text: from __future__ import print_function, division
  
  from sympy.core.add import Add
  from sympy.core.expr import Expr
  from sympy.core.mul import Mul
  from sympy.core.relational import Equality
  from sympy.sets.sets import Interval
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol
  from sympy.core.sympify import sympify
  from sympy.core.compatibility import is_sequence, range
  from sympy.core.containers import Tuple
  from sympy.functions.elementary.piecewise import piecewise_fold
  from sympy.utilities import flatten
  from sympy.utilities.iterables import sift
  
  
  
  
  def _process_limits(*symbols):
      """Process the list of symbols and convert them to canonical limits,
      storing them as Tuple(symbol, lower, upper). The orientation of
      the function is also returned when the upper limit is missing
      so (x, 1, None) becomes (x, None, 1) and the orientation is changed.
      """
      limits = []
      orientation = 1
      for V in symbols:
          if isinstance(V, Symbol):
              limits.append(Tuple(V))
              continue
          elif is_sequence(V, Tuple):
              V = sympify(flatten(V))
              if V[0].is_Symbol:
                  newsymbol = V[0]
                  if len(V) == 2 and isinstance(V[1], Interval):
                      V[1:] = [V[1].start, V[1].end]
  
                  if len(V) == 3:
                      if V[1] is None and V[2] is not None:
                          nlim = [V[2]]
                      elif V[1] is not None and V[2] is None:
                          orientation *= -1
                          nlim = [V[1]]
                      elif V[1] is None and V[2] is None:
                          nlim = []
                      else:
                          nlim = V[1:]
                      limits.append(Tuple(newsymbol, *nlim ))
                      continue
                  elif len(V) == 1 or (len(V) == 2 and V[1] is None):
                      limits.append(Tuple(newsymbol))
                      continue
                  elif len(V) == 2:
                      limits.append(Tuple(newsymbol, V[1]))
                      continue
  
          raise ValueError('Invalid limits given: %s' % str(symbols))
  
      return limits, orientation
  
  class ExprWithLimits(Expr):
      __slots__ = ['is_commutative']
  
      def __new__(cls, function, *symbols, **assumptions):
          # Any embedded piecewise functions need to be brought out to the
          # top level so that integration can go into piecewise mode at the
          # earliest possible moment.
          function = sympify(function)
          if hasattr(function, 'func') and function.func is Equality:
              lhs = function.lhs
              rhs = function.rhs
              return Equality(cls(lhs, *symbols, **assumptions), \
                  cls(rhs, *symbols, **assumptions))
          function = piecewise_fold(function)
  
          if function is S.NaN:
              return S.NaN
  
          if symbols:
              limits, orientation = _process_limits(*symbols)
          else:
              # symbol not provided -- we can still try to compute a general form
              free = function.free_symbols
              if len(free) != 1:
                  raise ValueError(
                      "specify dummy variables for %s" % function)
              limits, orientation = [Tuple(s) for s in free], 1
  
          # denest any nested calls
          while cls == type(function):
              limits = list(function.limits) + limits
              function = function.function
  
          # Only limits with lower and upper bounds are supported; the indefinite form
          # is not supported
          if any(len(l) != 3 or None in l for l in limits):
              raise ValueError('ExprWithLimits requires values for lower and upper bounds.')
  
          obj = Expr.__new__(cls, **assumptions)
          arglist = [function]
          arglist.extend(limits)
          obj._args = tuple(arglist)
          obj.is_commutative = function.is_commutative  # limits already checked
  
          return obj
  
      @property
      def function(self):
          """Return the function applied across limits.
  
          Examples
          ========
  
          >>> from sympy import Integral
          >>> from sympy.abc import x
          >>> Integral(x**2, (x,)).function
          x**2
  
          See Also
          ========
  
          limits, variables, free_symbols
          """
          return self._args[0]
  
      @property
      def limits(self):
          """Return the limits of expression.
  
          Examples
          ========
  
          >>> from sympy import Integral
          >>> from sympy.abc import x, i
          >>> Integral(x**i, (i, 1, 3)).limits
          ((i, 1, 3),)
  
          See Also
          ========
  
          function, variables, free_symbols
          """
          return self._args[1:]
  
      @property
      def variables(self):
          """Return a list of the dummy variables
  
          >>> from sympy import Sum
          >>> from sympy.abc import x, i
          >>> Sum(x**i, (i, 1, 3)).variables
          [i]
  
          See Also
          ========
  
          function, limits, free_symbols
          as_dummy : Rename dummy variables
          transform : Perform mapping on the dummy variable
          """
          return [l[0] for l in self.limits]
  
      @property
      def free_symbols(self):
          """
          This method returns the symbols in the object, excluding those
          that take on a specific value (i.e. the dummy symbols).
  
          Examples
          ========
  
          >>> from sympy import Sum
          >>> from sympy.abc import x, y
          >>> Sum(x, (x, y, 1)).free_symbols
          set([y])
          """
          # don't test for any special values -- nominal free symbols
          # should be returned, e.g. don't return set() if the
          # function is zero -- treat it like an unevaluated expression.
          function, limits = self.function, self.limits
          isyms = function.free_symbols
          for xab in limits:
              if len(xab) == 1:
                  isyms.add(xab[0])
                  continue
              # take out the target symbol
              if xab[0] in isyms:
                  isyms.remove(xab[0])
              # add in the new symbols
              for i in xab[1:]:
                  isyms.update(i.free_symbols)
          return isyms
  
      @property
      def is_number(self):
          """Return True if the Sum has no free symbols, else False."""
          return not self.free_symbols
  
      def as_dummy(self):
          """
          Replace instances of the given dummy variables with explicit dummy
          counterparts to make clear what are dummy variables and what
          are real-world symbols in an object.
  
          Examples
          ========
  
          >>> from sympy import Integral
          >>> from sympy.abc import x, y
          >>> Integral(x, (x, x, y), (y, x, y)).as_dummy()
          Integral(_x, (_x, x, _y), (_y, x, y))
  
          If the object supperts the "integral at" limit ``(x,)`` it
          is not treated as a dummy, but the explicit form, ``(x, x)``
          of length 2 does treat the variable as a dummy.
  
          >>> Integral(x, x).as_dummy()
          Integral(x, x)
          >>> Integral(x, (x, x)).as_dummy()
          Integral(_x, (_x, x))
  
          If there were no dummies in the original expression, then the
          the symbols which cannot be changed by subs() are clearly seen as
          those with an underscore prefix.
  
          See Also
          ========
  
          variables : Lists the integration variables
          transform : Perform mapping on the integration variable
          """
          reps = {}
          f = self.function
          limits = list(self.limits)
          for i in range(-1, -len(limits) - 1, -1):
              xab = list(limits[i])
              if len(xab) == 1:
                  continue
              x = xab[0]
              xab[0] = x.as_dummy()
              for j in range(1, len(xab)):
                  xab[j] = xab[j].subs(reps)
              reps[x] = xab[0]
              limits[i] = xab
          f = f.subs(reps)
          return self.func(f, *limits)
  
      def _eval_interval(self, x, a, b):
          limits = [( i if i[0] != x else (x,a,b) ) for i in self.limits]
          integrand = self.function
          return self.func(integrand, *limits)
  
      def _eval_subs(self, old, new):
          """
          Perform substitutions over non-dummy variables
          of an expression with limits.  Also, can be used
          to specify point-evaluation of an abstract antiderivative.
  
          Examples
          ========
  
          >>> from sympy import Sum, oo
          >>> from sympy.abc import s,n
          >>> Sum(1/n**s, (n, 1, oo)).subs(s, 2)
          Sum(n**(-2), (n, 1, oo))
  
          >>> from sympy import Integral
          >>> from sympy.abc import x,a
          >>> Integral(a*x**2,x).subs(x,4)
          Integral(a*x**2, (x, 4))
  
          See Also
          ========
  
          variables : Lists the integration variables
          transform : Perform mapping on the dummy variable for intgrals
          change_index : Perform mapping on the sum and product dummy variables
  
          """
          from sympy.core.function import AppliedUndef, UndefinedFunction
          func, limits = self.function, list(self.limits)
  
          # If one of the expressions we are replacing is used as a func index
          # one of two things happens.
          #   - the old variable first appears as a free variable
          #     so we perform all free substitutions before it becomes
          #     a func index.
          #   - the old variable first appears as a func index, in
          #     which case we ignore.  See change_index.
  
          # Reorder limits to match standard mathematical practice for scoping
          limits.reverse()
  
          if not isinstance(old, Symbol) or \
                  old.free_symbols.intersection(self.free_symbols):
              sub_into_func = True
              for i, xab in enumerate(limits):
                  if 1 == len(xab) and old == xab[0]:
                      xab = (old, old)
                  limits[i] = Tuple(xab[0], *[l._subs(old, new) for l in xab[1:]])
                  if len(xab[0].free_symbols.intersection(old.free_symbols)) != 0:
                      sub_into_func = False
                      break
              if isinstance(old, AppliedUndef) or isinstance(old, UndefinedFunction):
                  sy2 = set(self.variables).intersection(set(new.atoms(Symbol)))
                  sy1 = set(self.variables).intersection(set(old.args))
                  if not sy2.issubset(sy1):
                      raise ValueError(
                          "substitution can not create dummy dependencies")
                  sub_into_func = True
              if sub_into_func:
                  func = func.subs(old, new)
          else:
              # old is a Symbol and a dummy variable of some limit
              for i, xab in enumerate(limits):
                  if len(xab) == 3:
                      limits[i] = Tuple(xab[0], *[l._subs(old, new) for l in xab[1:]])
                      if old == xab[0]:
                          break
          # simplify redundant limits (x, x)  to (x, )
          for i, xab in enumerate(limits):
              if len(xab) == 2 and (xab[0] - xab[1]).is_zero:
                  limits[i] = Tuple(xab[0], )
  
          # Reorder limits back to representation-form
          limits.reverse()
  
          return self.func(func, *limits)
  
  class AddWithLimits(ExprWithLimits):
      r"""Represents unevaluated oriented additions.
          Parent class for Integral and Sum.
      """
  
      def __new__(cls, function, *symbols, **assumptions):
          # Any embedded piecewise functions need to be brought out to the
          # top level so that integration can go into piecewise mode at the
          # earliest possible moment.
          #
          # This constructor only differs from ExprWithLimits
          # in the application of the orientation variable.  Perhaps merge?
          function = sympify(function)
          if hasattr(function, 'func') and function.func is Equality:
              lhs = function.lhs
              rhs = function.rhs
              return Equality(cls(lhs, *symbols, **assumptions), \
                  cls(rhs, *symbols, **assumptions))
          function = piecewise_fold(function)
  
          if function is S.NaN:
              return S.NaN
  
          if symbols:
              limits, orientation = _process_limits(*symbols)
          else:
              # symbol not provided -- we can still try to compute a general form
              free = function.free_symbols
              if len(free) != 1:
                  raise ValueError(
                      " specify dummy variables for %s. If the integrand contains"
                      " more than one free symbol, an integration variable should"
                      " be supplied explicitly e.g., integrate(f(x, y), x)"
                      % function)
              limits, orientation = [Tuple(s) for s in free], 1
  
          # denest any nested calls
          while cls == type(function):
              limits = list(function.limits) + limits
              function = function.function
  
          obj = Expr.__new__(cls, **assumptions)
          arglist = [orientation*function]
          arglist.extend(limits)
          obj._args = tuple(arglist)
          obj.is_commutative = function.is_commutative  # limits already checked
  
          return obj
  
      def _eval_adjoint(self):
          if all([x.is_real for x in flatten(self.limits)]):
              return self.func(self.function.adjoint(), *self.limits)
          return None
  
      def _eval_conjugate(self):
          if all([x.is_real for x in flatten(self.limits)]):
              return self.func(self.function.conjugate(), *self.limits)
          return None
  
      def _eval_transpose(self):
          if all([x.is_real for x in flatten(self.limits)]):
              return self.func(self.function.transpose(), *self.limits)
          return None
  
      def _eval_factor(self, **hints):
          if 1 == len(self.limits):
              summand = self.function.factor(**hints)
              if summand.is_Mul:
                  out = sift(summand.args, lambda w: w.is_commutative \
                      and not w.has(*self.variables))
                  return Mul(*out[True])*self.func(Mul(*out[False]), \
                      *self.limits)
          else:
              summand = self.func(self.function, self.limits[0:-1]).factor()
              if not summand.has(self.variables[-1]):
                  return self.func(1, [self.limits[-1]]).doit()*summand
              elif isinstance(summand, Mul):
                  return self.func(summand, self.limits[-1]).factor()
          return self
  
      def _eval_expand_basic(self, **hints):
          summand = self.function.expand(**hints)
          if summand.is_Add and summand.is_commutative:
              return Add(*[ self.func(i, *self.limits) for i in summand.args ])
          elif summand != self.function:
              return self.func(summand, *self.limits)
          return self
  