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# Parma_Polyhedra_Library::Constraint Class Reference

A linear equality or inequality. More...

List of all members.

## Public Types
enum | Type { EQUALITY, NONSTRICT_INEQUALITY, STRICT_INEQUALITY }
| The constraint type. More...
## Public Member Functions
| Constraint (const Constraint &c)
| Ordinary copy-constructor.
| ~Constraint ()
| Destructor.
Constraint & | operator= (const Constraint &c)
| Assignment operator.
dimension_type | space_dimension () const
| Returns the dimension of the vector space enclosing `*this`.
Type | type () const
| Returns the constraint type of `*this`.
bool | is_equality () const
| Returns `true` if and only if `*this` is an equality constraint.
bool | is_inequality () const
| Returns `true` if and only if `*this` is an inequality constraint (either strict or non-strict).
bool | is_nonstrict_inequality () const
| Returns `true` if and only if `*this` is a non-strict inequality constraint.
bool | is_strict_inequality () const
| Returns `true` if and only if `*this` is a strict inequality constraint.
const Integer & | coefficient (Variable v) const
| Returns the coefficient of `v` in `*this`.
const Integer & | inhomogeneous_term () const
| Returns the inhomogeneous term of `*this`.
bool | OK () const
| Checks if all the invariants are satisfied.
## Static Public Member Functions
const Constraint & | zero_dim_false ()
| The unsatisfiable (zero-dimension space) constraint .
const Constraint & | zero_dim_positivity ()
| The true (zero-dimension space) constraint , also known as positivity constraint.
## Related Functions
(Note that these are not member functions.)
std::ostream & | operator<< (std::ostream &s, const Constraint &c)
| Output operator.
Constraint | operator== (const LinExpression &e1, const LinExpression &e2)
| Returns the constraint `e1` = `e2`.
Constraint | operator== (const LinExpression &e, const Integer &n)
| Returns the constraint `e` = `n`.
Constraint | operator== (const Integer &n, const LinExpression &e)
| Returns the constraint `n` = `e`.
Constraint | operator<= (const LinExpression &e1, const LinExpression &e2)
| Returns the constraint `e1` <= `e2`.
Constraint | operator<= (const LinExpression &e, const Integer &n)
| Returns the constraint `e` <= `n`.
Constraint | operator<= (const Integer &n, const LinExpression &e)
| Returns the constraint `n` <= `e`.
Constraint | operator>= (const LinExpression &e1, const LinExpression &e2)
| Returns the constraint `e1` >= `e2`.
Constraint | operator>= (const LinExpression &e, const Integer &n)
| Returns the constraint `e` >= `n`.
Constraint | operator>= (const Integer &n, const LinExpression &e)
| Returns the constraint `n` >= `e`.
Constraint | operator< (const LinExpression &e1, const LinExpression &e2)
| Returns the constraint `e1` < `e2`.
Constraint | operator< (const LinExpression &e, const Integer &n)
| Returns the constraint `e` < `n`.
Constraint | operator< (const Integer &n, const LinExpression &e)
| Returns the constraint `n` < `e`.
Constraint | operator> (const LinExpression &e1, const LinExpression &e2)
| Returns the constraint `e1` > `e2`.
Constraint | operator> (const LinExpression &e, const Integer &n)
| Returns the constraint `e` > `n`.
Constraint | operator> (const Integer &n, const LinExpression &e)
| Returns the constraint `n` > `e`.
void | swap (Parma_Polyhedra_Library::Constraint &x, Parma_Polyhedra_Library::Constraint &y)
| Specializes `std::swap`.

## Detailed Description

A linear equality or inequality.

An object of the class Constraint is either:

- an equality: ;
- a non-strict inequality: ; or
- a strict inequality: ;

where is the dimension of the space, is the integer coefficient of variable and is the integer inhomogeneous term.

How to build a constraint
  Constraints are typically built by applying a relation symbol to a pair of linear expressions. Available relation symbols are equality (`==`), non-strict inequalities (`>=` and `<=`) and strict inequalities (`<` and `>`). The space-dimension of a constraint is defined as the maximum space-dimension of the arguments of its constructor.

  In the following examples it is assumed that variables `x`, `y` and `z` are defined as follows:

        Variable x(0);
        Variable y(1);
        Variable z(2);

Example 1
  The following code builds the equality constraint , having space-dimension :

        Constraint eq_c(3*x + 5*y - z == 0);

  The following code builds the (non-strict) inequality constraint , having space-dimension :

        Constraint ineq_c(4*x >= 2*y - 13);

  The corresponding strict inequality constraint is obtained as follows:

        Constraint strict_ineq_c(4*x > 2*y - 13);

  An unsatisfiable constraint on the zero-dimension space can be specified as follows:

        Constraint false_c = Constraint::zero_dim_false();

  Equivalent, but more involved ways are the following:

        Constraint false_c1(LinExpression::zero() == 1);
        Constraint false_c2(LinExpression::zero() >= 1);
        Constraint false_c3(LinExpression::zero() > 0);

  In contrast, the following code defines an unsatisfiable constraint having space-dimension :

        Constraint false_c(0*z == 1);

How to inspect a constraint
  Several methods are provided to examine a constraint and extract all the encoded information: its space-dimension, its type (equality, non-strict inequality, strict inequality) and the value of its integer coefficients.

Example 2
  The following code shows how it is possible to access each single coefficient of a constraint. Given an inequality constraint (in this case ), we construct a new constraint corresponding to its complement (thus, in this case we want to obtain the strict inequality constraint ).

        Constraint c1(x - 5*y + 3*z <= 4);
        cout << "Constraint c1: " << c1 << endl;
        if (c1.is_equality())
          cout << "Constraint c1 is not an inequality." << endl;
        else {
          LinExpression e;
          for (int i = c1.space_dimension() - 1; i >= 0; i--)
            e += c1.coefficient(Variable(i)) * Variable(i);
          e += c1.inhomogeneous_term();
          Constraint c2 = c1.is_strict_inequality() ? (e <= 0) : (e < 0);
          cout << "Complement c2: " << c2 << endl;
        }

  The actual output is the following:

        Constraint c1: -A + 5*B - 3*C >= -4
        Complement c2: A - 5*B + 3*C > 4

  Note that, in general, the particular output obtained can be syntactically different from the (semantically equivalent) constraint considered.

## Member Enumeration Documentation

enum Parma_Polyhedra_Library::Constraint::Type

  | The constraint type. Enumeration values:
  EQUALITY | The constraint is an equality.
  NONSTRICT_INEQUALITY | The constraint is a non-strict inequality.
  STRICT_INEQUALITY | The constraint is a strict inequality.

## Member Function Documentation

const Integer& Parma_Polyhedra_Library::Constraint::coefficient | ( | Variable | v | ) | const

  | Returns the coefficient of `v` in `*this`. Exceptions:
  | std::invalid_argument | thrown if the index of `v` is greater than or equal to the space-dimension of `*this`.