interval.py

# from http://pypi.python.org/pypi/interval/1.0.0

"""Provides the Interval and IntervalSet classes

The interval module provides the Interval and IntervalSet data types.   
Intervals describe continuous ranges that can be open, closed, half-open,
or infinite.  IntervalSets contain zero to many disjoint sets of 
Intervals.

Intervals don't have to pertain to numbers.  They can contain any data 
that is comparable via the Python operators <, <=, ==, >=, and >.  Here's 
an example of how strings can be used with Intervals:

>>> volume1 = Interval.between("A", "Foe")
>>> volume2 = Interval.between("Fog", "McAfee")
>>> volume3 = Interval.between("McDonalds", "Space")
>>> volume4 = Interval.between("Spade", "Zygote")
>>> encyclopedia = IntervalSet([volume1, volume2, volume3, volume4])
>>> mySet = IntervalSet([volume1, volume3, volume4])
>>> "Meteor" in encyclopedia
True
>>> "Goose" in encyclopedia
True
>>> "Goose" in mySet
False
>>> volume2 in (encyclopedia ^ mySet)
True

Here's an example of how times can be used with Intervals:

>>> officeHours = IntervalSet.between("08:00", "17:00")
>>> myLunch = IntervalSet.between("11:30", "12:30")
>>> myHours = IntervalSet.between("08:30", "19:30") - myLunch
>>> myHours.issubset(officeHours)
False
>>> "12:00" in myHours
False
>>> "15:30" in myHours
True
>>> inOffice = officeHours & myHours
>>> print inOffice
['08:30'..'11:30'),('12:30'..'17:00']
>>> overtime = myHours - officeHours
>>> print overtime
('17:00'..'19:30']
"""

import copy

class Smallest:
  """Represents the smallest value
  
  This type doesn't do much; it implements a pseudo-value that's smaller
  than everything but itself.
  
  >>> negInf = Smallest()
  >>> smallest = Smallest()
  >>> -264 < negInf
  False
  >>> -264 == negInf
  False
  >>> -264 > negInf
  True
  >>> negInf < negInf
  False
  >>> negInf == smallest
  True
  """
  
  def __neg__(self):
    """Returns the largest value
    
    The opposite of negative infinity is infinity, the largest value.

    >>> print -Smallest()
    ~
    """
    return Largest()
    
  def __cmp__(self, other):
    """Compares this with another object
    
    Always indicates that self is less than other, unless both are of 
    type Smallest, in which case they are equal.
    
    >>> 0 < Smallest()
    False
    >>> -9999999 < Smallest()
    False
    >>> Smallest() < -9999999
    True
    >>> Smallest() < Smallest()
    False
    >>> Smallest() == Smallest()
    True
    """
    if other.__class__ == self.__class__:
        retval = 0
    else:
        retval = -1
    return retval
    
  def __str__(self):
      """Returns a printable representation of this value
      
      The string for the smallest number is -~, which means negative infinity.
      
      >>> print Smallest()
      -~
      """
      return "-~"
    
  def __repr__(self):
    """Returns an evaluable representation of the object
    
    The representation of the smallest number is -Inf, which means 
    negative infinity.

    >>> Smallest()
    -Inf
    """
    return "-Inf"
    
  def __hash__(self):
    "Returns a value that can be used for generating hashes"
    return 0x55555555


class Largest:
  """Class representing the universal largest value
  
  This type doesn't do much; it implements a pseudo-value that's larger
  than everything but itself.
  
  >>> infinity = Largest()
  >>> greatest = Largest()
  >>> 6234 < infinity
  True
  >>> 6234 == infinity
  False
  >>> 6234 > infinity
  False
  >>> infinity > infinity
  False
  >>> infinity == greatest
  True
  """
  
  def __neg__(self):
    """Returns the smallest universal value
    
    The opposite of infinity is negative infinity, the smallest value.

    >>> print -Largest()
    -~
    """
    return Smallest()
    
  def __cmp__(self, other):
    """Compares object with another object
    
    Always indicates that self is greater than other, unless both are of
    type Largest, in which case they are equal.
    
    >>> 0 > Largest()
    False
    >>> Largest() < 9999999
    False
    >>> Largest() > 9999999
    True
    >>> Largest() < Largest()
    False
    >>> Largest() == Largest()
    True
    """
    if other.__class__ == self.__class__:
        retval = 0
    else:
        retval = 1
    return retval

  def __str__(self):
      """Returns a string representation of the object
      
      The largest number is displayed as ~ (it sort of looks like infinity...)
      
      >>> print Largest()
      ~
      """
      return "~"
    
  def __repr__(self):
    """Returns an evaluable expression representing this object
    
    >>> Largest()
    Inf
    """
    return "Inf"

  def __hash__(self):
    "Returns a value that can be used for generating hashes"
    return -0x55555555
    

Inf = Largest()
# Use -Inf for the smallest value      

class Interval:
    """Represents a continuous range of values
    
    An Interval is composed of the lower bound, a closed lower bound 
    flag, an upper bound, and a closed upper bound flag.  The attributes
    are called lower_bound, lower_closed, upper_bound, and upper_closed,
    respectively.  For an infinite interval, the bound is set to inf or 
    -inf.  IntervalSets are composed of zero to many Intervals.
    """

    def __init__(self, lower_bound=-Inf, upper_bound=Inf, **kwargs):
        """Initializes an interval
        
        Parameters
        ==========
        - lower_bound: The lower bound of an interval (default -Inf)
        - upper_bound: The upper bound of an interval (default Inf)
        - closed: Boolean telling whether both ends of the interval are closed
          (default True).  Setting this sets both lower_closed and upper_closed
        - lower_closed: Boolean telling whether the lower end of the interval
          is closed (default True)
        - upper_closed: Boolean telling whether the upper end of the interval
          is closed (default True)
        
        An Interval can represent an infinite set.

        >>> r = Interval(-Inf, Inf) # All values

        An Interval can represent sets unbounded on an end.

        >>> r = Interval(upper_bound=62, closed=False)
        >>> r = Interval(upper_bound=37)
        >>> r = Interval(lower_bound=246)
        >>> r = Interval(lower_bound=2468, closed=False)

        An Interval can represent a set of values up to, but not including a
        value.

        >>> r = Interval(25, 28, closed=False)

        An Interval can represent a set of values that have an inclusive
        boundary.

        >>> r = Interval(29, 216)

        An Interval can represent a single value

        >>> r = Interval(82, 82)

        Intervals that are not normalized, i.e. that have a lower bound
        exceeding an upper bound, are silently normalized.

        >>> print Interval(5, 2, lower_closed=False)
        [2..5)

        Intervals can represent an empty set.

        >>> r = Interval(5, 5, closed=False)
        
        Intervals can only contain hashable (immutable) objects.

        >>> r = Interval([], 12)
        Traceback (most recent call last):
        ...
        TypeError: lower_bound is not hashable.
        >>> r = Interval(12, [])
        Traceback (most recent call last):
        ...
        TypeError: upper_bound is not hashable.
        """
        try:
            h = hash(lower_bound)
        except TypeError:
            raise TypeError("lower_bound is not hashable.")
            
        try:
            h = hash(upper_bound)
        except TypeError:
            raise TypeError("upper_bound is not hashable.")
        
        lower_closed = not (
            isinstance(lower_bound, Smallest) 
            or isinstance(lower_bound, Largest)) \
          and kwargs.get("lower_closed", kwargs.get("closed", True))
        upper_closed = not (
            isinstance(upper_bound, Smallest)
            or isinstance(upper_bound, Largest)) \
          and kwargs.get("upper_closed", kwargs.get("closed", True))
        
        if upper_bound < lower_bound:
            lower_bound, lower_closed, upper_bound, upper_closed = \
            upper_bound, upper_closed, lower_bound, lower_closed 
        if ((lower_bound == -Inf) and lower_closed) \
          or ((upper_bound == Inf) and upper_closed):
            raise ValueError(
                "Unbound ends cannot be included in an interval.")

        self.lower_bound  = lower_bound 
        self.lower_closed = lower_closed
        self.upper_bound  = upper_bound
        self.upper_closed = upper_closed

    def __hash__(self):
        """Returns a hashed value of the object
        
        Intervals are to be considered immutable.  Thus, a 32-bit hash can
        be generated for them.
        
        >>> h = hash(Interval.less_than(5))
        """
        return hash((
            self.lower_bound, self.lower_closed, 
            self.upper_bound, self.upper_closed))
        
    def __repr__(self):
        """Returns an evaluable expression that can reproduce the object
        
        >>> Interval(3, 6)
        Interval(3, 6, lower_closed=True, upper_closed=True)
        >>> Interval(3, 6, closed=False)
        Interval(3, 6, lower_closed=False, upper_closed=False)
        >>> Interval(3, 6, lower_closed=False)
        Interval(3, 6, lower_closed=False, upper_closed=True)
        >>> Interval()
        Interval(-Inf, Inf, lower_closed=False, upper_closed=False)
        """
        return "Interval(%s, %s, lower_closed=%s, upper_closed=%s)" % (
            repr(self.lower_bound), repr(self.upper_bound), 
            self.lower_closed, self.upper_closed)
      
    def __str__(self):
        """Returns a string representation of the object
        
        This function yields a graphical representation of an Interval.  
        It is included in the __str__ of an IntervalSet.  Non-inclusive 
        boundaries are bordered by a ( or ).  Inclusive boundaries are 
        bordered by [ or ].  Unbound values are shown as ....  Intervals 
        consisting of only a single value are shown as that value.  Empty 
        intervals are shown as the string <Empty>
        
        >>> print Interval.all()
        (...)
        >>> print Interval.less_than(100)
        (...100)
        >>> print Interval.less_than_or_equal_to(2593)
        (...2593]
        >>> print Interval.greater_than(2378)
        (2378...)
        >>> print Interval.between(26, 8234, False)
        (26..8234)
        >>> print Interval(237, 2348, lower_closed=False)
        (237..2348]
        >>> print Interval.greater_than_or_equal_to(347)
        [347...)
        >>> print Interval(237, 278, upper_closed=False)
        [237..278)
        >>> print Interval.between(723, 2378)
        [723..2378]
        >>> print Interval.equal_to(5)
        5
        >>> print Interval.none()
        <Empty>
        """
        if self.lower_bound == self.upper_bound:
            if self.lower_closed or self.upper_closed:
                retval = repr(self.lower_bound)
            else:
                retval = "<Empty>"
        else:
            between = ".."
            
            if self.lower_closed:
                lbchar = '['
            else:
                lbchar = '('
                
            if self.lower_bound == -Inf:
                lstr = ""
                between = "..."
            else:
                lstr = repr(self.lower_bound)
                
            if self.upper_closed:
                ubchar = ']'
            else:
                ubchar = ')'
                
            if self.upper_bound == Inf:
                ustr = ""
                between = "..."
            else:
                ustr = repr(self.upper_bound)
                
            retval = "".join([lbchar, lstr, between, ustr, ubchar])
        return retval

    def __nonzero__(self):
        """Tells whether the interval is empty
        
        >>> if Interval(12, 12, closed=False):
        ...   print "Non-empty"
        >>> if Interval(12, 12, upper_closed=False):
        ...   print "Non-empty"
        >>> if Interval(12, 12):
        ...   print "Non-empty"
        Non-empty
        """
        return self.lower_bound != self.upper_bound \
            or (self.upper_closed and self.lower_closed)

    def __cmp__(self, other):
        """Compares two intervals for ordering purposes
        
        >>> Interval.equal_to(-1) < Interval.equal_to(2)
        True
        >>> Interval.equal_to(-1) == Interval.equal_to(2)
        False
        >>> Interval.equal_to(-1) > Interval.equal_to(2)
        False
        >>> Interval.between(2, 5) > Interval.between(2, 4)
        True
        >>> Interval.between(2, 5) == Interval.between(2, 4)
        False
        >>> Interval.between(2, 5) == Interval.between(2, 5)
        True
        >>> Interval.between(2, 5) >= Interval.between(2, 5)
        True
        """
        if self == other:
            result = 0
        elif self.comes_before(other):
            result = -1
        else:
            result = 1
        return result
            
    def __and__(self, other):
        """Returns the intersection of two intervals

        >>> print Interval.greater_than(3) & Interval.greater_than(5)
        (5...)
        >>> print Interval.greater_than(3) & Interval.equal_to(3)
        <Empty>
        >>> print Interval.greater_than_or_equal_to(3) & Interval.equal_to(3)
        3
        >>> print Interval.all() & Interval.all()
        (...)
        >>> print Interval.greater_than(3) & Interval.less_than(10)
        (3..10)
        """
        if self == other:
            result = Interval()
            result.lower_bound = self.lower_bound
            result.upper_bound = self.upper_bound
            result.lower_closed = self.lower_closed
            result.upper_closed = self.upper_closed
        elif self.comes_before(other):
            if self.overlaps(other):
                if self.lower_bound == other.lower_bound:
                    lower = self.lower_bound
                    lower_closed = min(self.lower_closed, other.lower_closed)
                elif self.lower_bound > other.lower_bound:
                    lower = self.lower_bound
                    lower_closed = self.lower_closed
                else:
                    lower = other.lower_bound
                    lower_closed = other.lower_closed
                    
                if self.upper_bound == other.upper_bound:
                    upper = self.upper_bound
                    upper_closed = min(self.upper_closed, other.upper_closed)
                elif self.upper_bound < other.upper_bound:
                    upper = self.upper_bound
                    upper_closed = self.upper_closed
                else:
                    upper = other.upper_bound
                    upper_closed = other.upper_closed
                    
                result = Interval(
                    lower, upper, 
                    lower_closed=lower_closed, upper_closed=upper_closed)
            else:
                result = Interval.none()
        else:
            result = other & self
        return result
            
    @classmethod
    def none(cls):
        """Returns an empty interval
        
        >>> print Interval.none()
        <Empty>
        """
        return cls(0, 0, closed=False)
            
    @classmethod
    def all(cls):
      """Returns an interval encompassing all values
      
      >>> print Interval.all()
      (...)
      """
      return cls()

    @classmethod
    def between(cls, a, b, closed=True):
      """Returns an interval between two values
      
      Returns an interval between values a and b.  If closed is True,
      then the endpoints are included.  Otherwise, the endpoints are
      excluded.

      >>> print Interval.between(2, 4)
      [2..4]
      >>> print Interval.between(2, 4, False)
      (2..4)
      """
      return cls(a, b, closed=closed)
      
    @classmethod
    def equal_to(cls, a):
      """Returns an point interval
      
      Returns an interval containing only a.

      >>> print Interval.equal_to(32)
      32
      """
      return cls(a, a)
      
    @classmethod
    def less_than(cls, a):
      """Returns interval of all values less than the given value
      
      Returns an interval containing all values less than a.  If closed
      is True, then all values less than or equal to a are returned.

      >>> print Interval.less_than(32)
      (...32)
      """
      return cls(upper_bound=a, upper_closed=False)
    
    @classmethod
    def less_than_or_equal_to(cls, a):
        """Returns an interval containing the given values and everything less

        >>> print Interval.less_than_or_equal_to(32)
        (...32]
        """
        return cls(upper_bound=a, upper_closed=True)
      
    @classmethod
    def greater_than(cls, a):
      """Returns interval of all values greater than the given value
      
      >>> print Interval.greater_than(32)
      (32...)
      """
      return cls(lower_bound=a, lower_closed=False)
    
    @classmethod
    def greater_than_or_equal_to(cls, a):
      """Returns interval of all values greater than or equal to the given value
      
      >>> print Interval.greater_than_or_equal_to(32)
      [32...)
      """
      return cls(lower_bound=a, lower_closed=True)
    
    def comes_before(self, other):
        """Tells whether an interval lies before the object
        
        self comes before other when sorted if its lower bound is less 
        than other's smallest value.  If the smallest value is the same, 
        then the Interval with the smallest upper bound comes first.  
        Otherwise, they are equal.

        >>> Interval.equal_to(1).comes_before(Interval.equal_to(4))
        True
        >>> Interval.less_than_or_equal_to(1).comes_before(Interval.equal_to(4))
        True
        >>> Interval.less_than_or_equal_to(5).comes_before(
        ...   Interval.less_than(5))
        False
        >>> Interval.less_than(5).comes_before(
        ...   Interval.less_than_or_equal_to(5))
        True
        >>> Interval.all().comes_before(Interval.all())
        False
        """
        if self == other:
            result = False
        elif self.lower_bound < other.lower_bound:
            result = True
        elif self.lower_bound > other.lower_bound:
            result = False
        elif self.lower_closed == other.lower_closed:
            if self.upper_bound < other.upper_bound:
                result = True
            elif self.upper_bound > other.upper_bound \
              or self.upper_closed == other.upper_closed \
              or self.upper_closed:
                result = False
            else:
                result = True
        elif self.lower_closed:
            result = True
        else:
            result = False
            
        return result

    def join(self, other):
        """Combines two continous Intervals
        
        Combines two continuous Intervals into one Interval.  If the two
        Intervals are disjoint, then an exception is raised.

        >>> r1  = Interval.less_than(-100)
        >>> r2  = Interval.less_than_or_equal_to(-100)
        >>> r3  = Interval.less_than(100)
        >>> r4  = Interval.less_than_or_equal_to(100)
        >>> r5  = Interval.all()
        >>> r6  = Interval.between(-100, 100, False)
        >>> r7  = Interval(-100, 100, lower_closed=False)
        >>> r8  = Interval.greater_than(-100)
        >>> r9  = Interval.equal_to(-100)
        >>> r10 = Interval(-100, 100, upper_closed=False)
        >>> r11 = Interval.between(-100, 100)
        >>> r12 = Interval.greater_than_or_equal_to(-100)
        >>> r13 = Interval.greater_than(100)
        >>> r14 = Interval.equal_to(100)
        >>> r15 = Interval.greater_than_or_equal_to(100)
        >>> print r13.join(r15)
        [100...)
        >>> print r7.join(r6)
        (-100..100]
        >>> print r11.join(r2)
        (...100]
        >>> print r4.join(r15)
        (...)
        >>> print r8.join(r8)
        (-100...)
        >>> print r3.join(r7)
        (...100]
        >>> print r5.join(r10)
        (...)
        >>> print r9.join(r1)
        (...-100]
        >>> print r12.join(r5)
        (...)
        >>> print r13.join(r1)
        Traceback (most recent call last):
        ...
        ArithmeticError: The Intervals are disjoint.
        >>> print r14.join(r2)
        Traceback (most recent call last):
        ...
        ArithmeticError: The Intervals are disjoint.
        """
        if self.overlaps(other) or self.adjacent_to(other):
            if self.lower_bound < other.lower_bound:
                lbound = self.lower_bound
                linc = self.lower_closed
            elif self.lower_bound == other.lower_bound:
                lbound = self.lower_bound
                linc = max(self.lower_closed, other.lower_closed)
            else:
                lbound = other.lower_bound
                linc = other.lower_closed

            if self.upper_bound > other.upper_bound:
                ubound = self.upper_bound
                uinc = self.upper_closed
            elif self.upper_bound == other.upper_bound:
                ubound = self.upper_bound
                uinc = max(self.upper_closed, other.upper_closed)
            else:
                ubound = other.upper_bound
                uinc = other.upper_closed
            return Interval(
                lbound, ubound, upper_closed=uinc, lower_closed=linc)
        else:
            raise ArithmeticError("The Intervals are disjoint.")    
        
    def __contains__(self, obj):
        """Returns True if obj lies wholly within the Interval.

        >>> all    = Interval.all()
        >>> lt     = Interval.less_than(10)
        >>> le     = Interval.less_than_or_equal_to(10)
        >>> some   = Interval(10, 20, lower_closed=False)
        >>> single = Interval.equal_to(10)
        >>> ge     = Interval.greater_than_or_equal_to(10)
        >>> gt     = Interval.greater_than(10)
        >>> ne     = Interval.equal_to(17)
        >>> 10 in all
        True
        >>> 10 in lt
        False
        >>> 10 in le
        True
        >>> 10 in some
        False
        >>> 10 in single
        True
        >>> 10 in ge
        True
        >>> 10 in gt
        False
        >>> 10 in ne
        False
        >>> all in some
        False
        >>> lt in all
        True
        >>> lt in some
        False
        >>> single in ge
        True
        >>> ne in some
        True
        """
        if isinstance(obj, Interval):
            if obj.lower_bound < self.lower_bound:
                insideLower = False
            elif obj.lower_bound == self.lower_bound:
                insideLower = (obj.lower_closed <= self.lower_closed)
            else:
                insideLower = True

            if obj.upper_bound > self.upper_bound:
                insideUpper = False
            elif obj.upper_bound == self.upper_bound:
                insideUpper = (obj.upper_closed <= self.upper_closed)
            else:
                insideUpper = True

            result = insideLower and insideUpper
        else:
            result = Interval.equal_to(obj) in self
        return result

    def overlaps(self, other):
        """Tells whether the given interval overlaps the object
        
        Returns True if the one Interval overlaps another.  If they are
        immediately adjacent, then this returns False.  Use the adjacent_to
        function for testing for adjacent Intervals.

        >>> r1  = Interval.less_than(-100)
        >>> r2  = Interval.less_than_or_equal_to(-100)
        >>> r3  = Interval.less_than(100)
        >>> r4  = Interval.less_than_or_equal_to(100)
        >>> r5  = Interval.all()
        >>> r6  = Interval.between(-100, 100, False)
        >>> r7  = Interval(-100, 100, lower_closed=False)
        >>> r8  = Interval.greater_than(-100)
        >>> r9  = Interval.equal_to(-100)
        >>> r10 = Interval(-100, 100, upper_closed=False)
        >>> r11 = Interval.between(-100, 100)
        >>> r12 = Interval.greater_than_or_equal_to(-100)
        >>> r13 = Interval.greater_than(100)
        >>> r14 = Interval.equal_to(100)
        >>> r15 = Interval.greater_than_or_equal_to(100)
        >>> r8.overlaps(r9)
        False
        >>> r12.overlaps(r6)
        True
        >>> r7.overlaps(r8)
        True
        >>> r8.overlaps(r4)
        True
        >>> r14.overlaps(r11)
        True
        >>> r10.overlaps(r13)
        False
        >>> r5.overlaps(r1)
        True
        >>> r5.overlaps(r2)
        True
        >>> r15.overlaps(r6)
        False
        >>> r3.overlaps(r1)
        True
        """
        if self == other:
            result = True
        elif other.comes_before(self):
            result = other.overlaps(self)
        elif other.lower_bound < self.upper_bound:
            result = True
        elif other.lower_bound == self.upper_bound:
            result = (other.lower_closed and self.upper_closed)
        else:
            result = False
        return result           

    def adjacent_to(self, other):
        """Tells whether an Interval is adjacent to the object without overlap
        
        Returns True if self is adjacent to other, meaning that if they 
        were joined, there would be no discontinuity.  They cannot 
        overlap.

        >>> r1  = Interval.less_than(-100)
        >>> r2  = Interval.less_than_or_equal_to(-100)
        >>> r3  = Interval.less_than(100)
        >>> r4  = Interval.less_than_or_equal_to(100)
        >>> r5  = Interval.all()
        >>> r6  = Interval.between(-100, 100, False)
        >>> r7  = Interval(-100, 100, lower_closed=False)
        >>> r8  = Interval.greater_than(-100)
        >>> r9  = Interval.equal_to(-100)
        >>> r10 = Interval(-100, 100, upper_closed=False)
        >>> r11 = Interval.between(-100, 100)
        >>> r12 = Interval.greater_than_or_equal_to(-100)
        >>> r13 = Interval.greater_than(100)
        >>> r14 = Interval.equal_to(100)
        >>> r15 = Interval.greater_than_or_equal_to(100)
        >>> r1.adjacent_to(r6)
        False
        >>> r6.adjacent_to(r11)
        False
        >>> r7.adjacent_to(r9)
        True
        >>> r3.adjacent_to(r10)
        False
        >>> r5.adjacent_to(r14)
        False
        >>> r6.adjacent_to(r15)
        True
        >>> r1.adjacent_to(r8)
        False
        >>> r12.adjacent_to(r14)
        False
        >>> r6.adjacent_to(r13)
        False
        >>> r2.adjacent_to(r15)
        False
        >>> r1.adjacent_to(r4)
        False
        """ 
        if self.comes_before(other):
            if self.upper_bound == other.lower_bound:
                result = (self.upper_closed != other.lower_closed)
            else:
                result = False
        elif self == other:
            result = False
        else:
            result = other.adjacent_to(self)
        return result
        
    def __eq__(self, other):
        """Test if an interval is equivalent to the object
        
        >>> Interval.all() == Interval.none()
        False
        >>> Interval.equal_to(4) == Interval(4, 4)
        True
        >>> Interval(2, 2, closed=False) == Interval(0, 0, closed=False)
        True
        """
        return (\
            self.lower_bound == self.upper_bound and (\
                not self.lower_closed or not self.upper_closed)\
            and other.lower_bound == other.upper_bound and (\
                not other.lower_closed or not other.upper_closed))\
            or (\
                self.lower_bound == other.lower_bound \
                and self.upper_bound == other.upper_bound \
                and self.lower_closed == other.lower_closed \
                and self.upper_closed == other.upper_closed)
            

class BaseIntervalSet(object):
    "Base class for IntervalSet and FrozenIntervalSet."
    
    def __init__(self, items=[]):
        """Initializes a BaseIntervalSet
        
        This function initializes an IntervalSet.  It takes an iterable
        object, such as a set, list, or generator.  The elements returned 
        by the iterator are interpreted as intervals for Interval objects 
        and discrete values for all other values.

        If no parameters are provided, then an empty IntervalSet is 
        constructed.

        >>> print IntervalSet() # An empty set
        <Empty>

        Interval objects arguments are added directly to the IntervalSet.

        >>> print IntervalSet([Interval(4, 6, lower_closed=False)])
        (4..6]
        >>> print IntervalSet([Interval.less_than_or_equal_to(2)])
        (...2]

        Each non-Interval value of an iterator is added as a discrete 
        value.

        >>> print IntervalSet(set([3, 7, 2, 1]))
        1,2,3,7
        >>> print IntervalSet(["Bob", "Fred", "Mary"])
        'Bob','Fred','Mary'
        >>> print IntervalSet(range(10))
        0,1,2,3,4,5,6,7,8,9
        >>> print IntervalSet(
        ...   Interval.between(l, u) for l, u in [(10, 20), (30, 40)])
        [10..20],[30..40]
        """
        self.intervals = []
        for i in items:
            self._add(i)
        self.intervals.sort()

    def __len__(self):
        """Returns the number of intervals contained in the object

        >>> len(IntervalSet.empty())
        0
        >>> len(IntervalSet.all())
        1
        >>> len(IntervalSet([2, 6, 34]))
        3
        >>> len(IntervalSet.greater_than(0))
        1
        >>> nonempty = IntervalSet([3])
        >>> if IntervalSet.empty():
        ...     print "Non-empty"
        >>> if nonempty:
        ...     print "Non-empty"
        Non-empty
        """
        return len(self.intervals)

    def __str__(self):
        """Returns a string representation of the object
        
        This function shows a string representation of an IntervalSet.  
        The string is shown sorted, with all intervals normalized.

        >>> print IntervalSet()
        <Empty>
        >>> print IntervalSet([62])
        62
        >>> print IntervalSet([62, 56])
        56,62
        >>> print IntervalSet([23, Interval(26, 32, upper_closed=False)])
        23,[26..32)
        >>> print IntervalSet.less_than(3) + IntervalSet.greater_than(3)
        (...3),(3...)
        >>> print IntervalSet([Interval.less_than_or_equal_to(6)])
        (...6]
        """
        if len(self.intervals) == 0:
            rangeStr = "<Empty>"
        else:
            def sortFn(x, y):
                if x.comes_before(y):
                    retval = -1
                elif y.comes_before(x):
                    retval = 1
                else:
                    retval = 0                
                return retval            
            rangeStr = ",".join([str(r) for r in self.intervals])
        return rangeStr

    def __getitem__(self, index):
        """Gets the interval at the given index

        Unlike sets, which do not have ordering, BaseIntervalSets do.  Therefore,
        indexing was implemented.  Intervals are stored in order, starting with
        that with the left-most lower bound to that with the right-most.

        >>> IntervalSet()[0]
        Traceback (most recent call last):
            ...
        IndexError: Index is out of range
        >>> interval = IntervalSet.greater_than(5)
        >>> print interval[0]
        (5...)
        >>> print interval[1]
        Traceback (most recent call last):
            ...
        IndexError: Index is out of range
        >>> print interval[-1]
        (5...)
        >>> interval = IntervalSet([3, 6])
        >>> print interval[1]
        6
        >>> print interval[0]
        3
        >>> print interval[2]
        Traceback (most recent call last):
            ...
        IndexError: Index is out of range
        """
        try:
            return self.intervals[index]
        except IndexError:
            raise IndexError("Index is out of range")

    def __iter__(self):
        """Returns an iterator to iterate through the intervals

        Unlike sets, which do not have ordering, BaseIntervalSets do.  Therefore,
        iterating was implemented.  Intervals are stored in order, starting with
        that with the left-most lower bound to that with the right-most.

        >>> for i in IntervalSet():
        ...     print i
        ...
        >>> for i in IntervalSet.between(3, 5):
        ...     print i
        ...
        [3..5]
        >>> for i in IntervalSet([2, 5, 3]):
        ...     print i
        ...
        2
        3
        5
        """
        return self.intervals.__iter__()

    def lower_bound(self):
        """Returns the lower boundary of the BaseIntervalSet
        
        >>> IntervalSet([Interval.between(4, 6), 2, 12]).lower_bound()
        2
        >>> IntervalSet().lower_bound()
        Traceback (most recent call last):
            ...
        IndexError: The BaseIntervalSet is empty
        >>> IntervalSet.all().lower_bound()
        -Inf
        """
        if len(self.intervals) > 0:
            return self.intervals[0].lower_bound
        else:
            raise IndexError("The BaseIntervalSet is empty")
                
    def upper_bound(self):
        """Returns the upper boundary of the BaseIntervalSet
        
        >>> IntervalSet([Interval.between(4, 6), 2, 12]).upper_bound()
        12
        >>> IntervalSet().upper_bound()
        Traceback (most recent call last):
            ...
        IndexError: The BaseIntervalSet is empty
        >>> IntervalSet.all().upper_bound()
        Inf
        """
        if len(self.intervals) > 0:
            return self.intervals[-1].upper_bound
        else:
            raise IndexError("The BaseIntervalSet is empty")
    
    def lower_closed(self):
        """Returns a boolean telling whether the lower bound is closed or not
        
        >>> IntervalSet([Interval.between(4, 6), 2, 12]).lower_closed()
        True
        >>> IntervalSet().lower_closed()
        Traceback (most recent call last):
            ...
        IndexError: The BaseIntervalSet is empty
        >>> IntervalSet.all().lower_closed()
        False
        """
        if len(self.intervals) > 0:
            return self.intervals[0].lower_closed
        else:
            raise IndexError("The BaseIntervalSet is empty")
                
    def upper_closed(self):
        """Returns a boolean telling whether the upper bound is closed or not
        
        >>> IntervalSet([Interval.between(4, 6), 2, 12]).upper_closed()
        True
        >>> IntervalSet().upper_closed()
        Traceback (most recent call last):
            ...
        IndexError: The BaseIntervalSet is empty
        >>> IntervalSet.all().upper_closed()
        False
        """
        if len(self.intervals) > 0:
            return self.intervals[0].upper_closed
        else:
            raise IndexError("The BaseIntervalSet is empty")
    
    def bounds(self):
        """Returns an interval that encompasses the entire BaseIntervalSet
        
        >>> print IntervalSet([Interval.between(4, 6), 2, 12]).bounds()
        [2..12]
        >>> print IntervalSet().bounds()
        <Empty>
        >>> print IntervalSet.all().bounds()
        (...)
        """
        if len(self.intervals) == 0:
            result = Interval.none()
        else:
            result =  Interval(
                self.lower_bound(), self.upper_bound(), 
                lower_closed=self.lower_closed(), 
                upper_closed=self.upper_closed())
        return result
        
    def issubset(self, other):
        """Tells if the given object is a subset of the object
        
        Returns true if self is a subset of other.  other can be any 
        iterable object.
        
        >>> zero = IntervalSet([0])
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals = IntervalSet.greater_than_or_equal_to(0)
        >>> negatives = IntervalSet.less_than(0)
        >>> r = IntervalSet.between(3, 6)
        >>> r2 = IntervalSet.between(-8, -2)
        >>> zero.issubset(positives)
        False
        >>> zero.issubset(naturals)
        True
        >>> positives.issubset(zero)
        False
        >>> r.issubset(zero)
        False
        >>> r.issubset(positives)
        True
        >>> positives.issubset(r)
        False
        >>> negatives.issubset(IntervalSet.all())
        True
        >>> r2.issubset(negatives)
        True
        >>> negatives.issubset(positives)
        False
        >>> zero.issubset([0, 1, 2, 3])
        True
        """
        if isinstance(other, BaseIntervalSet):
            operand = other
        else:
            operand = self.__class__(other)
        return self <= operand

    def issuperset(self, other):
        """Tells whether the given object is a superset of the object
        
        Returns true if self is a superset of other.  other can be any
        iterable object.
        
        >>> zero = IntervalSet([0])
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals = IntervalSet.greater_than_or_equal_to(0)
        >>> negatives = IntervalSet.less_than(0)
        >>> r = IntervalSet.between(3, 6)
        >>> r2 = IntervalSet.between(-8, -2)
        >>> zero.issuperset(positives)
        False
        >>> zero.issuperset(naturals)
        False
        >>> positives.issuperset(zero)
        False
        >>> r.issuperset(zero)
        False
        >>> r.issuperset(positives)
        False
        >>> positives.issuperset(r)
        True
        >>> negatives.issuperset(IntervalSet.all())
        False
        >>> r2.issuperset(negatives)
        False
        >>> negatives.issuperset(positives)
        False
        >>> negatives.issuperset([-2, -632])
        True
        """
        if isinstance(other, BaseIntervalSet):
            operand = other
        else:
            operand = self.__class__(other)
        return self >= operand

    def __contains__(self, obj):
        """Tells whether the BaseIntervalSet contains the given value
        
        Returns True if obj is contained in self.  obj can be either a 
        discrete value, a sequence, or an Interval.

        >>> some = IntervalSet([
        ...   2, 8, Interval(12, 17, upper_closed=False), 
        ...   Interval.greater_than(17)])
        >>> all = IntervalSet.all()
        >>> empty = IntervalSet.empty()
        >>> 17 in empty
        False
        >>> 17 in all
        True
        >>> 17 in some
        False
        >>> r = Interval(100, 400, upper_closed=False)
        >>> r in empty
        False
        >>> r in all
        True
        >>> r in some
        True
        """
        result = False                    
        for r in self.intervals:
            if obj in r:
                result = True
                break
        return result

    def __iter__(self):
        """Returns an iterator over the intervals in the set
        
        >>> s = IntervalSet(
        ...   [2, 7, 2, 87, 4, 3, Interval.greater_than(12),
        ...   Interval.less_than(-2)])
        >>> l = set()
        >>> for i in s:
        ...   l.add(str(i))
        ...
        >>> print len(l)
        6
        >>> "2" in l
        True
        >>> "7" in l
        True
        >>> "87" in l
        False
        >>> "4" in l
        True
        >>> "3" in l
        True
        >>> "(12...)" in l
        True
        >>> "(...-2)" in l
        True

        Though an IntervalSet contains only Interval objects, we pretend
        that they contain discrete objects as well.

        >>> s = IntervalSet([2])
        >>> for i in s:
        ...   i == 2
        ...
        True
        """
        for i in self.intervals:
            # Return point Intervals as the values they abstract
            if i.lower_bound == i.upper_bound:
                yield i.lower_bound
            else:
                yield i
        
    def __add__(self, other):
        """Returns the union of two IntervalSets

        >>> empty     = IntervalSet()
        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> empty     = IntervalSet.empty()
        >>> print evens + positives
        -8,-6,-4,-2,[0...)
        >>> print negatives + zero
        (...0]
        >>> print empty + negatives
        (...0)
        >>> print empty + naturals
        [0...)
        >>> print nonzero + evens
        (...)
        """
        return self.__or__(other)

    def __sub__(self, other):
        """Subtracts intervals in the given object from the object and returns
        the result
        
        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> empty     = IntervalSet.empty()
        >>> all       = IntervalSet.all()
        >>> print evens - nonzero
        0
        >>> print empty - naturals
        <Empty>
        >>> print zero - naturals
        <Empty>
        >>> print positives - zero
        (0...)
        >>> print naturals - negatives
        [0...)
        >>> print all - zero
        (...0),(0...)
        >>> all - zero == nonzero
        True
        >>> naturals - [0]
        Traceback (most recent call last):
            ...
        TypeError: unsupported operand type(s) for -: expected BaseIntervalSet
        """
        if isinstance(other, BaseIntervalSet):
            result = IntervalSet(self)
            for j in other.intervals:
                temp = IntervalSet()
                for i in result.intervals:
                    if i.overlaps(j):
                        if i in j:
                            pass
                        elif j in i:
                            if j.lower_bound != None:
                                temp.add(Interval(
                                    i.lower_bound, j.lower_bound, 
                                    lower_closed=i.lower_closed, 
                                    upper_closed=not j.lower_closed))
                            if j.upper_bound != None:
                                temp.add(Interval(
                                    j.upper_bound, i.upper_bound, 
                                    lower_closed=not j.upper_closed, 
                                    upper_closed=i.upper_closed))
                        elif j.comes_before(i):
                            temp.add(Interval(
                                j.upper_bound, i.upper_bound, 
                                lower_closed=not j.upper_closed, 
                                upper_closed=i.upper_closed))
                        else:
                            temp.add(Interval(
                                i.lower_bound, j.lower_bound, 
                                lower_closed=i.lower_closed, 
                                upper_closed=not j.lower_closed))
                    else:
                        temp.add(copy.deepcopy(i))
                result = temp
        else:
            raise TypeError(
                "unsupported operand type(s) for -: expected BaseIntervalSet")
        return self.__class__(result)

    def difference(self, other):
        """Returns the difference between the object and the given object
        
        Returns all values of self minus all matching values in other.  It
        is identical to the - operator, only it accepts any iterable as 
        the operand.
        
        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> empty     = IntervalSet.empty()
        >>> all       = IntervalSet.all()
        >>> print evens.difference(nonzero)
        0
        >>> print empty.difference(naturals)
        <Empty>
        >>> print zero.difference(naturals)
        <Empty>
        >>> print positives.difference(zero)
        (0...)
        >>> print naturals.difference(negatives)
        [0...)
        >>> print all.difference(zero)
        (...0),(0...)
        >>> all.difference(zero) == nonzero
        True
        >>> naturals.difference([0]) == positives
        True
        """
        if isinstance(other, BaseIntervalSet):
            operand = other
        else:
            operand = self.__class__(other)
        return self - operand

    def __and__(self, other):
        """Returns the intersection of self and other.

        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> empty     = IntervalSet.empty()
        >>> print naturals and naturals
        [0...)
        >>> print evens & zero
        0
        >>> print negatives & zero
        <Empty>
        >>> print nonzero & positives
        (0...)
        >>> print empty & zero
        <Empty>
        >>> positives & [0]
        Traceback (most recent call last):
            ...
        TypeError: unsupported operand type(s) for &: expected BaseIntervalSet
        """
        if isinstance(other, BaseIntervalSet):
            result = IntervalSet()
            for j in other.intervals:
                for i in self.intervals:
                    if i.overlaps(j):
                        if i in j:
                            result.add(copy.deepcopy(i))
                        elif j in i:
                            result.add(copy.deepcopy(j))
                        elif j.comes_before(i):
                            result.add(Interval(
                                i.lower_bound, j.upper_bound, 
                                lower_closed=i.lower_closed, 
                                upper_closed=j.upper_closed))
                        else:
                            result.add(Interval(
                                j.lower_bound, i.upper_bound, 
                                lower_closed=j.lower_closed, 
                                upper_closed=i.upper_closed))
            # Convert IntervalSet to correct type
            result = self.__class__(result)
        else:
            raise TypeError(
                "unsupported operand type(s) for &: expected BaseIntervalSet")
        return result

    def intersection(self, other):
        """Returns the intersection between the object and the given value
        
        This function returns the intersection of self and other.  It is
        identical to the & operator, except this function accepts any 
        iterable as an operand, and & accepts only another 
        BaseIntervalSet.

        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> empty     = IntervalSet.empty()
        >>> print naturals.intersection(naturals)
        [0...)
        >>> print evens.intersection(zero)
        0
        >>> print negatives.intersection(zero)
        <Empty>
        >>> print nonzero.intersection(positives)
        (0...)
        >>> print empty.intersection(zero)
        <Empty>
        """
        if isinstance(other, BaseIntervalSet):
            operand = other
        else:
            operand = self.__class__(other)
        return self & operand
        
    def __or__(self, other):
        """Returns the union of two IntervalSets.

        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> empty     = IntervalSet.empty()
        >>> all       = IntervalSet.all()
        >>> print evens | positives
        -8,-6,-4,-2,[0...)
        >>> print negatives | zero
        (...0]
        >>> print empty | negatives
        (...0)
        >>> print empty | naturals
        [0...)
        >>> print nonzero | evens
        (...)
        >>> print negatives | range(5)
        Traceback (most recent call last):
            ...
        TypeError: unsupported operand type(s) for |: expected BaseIntervalSet
        """
        if isinstance(other, BaseIntervalSet):
            union = IntervalSet(self)
            for r in other.intervals:
                union.add(r)
        else:
            raise TypeError(
                "unsupported operand type(s) for |: expected BaseIntervalSet")
        return self.__class__(union)

    def union(self, other):
        """Returns the union of the given value with the object
        
        This function returns the union of a BaseIntervalSet and an 
        iterable object.  It is identical to the | operator, except that 
        | only accepts a BaseIntervalSet operand and union accepts any 
        iterable.
        
        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> empty     = IntervalSet.empty()
        >>> all       = IntervalSet.all()
        >>> print evens.union(positives)
        -8,-6,-4,-2,[0...)
        >>> print negatives.union(zero)
        (...0]
        >>> print empty.union(negatives)
        (...0)
        >>> print empty.union(naturals)
        [0...)
        >>> print nonzero.union(evens)
        (...)
        >>> print negatives.union(range(5))
        (...0],1,2,3,4
        """
        if isinstance(other, BaseIntervalSet):
           operand = other
        else:
            operand = self.__class__(other)
        return self | operand
        
    def __xor__(self, other):
        """Returns the exclusive or of two IntervalSets.  
        
        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> empty     = IntervalSet.empty()
        >>> print nonzero ^ naturals
        (...0]
        >>> print zero ^ negatives
        (...0]
        >>> print positives ^ empty
        (0...)
        >>> print evens ^ zero
        -8,-6,-4,-2,2,4,6,8
        >>> negatives ^ [0]
        Traceback (most recent call last):
            ...
        TypeError: unsupported operand type(s) for ^: expected BaseIntervalSet
        """
        if isinstance(other, BaseIntervalSet):
            return (self | other) - (self & other)
        else:
            raise TypeError(
                "unsupported operand type(s) for ^: expected BaseIntervalSet")

    def symmetric_difference(self, other):
        """Returns the exclusive or of the given value with the object
        
        This function returns the exclusive or of two IntervalSets.  
        It is identical to the ^ operator, except it accepts any iterable
        object for the operand.
        
        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> empty     = IntervalSet.empty()
        >>> print nonzero.symmetric_difference(naturals)
        (...0]
        >>> print zero.symmetric_difference(negatives)
        (...0]
        >>> print positives.symmetric_difference(empty)
        (0...)
        >>> print evens.symmetric_difference(zero)
        -8,-6,-4,-2,2,4,6,8
        >>> print evens.symmetric_difference(range(0, 9, 2))
        -8,-6,-4,-2
        """
        if isinstance(other, BaseIntervalSet):
            operand = other
        else:
            operand = self.__class__(other)
        return self ^ operand
        
    def __invert__(self):
        """Returns the disjoint set of self

        >>> negatives = IntervalSet.less_than(0)
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals  = IntervalSet.greater_than_or_equal_to(0)
        >>> evens     = IntervalSet([-8, -6, -4, -2, 0, 2, 4, 6, 8])
        >>> zero      = IntervalSet([0])
        >>> nonzero   = IntervalSet.not_equal_to(0)
        >>> print ~(IntervalSet.empty())
        (...)
        >>> ~negatives == naturals
        True
        >>> print ~positives
        (...0]
        >>> ~naturals == negatives
        True
        >>> print ~evens
        (...-8),(-8..-6),(-6..-4),(-4..-2),(-2..0),(0..2),(2..4),(4..6),(6..8),(8...)
        >>> ~zero == nonzero
        True
        >>> ~nonzero == zero
        True
        """
        return self.__class__.all() - self

    def __cmp__(self, other):
        """Compares two BaseIntervalSets
        
        Like set, raises a TypeError when invoked.
        >>> IntervalSet().__cmp__(IntervalSet())
        Traceback (most recent call last):
        ...
        TypeError: cannot compare BaseIntervalSets using cmp()
        """
        raise TypeError("cannot compare BaseIntervalSets using cmp()")        
        
    def __eq__(self, other):
        """Tests if two BaseIntervalSets are equivalent
        
        Two IntervalSets are identical if they contain the exact same 
        sets.

        >>> IntervalSet([4]) == IntervalSet([1])
        False
        >>> IntervalSet([5]) == IntervalSet([5])
        True
        >>> s1 = IntervalSet.between(4, 7)
        >>> s2 = IntervalSet([Interval(4, 7, upper_closed=False)])
        >>> s1 == s2
        False
        >>> s2.add(7)
        >>> s1 == s2
        True
        >>> s1.clear()
        >>> s1 == IntervalSet.empty()
        True
        
        An IntervalSet, when compared to a non-BaseIntervalSet, yields 
        False

        >>> s1 == [0, 2, 7, 4]
        False
        >>> IntervalSet([3, 4, 5]) == set([3, 4, 5])
        False
        >>> IntervalSet([3, 4, 5]) == [3, 4, 5, 3]
        False
        >>> IntervalSet([3]) == 3
        False
        """
        if isinstance(other, BaseIntervalSet):
            # If len(other) != len(operand), then that means extra objects were
            # discarded from other.  Thus it can't be equal to any sort of set.
            result = self.issubset(other) and other.issubset(self)
        else:
            result = False
        return result

    def __ne__(self, other):
        """Tests if two BaseIntervalSets are not equivalent
        
        Two IntervalSets are not identical if they contain different 
        values or Intervals.

        >>> IntervalSet([4]) != IntervalSet([1])
        True
        >>> IntervalSet([5]) != IntervalSet([5])
        False
        >>> s1 = IntervalSet.between(4, 7)
        >>> s2 = IntervalSet([Interval(4, 7, upper_closed=False)])
        >>> s1 != s2
        True
        >>> s2.add(7)
        >>> s1 != s2
        False
        >>> s1.clear()
        >>> s1 != IntervalSet.empty()
        False
        
        An IntervalSet can also be compared to any other value.  The result
        is always True.

        >>> s1 != [0, 2, 7, 4]
        True
        >>> IntervalSet([3, 4, 5]) != set([3, 4, 5])
        True
        >>> IntervalSet([3, 4, 5]) != [3, 4, 5, 3]
        True
        """
        return not (self == other)

    def __lt__(self, other):
        """Tests if the given operand is a subset of the object
        
        To test if a set is a subset that's not equal to another, you can 
        use the < operator.  I don't like this, personally, but in my 
        attempt to implement a set-like object, I've duplicated this 
        functionality.
        
        >>> zero = IntervalSet([0])
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals = IntervalSet.greater_than_or_equal_to(0)
        >>> negatives = IntervalSet.less_than(0)
        >>> r = IntervalSet.between(3, 6)
        >>> r2 = IntervalSet.between(-8, -2)
        >>> zero < positives
        False
        >>> zero < naturals
        True
        >>> positives < zero
        False
        >>> r < zero
        False
        >>> r < positives
        True
        >>> positives < r
        False
        >>> negatives < IntervalSet.all()
        True
        >>> r2 < negatives
        True
        >>> negatives < positives
        False
        >>> zero < [0, 2, 6, 7]
        Traceback (most recent call last):
            ...
        TypeError: unsupported operand type(s) for <: expected BaseIntervalSet
        >>> positives < positives
        False
        """
        if isinstance(other, BaseIntervalSet):
            return (self <= other and not self == other)
        else:
            raise TypeError(
                "unsupported operand type(s) for <: expected BaseIntervalSet")
        
    def __le__(self, other):
        """Tests if the given operand is a subset or is equal to the object
        
        To test if a set is a subset of another, you can use the <= 
        operator.  I don't like this, personally, but in my attempt
        to implement a set-like object, I've duplicated this 
        functionality.
        
        >>> zero = IntervalSet([0])
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals = IntervalSet.greater_than_or_equal_to(0)
        >>> negatives = IntervalSet.less_than(0)
        >>> r = IntervalSet.between(3, 6)
        >>> r2 = IntervalSet.between(-8, -2)
        >>> zero <= positives
        False
        >>> zero <= naturals
        True
        >>> positives <= zero
        False
        >>> r <= zero
        False
        >>> r <= positives
        True
        >>> positives <= r
        False
        >>> negatives <= IntervalSet.all()
        True
        >>> r2 <= negatives
        True
        >>> negatives <= positives
        False
        >>> zero <= [0, 2, 6, 7]
        Traceback (most recent call last):
            ...
        TypeError: unsupported operand type(s) for <=: expected BaseIntervalSet
        >>> positives <= positives
        True
        """
        if isinstance(other, BaseIntervalSet):
            result = True
            for i in self.intervals:
                if i not in other:
                    result = False
                    break
        else:
            raise TypeError(
                "unsupported operand type(s) for <=: expected BaseIntervalSet")
        return result

    def __ge__(self, other):
        """Tests if the given operand is a superset or is equal to the object
        
        To test if a set is a superset of another, you can use the >=
        operator.  I don't like this, personally, but in my attempt
        to implement a set-like object, I've duplicated this 
        functionality.
        
        >>> zero = IntervalSet([0])
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals = IntervalSet.greater_than_or_equal_to(0)
        >>> negatives = IntervalSet.less_than(0)
        >>> r = IntervalSet.between(3, 6)
        >>> r2 = IntervalSet.between(-8, -2)
        >>> zero >= positives
        False
        >>> zero >= naturals
        False
        >>> positives >= zero
        False
        >>> r >= zero
        False
        >>> r >= positives
        False
        >>> positives >= r
        True
        >>> negatives >= IntervalSet.all()
        False
        >>> r2 >= negatives
        False
        >>> negatives >= positives
        False
        >>> negatives >= [-2, -63]
        Traceback (most recent call last):
            ...
        TypeError: unsupported operand type(s) for >=: expected BaseIntervalSet
        """
        if isinstance(other, BaseIntervalSet):
            result = True
            for i in other.intervals:
                if i not in self:
                    result = False
                    break
        else:
            raise TypeError(
                "unsupported operand type(s) for >=: expected BaseIntervalSet")
        return result
                
    def __gt__(self, other):
        """Tests if the given operand is a superset of the object
        
        To test if a set is a superset of another, but not equal to it,
        you can use the > operator.  I don't like this, personally, but in 
        my attempt to implement a set-like object, I've duplicated this 
        functionality.
        
        >>> zero = IntervalSet([0])
        >>> positives = IntervalSet.greater_than(0)
        >>> naturals = IntervalSet.greater_than_or_equal_to(0)
        >>> negatives = IntervalSet.less_than(0)
        >>> r = IntervalSet.between(3, 6)
        >>> r2 = IntervalSet.between(-8, -2)
        >>> zero > positives
        False
        >>> zero > naturals
        False
        >>> positives > zero
        False
        >>> r > zero
        False
        >>> r > positives
        False
        >>> positives > r
        True
        >>> negatives > IntervalSet.all()
        False
        >>> r2 > negatives
        False
        >>> negatives > positives
        False
        >>> positives > positives
        False
        >>> negatives > [-2, -63]
        Traceback (most recent call last):
            ...
        TypeError: unsupported operand type(s) for >: expected BaseIntervalSet
        """
        if isinstance(other, BaseIntervalSet):
            return self >= other and not self == other
        else:
            raise TypeError(
                "unsupported operand type(s) for >: expected BaseIntervalSet")

    def _add(self, obj):
        "Appends an Interval or BaseIntervalSet to the object"
        if isinstance(obj, Interval):
            r = obj
        else:
            r = Interval.equal_to(obj)

        if r:   # Don't bother appending an empty Interval
            # If r continuously joins with any of the other
            newIntervals = []
            for i in self.intervals:
                if i.overlaps(r) or i.adjacent_to(r):
                    r = r.join(i)
                else:
                    newIntervals.append(i)
            newIntervals.append(r)
            self.intervals = newIntervals
            self.intervals.sort()
    
    def copy(self):
        """Returns a copy of the object
        
        >>> s = IntervalSet(
        ...   [7, 2, 3, 2, 6, 2, Interval.greater_than(3)])
        >>> s2 = s.copy()
        >>> s == s2
        True
        >>> s = FrozenIntervalSet(
        ...   [7, 2, 3, 2, 6, 2, Interval.greater_than(3)])
        >>> s2 = s.copy()
        >>> s == s2
        True
        """                
        return copy.copy(self)
                                    
    @classmethod
    def less_than(cls, n):
        """Returns an IntervalSet containing values less than the given value
        
        >>> print IntervalSet.less_than(0)
        (...0)
        >>> print IntervalSet.less_than(-23)
        (...-23)
        """
        return cls([Interval.less_than(n)])
      
    @classmethod
    def less_than_or_equal_to(cls, n, closed=False):
        """Returns an IntervalSet containing values less than or equal to the 
        given value
        
        >>> print IntervalSet.less_than_or_equal_to(0)
        (...0]
        >>> print IntervalSet.less_than_or_equal_to(-23)
        (...-23]
        """
        return cls([Interval.less_than_or_equal_to(n)])
      
    @classmethod
    def greater_than(cls, n):
        """Returns an IntervalSet containing values greater than the given value
        
        >>> print IntervalSet.greater_than(0)
        (0...)
        >>> print IntervalSet.greater_than(-23)
        (-23...)
        """
        return cls([Interval.greater_than(n)])
      
    @classmethod
    def greater_than_or_equal_to(cls, n):
        """Returns an IntervalSet containing values greater than or equal to 
        the given value
        
        >>> print IntervalSet.greater_than_or_equal_to(0)
        [0...)
        >>> print IntervalSet.greater_than_or_equal_to(-23)
        [-23...)
        """
        return cls([Interval.greater_than_or_equal_to(n)])
      
    @classmethod
    def not_equal_to(cls, n):
        """Returns an IntervalSet of all values not equal to n
    
        >>> print IntervalSet.not_equal_to(0)
        (...0),(0...)
        >>> print IntervalSet.not_equal_to(-23)
        (...-23),(-23...)
        """
        return cls([Interval.less_than(n), Interval.greater_than(n)])
  
    @classmethod
    def between(cls, a, b, closed=True):
        """Returns an IntervalSet of all values between a and b.
        
        If closed is True, then the endpoints are included; otherwise, they 
        aren't.
    
        >>> print IntervalSet.between(0, 100)
        [0..100]
        >>> print IntervalSet.between(-1, 1)
        [-1..1]
        """
        return cls([Interval.between(a, b, closed)])
  
    @classmethod
    def all(cls):
        """Returns an interval set containing all values
        
        >>> print IntervalSet.all()
        (...)
        """
        return cls([Interval.all()])
      
    @classmethod
    def empty(cls):
        """Returns an interval set containing no values.
        
        >>> print IntervalSet.empty()
        <Empty>
        """
        return cls()

class IntervalSet(BaseIntervalSet):
    """The mutable version of BaseIntervalSet
    
    IntervalSet is a class representing sets of continuous values, as 
    opposed to a discrete set, which is already implemented by the set 
    type in Python.

    IntervalSets can be bounded, unbounded, and non-continuous.  They were 
    designed to accomodate any sort of mathematical set dealing with
    continuous values.  This will usually mean numbers, but any Python 
    type that has valid comparison operations can be used in an 
    IntervalSet.
    
    Because IntervalSets are mutable, it cannot be used as a dictionary 
    key.
    
    >>> {IntervalSet([3, 66]) : 52}
    Traceback (most recent call last):
        ...
    TypeError: unhashable instance
    """
    def __init__(self, items=[]):
        "Initializes the IntervalSet"
        BaseIntervalSet.__init__(self, items)

    def __repr__(self):
        """Returns an evaluable representation of the object
        
        >>> IntervalSet([Interval()])
        IntervalSet([Interval(-Inf, Inf, lower_closed=False, upper_closed=False)])
        >>> IntervalSet()
        IntervalSet([])
        >>> IntervalSet([2, 4])
        IntervalSet([Interval(2, 2, lower_closed=True, upper_closed=True), Interval(4, 4, lower_closed=True, upper_closed=True)])
        """
        return "IntervalSet([%s])" % (
            ", ".join(repr(i) for i in self.intervals),)

    def __delitem__(self, index):
        """Removes the interval at the given index from the IntervalSet

        >>> interval = IntervalSet()
        >>> del interval[0]
        Traceback (most recent call last):
            ...
        IndexError: Index is out of range
        >>> interval = IntervalSet.between(-12, 2)
        >>> del interval[1]
        Traceback (most recent call last):
            ...
        IndexError: Index is out of range
        >>> len(interval)
        1
        >>> del interval[0]
        >>> len(interval)
        0
        >>> interval = IntervalSet([2, 7, -2])
        >>> len(interval)
        3
        >>> del interval[1]
        >>> len(interval)
        2
        >>> print interval
        -2,7
        """
        try:
            del self.intervals[index]
        except IndexError:
            raise IndexError("Index is out of range")

    def add(self, obj):
        """Adds an Interval or discrete value to the object

        >>> r = IntervalSet()
        >>> r.add(4)
        >>> print r
        4
        >>> r.add(Interval(23, 39, lower_closed=False))
        >>> print r
        4,(23..39]
        >>> r.add(Interval.less_than(25))
        >>> print r
        (...39]
        """
        BaseIntervalSet._add(self, obj)

    def remove(self, obj):
        """Removes a value from the object
        
        This function removes an Interval, discrete value, or set 
        from an IntervalSet.  If the object is not in the set, a KeyError
        is raised.

        >>> r = IntervalSet.all()
        >>> r.remove(4)
        >>> print r
        (...4),(4...)
        >>> r.remove(Interval(23, 39, lower_closed=False))
        >>> print r
        (...4),(4..23],(39...)
        >>> r.remove(Interval.less_than(25))
        Traceback (most recent call last):
            ...
        KeyError: '(...25)'
        """
        if obj in self:
            self.discard(obj)
        else:
            raise KeyError(str(obj))

    def discard(self, obj):
        """Removes a value from the object
        
        This function removes an Interval or discrete value from an
        IntervalSet.

        >>> r = IntervalSet.all()
        >>> r.discard(4)
        >>> print r
        (...4),(4...)
        >>> r.discard(Interval(23, 39, lower_closed=False))
        >>> print r
        (...4),(4..23],(39...)
        >>> r.discard(Interval.less_than(25))
        >>> print r
        (39...)
        """
        diff = self - IntervalSet([obj])
        self.intervals = diff.intervals
                            
    def difference_update(self, other):
        """Removes any elements in the given value from the object
        
        This function removes the elements in other from self.  other can
        be any iterable object.

        >>> r = IntervalSet.all()
        >>> r.difference_update([4])
        >>> print r
        (...4),(4...)
        >>> r.difference_update(
        ...   IntervalSet([Interval(23, 39, lower_closed=False)]))
        >>> print r
        (...4),(4..23],(39...)
        >>> r.difference_update(IntervalSet.less_than(25))
        >>> print r
        (39...)
        >>> r2 = IntervalSet.all()
        >>> r.difference_update(r2)
        >>> print r
        <Empty>
        """
        diff = self.difference(other)
        self.intervals = diff.intervals
                
    def clear(self):
        """Removes all Intervals from the object
        
        >>> s = IntervalSet([2, 7, Interval.greater_than(8), 2, 6, 34])
        >>> print s
        2,6,7,(8...)
        >>> s.clear()
        >>> print s
        <Empty>
        """
        self.intervals = []
        
    def update(self, other):
        """Adds elements from the given value to the object
        
        Adds elements from other to self.  other can be any iterable 
        object.

        >>> r = IntervalSet()
        >>> r.update([4])
        >>> print r
        4
        >>> r.update(IntervalSet([Interval(23, 39, lower_closed=False)]))
        >>> print r
        4,(23..39]
        >>> r.update(IntervalSet.less_than(25))
        >>> print r
        (...39]
        >>> r2 = IntervalSet.all()
        >>> r.update(r2)
        >>> print r
        (...)
        """
        union = self.union(other)
        self.intervals = union.intervals
                
    def intersection_update(self, other):
        """Removes values not found in the parameter
        
        Removes elements not found in other.  other can be any iterable
        object

        >>> r = IntervalSet.all()
        >>> r.intersection_update([4])
        >>> print r
        4
        >>> r = IntervalSet.all()
        >>> r.intersection_update(
        ...   IntervalSet([Interval(23, 39, lower_closed=False)]))
        >>> print r
        (23..39]
        >>> r.intersection_update(IntervalSet.less_than(25))
        >>> print r
        (23..25)
        >>> r2 = IntervalSet.all()
        >>> r.intersection_update(r2)
        >>> print r
        (23..25)
        """
        intersection = self.intersection(other)
        self.intervals = intersection.intervals
                
    def symmetric_difference_update(self, other):
        """Updates the object as though doing an xor with the parameter
        
        Removes elements found in other and adds elements in other that 
        are not in self.  other can be any iterable object.

        >>> r = IntervalSet.empty()
        >>> r.symmetric_difference_update([4])
        >>> print r
        4
        >>> r.symmetric_difference_update(
        ...   IntervalSet([Interval(23, 39, lower_closed=False)]))
        >>> print r
        4,(23..39]
        >>> r.symmetric_difference_update(IntervalSet.less_than(25))
        >>> print r
        (...4),(4..23],[25..39]
        >>> r2 = IntervalSet.all()
        >>> r.symmetric_difference_update(r2)
        >>> print r
        4,(23..25),(39...)
        """
        xor = self.symmetric_difference(other)
        self.intervals = xor.intervals
    
    def pop(self):
        """Returns and discards an Interval or value from the IntervalSet
        
        >>> s = IntervalSet([7, Interval.less_than(2), 2, 0])
        >>> l = []
        >>> l.append(str(s.pop()))
        >>> l.append(str(s.pop()))
        >>> "(...2)" in l
        False
        >>> "(...2]" in l
        True
        >>> "7" in l
        True
        >>> print s
        <Empty>
        >>> i = s.pop()
        Traceback (most recent call last):
            ...
        KeyError: 'pop from an empty IntervalSet'
        """
        if len(self.intervals) > 0:
            i = self.intervals.pop()
            if i.lower_bound == i.upper_bound:
                i = i.lower_bound
        else:
            raise KeyError("pop from an empty IntervalSet")
        return i
        
    def __hash__(self):
        "Raises an error since IntervalSets are mutable"
        raise TypeError("unhashable instance")

class FrozenIntervalSet(BaseIntervalSet):
    """An immutable version of BaseIntervalSet
    
    FrozenIntervalSet is like IntervalSet, only add and remove are not
    implemented, and hashes can be generated.
    
    >>> fs = FrozenIntervalSet([3, 6, 2, 4])
    >>> fs.add(12)
    Traceback (most recent call last):
      ...
    AttributeError: 'FrozenIntervalSet' object has no attribute 'add'
    >>> fs.remove(4)
    Traceback (most recent call last):
      ...
    AttributeError: 'FrozenIntervalSet' object has no attribute 'remove'
    >>> fs.clear()
    Traceback (most recent call last):
      ...
    AttributeError: 'FrozenIntervalSet' object has no attribute 'clear'
    
    Because FrozenIntervalSets are immutable, they can be used as a 
    dictionary key.
    
    >>> d = {
    ...   FrozenIntervalSet([3, 66]) : 52, 
    ...   FrozenIntervalSet.less_than(3) : 3}
    """
    
    def __new__(cls, items=[]):
        """Constructs a new FrozenInteralSet
        
        Object creation is just like with a regular IntervalSet, except for
        the special case where another FrozenIntervalSet is passed as the
        initializer iterable.  If it is, then the result points to the 
        same object.
        
        >>> fs1 = FrozenIntervalSet.greater_than(12)
        >>> fs2 = FrozenIntervalSet(fs1)
        >>> id(fs1) == id(fs2)
        True
        """
        if (cls == FrozenIntervalSet) and isinstance(items, FrozenIntervalSet):
            result = items
        else:
            s = IntervalSet(items)
            result = super(FrozenIntervalSet, cls).__new__(cls, items)
            result.intervals = s.intervals
        return result
    
    def __init__(self, items=[]):
        "Initializes the FrozenIntervalSet"
        pass

    def __repr__(self):
        """Returns an evaluable representation of the object
        
        >>> FrozenIntervalSet([Interval()])
        FrozenIntervalSet([Interval(-Inf, Inf, lower_closed=False, upper_closed=False)])
        >>> FrozenIntervalSet()
        FrozenIntervalSet([])
        >>> FrozenIntervalSet([2, 4])
        FrozenIntervalSet([Interval(2, 2, lower_closed=True, upper_closed=True), Interval(4, 4, lower_closed=True, upper_closed=True)])
        """
        return "FrozenIntervalSet([%s])" % (
            ", ".join(repr(i) for i in self.intervals),)
            
    def __hash__(self):
        """Generates a 32-bit hash key
        
        >>> fs = FrozenIntervalSet([4, 7, 3])
        >>> key = hash(fs)
        """
        h = 0
        for i in self.intervals:
            h = h ^ hash(i)
        return h

    def copy(self):
        """Duplicates the object
        
        For FrozenIntervalSet objects, since they're immutable, a 
        reference, not a copy, of self is returned.
        
        >>> s = FrozenIntervalSet(
        ...   [7, 2, 3, 2, 6, 2, Interval.greater_than(3)])
        >>> s2 = s.copy()
        >>> s == s2
        True
        >>> id(s) == id(s2)
        True
        """
        if self.__class__ == FrozenIntervalSet:
            return self
        else:
            copy.copy(self)
                                    

if __name__ == "__main__":
    import doctest
    doctest.testmod()