppl import Variable
      sage: x = Variable(123)
      sage: x.id()
      123
      sage: x
      x123

  Note that the “meaning” of an object of the class Variable is completely specified by the integer index provided to its constructor: be careful not to be mislead by C++ language variable names. For instance, in the following example the linear expressions `e1` and `e2` are equivalent, since the two variables `x` and `z` denote the same Cartesian axis:

      sage: x = Variable(0)
      sage: y = Variable(1)
      sage: z = Variable(0)
      sage: e1 = x + y; e1
      x0+x1
      sage: e2 = y + z; e2
      x0+x1
      sage: e1 - e2
      0

  `OK`()

    Checks if all the invariants are satisfied.

    OUTPUT:

    Boolean.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: x.OK()
        True

  `id`()

    Return the index of the Cartesian axis associated to the variable.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(123)
        sage: x.id()
        123

  `space_dimension`()

    Return the dimension of the vector space enclosing `self`.

    OUPUT:

    Integer. The returned value is `self.id()+1`.

    EXAMPLES:

        sage: from sage.libs.ppl import Variable
        sage: x = Variable(0)
        sage: x.space_dimension()
        1

`sage.libs.ppl.``closure_point`(expression=0, divisor=1)

  Constuct a closure point.

  See `Generator.closure_point()` for documentation.

  EXAMPLES:

      sage: from sage.libs.ppl import Variable, closure_point
      sage: y = Variable(1)
      sage: closure_point(2*y, 5)
      closure_point(0/5, 2/5)

`sage.libs.ppl.``equation`(expression)

  Constuct an equation.

  INPUT:

  - `expression` – a `Linear_Expression`.

  OUTPUT:

  The equation `expression` == 0.

  EXAMPLES:

      sage: from sage.libs.ppl import Variable, equation
      sage: y = Variable(1)
      sage: 2*y+1 == 0
      2*x1+1==0
      sage: equation(2*y+1)
      2*x1+1==0

`sage.libs.ppl.``inequality`(expression)

  Constuct an inequality.

  INPUT:

  - `expression` – a `Linear_Expression`.

  OUTPUT:

  The inequality `expression` >= 0.

  EXAMPLES:

      sage: from sage.libs.ppl import Variable, inequality
      sage: y = Variable(1)
      sage: 2*y+1 >= 0
      2*x1+1>=0
      sage: inequality(2*y+1)
      2*x1+1>=0

`sage.libs.ppl.``line`(expression)

  Constuct a line.

  See `Generator.line()` for documentation.

  EXAMPLES:

      sage: from sage.libs.ppl import Variable, line
      sage: y = Variable(1)
      sage: line(2*y)
      line(0, 1)

`sage.libs.ppl.``point`(expression=0, divisor=1)

  Constuct a point.

  See `Generator.point()` for documentation.

  EXAMPLES:

      sage: from sage.libs.ppl import Variable, point
      sage: y = Variable(1)
      sage: point(2*y, 5)
      point(0/5, 2/5)

`sage.libs.ppl.``ray`(expression)

  Constuct a ray.

  See `Generator.ray()` for documentation.

  EXAMPLES:

      sage: from sage.libs.ppl import Variable, ray
      sage: y = Variable(1)
      sage: ray(2*y)
      ray(0, 1)

`sage.libs.ppl.``strict_inequality`(expression)

  Constuct a strict inequality.

  INPUT:

  - `expression` – a `Linear_Expression`.

  OUTPUT:

  The inequality `expression` > 0.

  EXAMPLES:

      sage: from sage.libs.ppl import Variable, strict_inequality
      sage: y = Variable(1)
      sage: 2*y+1 > 0
      2*x1+1>0
      sage: strict_inequality(2*y+1)
      2*x1+1>0

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