@@ -37,8 +37,9 @@ __all__ = [
class Polyhedron(Domain):
"""
A convex polyhedron (or simply "polyhedron") is the space defined by a
-    system of linear equalities and inequalities. This space can be
-    unbounded.
+    system of linear equalities and inequalities. This space can be unbounded. A
+    Z-polyhedron (simply called "polyhedron" in LinPy) is the set of integer
+    points in a convex polyhedron.
"""

__slots__ = (
@@ -56,28 +57,31 @@ class Polyhedron(Domain):
0 <= x <= 2, 0 <= y <= 2 can be constructed with:

>>> x, y = symbols('x y')
-        >>> square = Polyhedron([], [x, 2 - x, y, 2 - y])
+        >>> square1 = Polyhedron([], [x, 2 - x, y, 2 - y])
+        >>> square1
+        And(0 <= x, x <= 2, 0 <= y, y <= 2)

It may be easier to use comparison operators LinExpr.__lt__(),
LinExpr.__le__(), LinExpr.__ge__(), LinExpr.__gt__(), or functions Lt(),
Le(), Eq(), Ge() and Gt(), using one of the following instructions:

>>> x, y = symbols('x y')
-        >>> square = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
-        >>> square = Le(0, x, 2) & Le(0, y, 2)
+        >>> square1 = (0 <= x) & (x <= 2) & (0 <= y) & (y <= 2)
+        >>> square1 = Le(0, x, 2) & Le(0, y, 2)

It is also possible to build a polyhedron from a string.

-        >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+        >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')

Finally, a polyhedron can be constructed from a GeometricObject
instance, calling the GeometricObject.aspolyedron() method. This way, it
is possible to compute the polyhedral hull of a Domain instance, i.e.,
the convex hull of two polyhedra:

-        >>> square = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
-        >>> square2 = Polyhedron('2 <= x <= 4, 2 <= y <= 4')
-        >>> Polyhedron(square | square2)
+        >>> square1 = Polyhedron('0 <= x <= 2, 0 <= y <= 2')
+        >>> square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')
+        >>> Polyhedron(square1 | square2)
+        And(0 <= x, 0 <= y, x <= y + 2, y <= x + 2, x <= 3, y <= 3)
"""
if isinstance(equalities, str):
if inequalities is not None:
@@ -90,15 +94,23 @@ class Polyhedron(Domain):
sc_equalities = []
if equalities is not None:
for equality in equalities:
-                if not isinstance(equality, LinExpr):
-                    raise TypeError('equalities must be linear expressions')
-                sc_equalities.append(equality.scaleint())
+                if isinstance(equality, LinExpr):
+                    sc_equalities.append(equality.scaleint())
+                elif isinstance(equality, numbers.Rational):
+                    sc_equalities.append(Rational(equality).scaleint())
+                else:
+                    raise TypeError('equalities must be linear expressions '
+                        'or rational numbers')
sc_inequalities = []
if inequalities is not None:
for inequality in inequalities:
-                if not isinstance(inequality, LinExpr):
-                    raise TypeError('inequalities must be linear expressions')
-                sc_inequalities.append(inequality.scaleint())
+                if isinstance(inequality, LinExpr):
+                    sc_inequalities.append(inequality.scaleint())
+                elif isinstance(inequality, numbers.Rational):
+                    sc_inequalities.append(Rational(inequality).scaleint())
+                else:
+                    raise TypeError('inequalities must be linear expressions '
+                        'or rational numbers')
symbols = cls._xsymbols(sc_equalities + sc_inequalities)
islbset = cls._toislbasicset(sc_equalities, sc_inequalities, symbols)
return cls._fromislbasicset(islbset, symbols)
@@ -144,6 +156,15 @@ class Polyhedron(Domain):
def aspolyhedron(self):
return self

+    def convex_union(self, *others):
+        """
+        Return the convex union of two or more polyhedra.
+        """
+        for other in others:
+            if not isinstance(other, Polyhedron):
+                raise TypeError('arguments must be Polyhedron instances')
+        return Polyhedron(self.union(*others))
+
def __contains__(self, point):
if not isinstance(point, Point):
raise TypeError('point must be a Point instance')
@@ -164,7 +185,11 @@ class Polyhedron(Domain):
for inequality in self.inequalities]
return Polyhedron(equalities, inequalities)

-    def _asinequalities(self):
+    def asinequalities(self):
+        """
+        Express the polyhedron using inequalities, given as a list of
+        expressions greater or equal to 0.
+        """
inequalities = list(self.equalities)
inequalities.extend([-expression for expression in self.equalities])
inequalities.extend(self.inequalities)
@@ -173,11 +198,14 @@ class Polyhedron(Domain):
def widen(self, other):
"""
Compute the standard widening of two polyhedra, à la Halbwachs.
+
+        In its current implementation, this method is slow and should not be
+        used on large polyhedra.
"""
if not isinstance(other, Polyhedron):
-            raise ValueError('argument must be a Polyhedron instance')
-        inequalities1 = self._asinequalities()
-        inequalities2 = other._asinequalities()
+            raise TypeError('argument must be a Polyhedron instance')
+        inequalities1 = self.asinequalities()
+        inequalities2 = other.asinequalities()
inequalities = []
for inequality1 in inequalities1:
if other <= Polyhedron(inequalities=[inequality1]):
@@ -265,27 +293,43 @@ class Polyhedron(Domain):
def __repr__(self):
strings = []
for equality in self.equalities:
-            strings.append('Eq({}, 0)'.format(equality))
+            left, right, swap = 0, 0, False
+            for i, (symbol, coefficient) in enumerate(equality.coefficients()):
+                if coefficient > 0:
+                    left += coefficient * symbol
+                else:
+                    right -= coefficient * symbol
+                    if i == 0:
+                        swap = True
+            if equality.constant > 0:
+                left += equality.constant
+            else:
+                right -= equality.constant
+            if swap:
+                left, right = right, left
+            strings.append('{} == {}'.format(left, right))
for inequality in self.inequalities:
-            strings.append('Ge({}, 0)'.format(inequality))
+            left, right = 0, 0
+            for symbol, coefficient in inequality.coefficients():
+                if coefficient < 0:
+                    left -= coefficient * symbol
+                else:
+                    right += coefficient * symbol
+            if inequality.constant < 0:
+                left -= inequality.constant
+            else:
+                right += inequality.constant
+            strings.append('{} <= {}'.format(left, right))
if len(strings) == 1:
return strings
else:
return 'And({})'.format(', '.join(strings))

-    def _repr_latex_(self):
-        strings = []
-        for equality in self.equalities:
-            strings.append('{} = 0'.format(equality._repr_latex_().strip('\$')))
-        for inequality in self.inequalities:
-            strings.append('{} \\ge 0'.format(inequality._repr_latex_().strip('\$')))
-        return '\$\${}\$\$'.format(' \\wedge '.join(strings))
-
@classmethod
-    def fromsympy(cls, expr):
-        domain = Domain.fromsympy(expr)
+    def fromsympy(cls, expression):
+        domain = Domain.fromsympy(expression)
if not isinstance(domain, Polyhedron):
-            raise ValueError('non-polyhedral expression: {!r}'.format(expr))
+            raise ValueError('non-polyhedral expression: {!r}'.format(expression))
return domain

def tosympy(self):
@@ -303,8 +347,6 @@ class EmptyType(Polyhedron):
The empty polyhedron, whose set of constraints is not satisfiable.
"""

-    __slots__ = Polyhedron.__slots__
-
def __new__(cls):
self = object().__new__(cls)
self._equalities = (Rational(1),)
@@ -321,9 +363,6 @@ class EmptyType(Polyhedron):
def __repr__(self):
return 'Empty'

-    def _repr_latex_(self):
-        return '\$\$\\emptyset\$\$'
-
Empty = EmptyType()

@@ -333,8 +372,6 @@ class UniverseType(Polyhedron):
i.e. is empty.
"""

-    __slots__ = Polyhedron.__slots__
-
def __new__(cls):
self = object().__new__(cls)
self._equalities = ()
@@ -346,69 +383,80 @@ class UniverseType(Polyhedron):
def __repr__(self):
return 'Universe'

-    def _repr_latex_(self):
-        return '\$\$\\Omega\$\$'
-
Universe = UniverseType()

-def _polymorphic(func):
+def _pseudoconstructor(func):
@functools.wraps(func)
-    def wrapper(left, right):
-        if not isinstance(left, LinExpr):
-            if isinstance(left, numbers.Rational):
-                left = Rational(left)
-            else:
-                raise TypeError('left must be a a rational number '
-                    'or a linear expression')
-        if not isinstance(right, LinExpr):
-            if isinstance(right, numbers.Rational):
-                right = Rational(right)
-            else:
-                raise TypeError('right must be a a rational number '
-                    'or a linear expression')
-        return func(left, right)
+    def wrapper(expression1, expression2, *expressions):
+        expressions = (expression1, expression2) + expressions
+        for expression in expressions:
+            if not isinstance(expression, LinExpr):
+                if isinstance(expression, numbers.Rational):
+                    expression = Rational(expression)
+                else:
+                    raise TypeError('arguments must be rational numbers '
+                        'or linear expressions')
+        return func(*expressions)
return wrapper

-@_polymorphic
-def Lt(left, right):
+@_pseudoconstructor
+def Lt(*expressions):
"""
Create the polyhedron with constraints expr1 < expr2 < expr3 ...
"""
-    return Polyhedron([], [right - left - 1])
+    inequalities = []
+    for left, right in zip(expressions, expressions[1:]):
+        inequalities.append(right - left - 1)
+    return Polyhedron([], inequalities)

-@_polymorphic
-def Le(left, right):
+@_pseudoconstructor
+def Le(*expressions):
"""
Create the polyhedron with constraints expr1 <= expr2 <= expr3 ...
"""
-    return Polyhedron([], [right - left])
+    inequalities = []
+    for left, right in zip(expressions, expressions[1:]):
+        inequalities.append(right - left)
+    return Polyhedron([], inequalities)

-@_polymorphic
-def Eq(left, right):
+@_pseudoconstructor
+def Eq(*expressions):
"""
Create the polyhedron with constraints expr1 == expr2 == expr3 ...
"""
-    return Polyhedron([left - right], [])
+    equalities = []
+    for left, right in zip(expressions, expressions[1:]):
+        equalities.append(left - right)
+    return Polyhedron(equalities, [])

-@_polymorphic
-def Ne(left, right):
+@_pseudoconstructor
+def Ne(*expressions):
"""
Create the domain such that expr1 != expr2 != expr3 ... The result is a
-    Domain, not a Polyhedron.
+    Domain object, not a Polyhedron.
"""
-    return ~Eq(left, right)
+    domain = Universe
+    for left, right in zip(expressions, expressions[1:]):
+        domain &= ~Eq(left, right)
+    return domain

-@_polymorphic
-def Gt(left, right):
+@_pseudoconstructor
+def Ge(*expressions):
"""
-    Create the polyhedron with constraints expr1 > expr2 > expr3 ...
+    Create the polyhedron with constraints expr1 >= expr2 >= expr3 ...
"""
-    return Polyhedron([], [left - right - 1])
+    inequalities = []
+    for left, right in zip(expressions, expressions[1:]):
+        inequalities.append(left - right)
+    return Polyhedron([], inequalities)

-@_polymorphic
-def Ge(left, right):
+@_pseudoconstructor
+def Gt(*expressions):
"""
-    Create the polyhedron with constraints expr1 >= expr2 >= expr3 ...
+    Create the polyhedron with constraints expr1 > expr2 > expr3 ...
"""
-    return Polyhedron([], [left - right])
+    inequalities = []
+    for left, right in zip(expressions, expressions[1:]):
+        inequalities.append(left - right - 1)
+    return Polyhedron([], inequalities)