MLPY/Lib/site-packages/sympy/concrete/expr_with_limits.py

from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import AppliedUndef, UndefinedFunction
from sympy.core.mul import Mul
from sympy.core.relational import Equality, Relational
from sympy.core.singleton import S
from sympy.core.symbol import Symbol, Dummy
from sympy.core.sympify import sympify
from sympy.functions.elementary.piecewise import (piecewise_fold,
    Piecewise)
from sympy.logic.boolalg import BooleanFunction
from sympy.matrices.matrixbase import MatrixBase
from sympy.sets.sets import Interval, Set
from sympy.sets.fancysets import Range
from sympy.tensor.indexed import Idx
from sympy.utilities import flatten
from sympy.utilities.iterables import sift, is_sequence
from sympy.utilities.exceptions import sympy_deprecation_warning


def _common_new(cls, function, *symbols, discrete, **assumptions):
    """Return either a special return value or the tuple,
    (function, limits, orientation). This code is common to
    both ExprWithLimits and AddWithLimits."""
    function = sympify(function)

    if isinstance(function, Equality):
        # This transforms e.g. Integral(Eq(x, y)) to Eq(Integral(x), Integral(y))
        # but that is only valid for definite integrals.
        limits, orientation = _process_limits(*symbols, discrete=discrete)
        if not (limits and all(len(limit) == 3 for limit in limits)):
            sympy_deprecation_warning(
                """
                Creating a indefinite integral with an Eq() argument is
                deprecated.

                This is because indefinite integrals do not preserve equality
                due to the arbitrary constants. If you want an equality of
                indefinite integrals, use Eq(Integral(a, x), Integral(b, x))
                explicitly.
                """,
                deprecated_since_version="1.6",
                active_deprecations_target="deprecated-indefinite-integral-eq",
                stacklevel=5,
            )

        lhs = function.lhs
        rhs = function.rhs
        return Equality(cls(lhs, *symbols, **assumptions), \
                        cls(rhs, *symbols, **assumptions))

    if function is S.NaN:
        return S.NaN

    if symbols:
        limits, orientation = _process_limits(*symbols, discrete=discrete)
        for i, li in enumerate(limits):
            if len(li) == 4:
                function = function.subs(li[0], li[-1])
                limits[i] = Tuple(*li[:-1])
    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

    # Any embedded piecewise functions need to be brought out to the
    # top level. We only fold Piecewise that contain the integration
    # variable.
    reps = {}
    symbols_of_integration = {i[0] for i in limits}
    for p in function.atoms(Piecewise):
        if not p.has(*symbols_of_integration):
            reps[p] = Dummy()
    # mask off those that don't
    function = function.xreplace(reps)
    # do the fold
    function = piecewise_fold(function)
    # remove the masking
    function = function.xreplace({v: k for k, v in reps.items()})

    return function, limits, orientation


def _process_limits(*symbols, discrete=None):
    """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.
    In the case that a limit is specified as (symbol, Range), a list of
    length 4 may be returned if a change of variables is needed; the
    expression that should replace the symbol in the expression is
    the fourth element in the list.
    """
    limits = []
    orientation = 1
    if discrete is None:
        err_msg = 'discrete must be True or False'
    elif discrete:
        err_msg = 'use Range, not Interval or Relational'
    else:
        err_msg = 'use Interval or Relational, not Range'
    for V in symbols:
        if isinstance(V, (Relational, BooleanFunction)):
            if discrete:
                raise TypeError(err_msg)
            variable = V.atoms(Symbol).pop()
            V = (variable, V.as_set())
        elif isinstance(V, Symbol) or getattr(V, '_diff_wrt', False):
            if isinstance(V, Idx):
                if V.lower is None or V.upper is None:
                    limits.append(Tuple(V))
                else:
                    limits.append(Tuple(V, V.lower, V.upper))
            else:
                limits.append(Tuple(V))
            continue
        if is_sequence(V) and not isinstance(V, Set):
            if len(V) == 2 and isinstance(V[1], Set):
                V = list(V)
                if isinstance(V[1], Interval):  # includes Reals
                    if discrete:
                        raise TypeError(err_msg)
                    V[1:] = V[1].inf, V[1].sup
                elif isinstance(V[1], Range):
                    if not discrete:
                        raise TypeError(err_msg)
                    lo = V[1].inf
                    hi = V[1].sup
                    dx = abs(V[1].step)  # direction doesn't matter
                    if dx == 1:
                        V[1:] = [lo, hi]
                    else:
                        if lo is not S.NegativeInfinity:
                            V = [V[0]] + [0, (hi - lo)//dx, dx*V[0] + lo]
                        else:
                            V = [V[0]] + [0, S.Infinity, -dx*V[0] + hi]
                else:
                    # more complicated sets would require splitting, e.g.
                    # Union(Interval(1, 3), interval(6,10))
                    raise NotImplementedError(
                        'expecting Range' if discrete else
                        'Relational or single Interval' )
            V = sympify(flatten(V))  # list of sympified elements/None
            if isinstance(V[0], (Symbol, Idx)) or getattr(V[0], '_diff_wrt', False):
                newsymbol = V[0]
                if len(V) == 3:
                    # general case
                    if V[2] is None and V[1] is not None:
                        orientation *= -1
                    V = [newsymbol] + [i for i in V[1:] if i is not None]

                lenV = len(V)
                if not isinstance(newsymbol, Idx) or lenV == 3:
                    if lenV == 4:
                        limits.append(Tuple(*V))
                        continue
                    if lenV == 3:
                        if isinstance(newsymbol, Idx):
                            # Idx represents an integer which may have
                            # specified values it can take on; if it is
                            # given such a value, an error is raised here
                            # if the summation would try to give it a larger
                            # or smaller value than permitted. None and Symbolic
                            # values will not raise an error.
                            lo, hi = newsymbol.lower, newsymbol.upper
                            try:
                                if lo is not None and not bool(V[1] >= lo):
                                    raise ValueError("Summation will set Idx value too low.")
                            except TypeError:
                                pass
                            try:
                                if hi is not None and not bool(V[2] <= hi):
                                    raise ValueError("Summation will set Idx value too high.")
                            except TypeError:
                                pass
                        limits.append(Tuple(*V))
                        continue
                    if lenV == 1 or (lenV == 2 and V[1] is None):
                        limits.append(Tuple(newsymbol))
                        continue
                    elif lenV == 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):
        from sympy.concrete.products import Product
        pre = _common_new(cls, function, *symbols,
            discrete=issubclass(cls, Product), **assumptions)
        if isinstance(pre, tuple):
            function, limits, _ = pre
        else:
            return pre

        # limits must have upper and lower bounds; the indefinite form
        # is not supported. This restriction does not apply to AddWithLimits
        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 kind(self):
        return self.function.kind

    @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 limit 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
        sympy.integrals.integrals.Integral.transform : Perform mapping on the dummy variable
        """
        return [l[0] for l in self.limits]

    @property
    def bound_symbols(self):
        """Return only variables that are dummy variables.

        Examples
        ========

        >>> from sympy import Integral
        >>> from sympy.abc import x, i, j, k
        >>> Integral(x**i, (i, 1, 3), (j, 2), k).bound_symbols
        [i, j]

        See Also
        ========

        function, limits, free_symbols
        as_dummy : Rename dummy variables
        sympy.integrals.integrals.Integral.transform : Perform mapping on the dummy variable
        """
        return [l[0] for l in self.limits if len(l) != 1]

    @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
        {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
        # mask off non-symbol integration variables that have
        # more than themself as a free symbol
        reps = {i[0]: i[0] if i[0].free_symbols == {i[0]} else Dummy()
            for i in self.limits}
        function = function.xreplace(reps)
        isyms = function.free_symbols
        for xab in limits:
            v = reps[xab[0]]
            if len(xab) == 1:
                isyms.add(v)
                continue
            # take out the target symbol
            if v in isyms:
                isyms.remove(v)
            # add in the new symbols
            for i in xab[1:]:
                isyms.update(i.free_symbols)
        reps = {v: k for k, v in reps.items()}
        return {reps.get(_, _) for _ in isyms}

    @property
    def is_number(self):
        """Return True if the Sum has no free symbols, else False."""
        return not self.free_symbols

    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 integrals
        change_index : Perform mapping on the sum and product dummy variables

        """
        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]:
                    if new._diff_wrt:
                        xab = (new,)
                    else:
                        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, 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 cannot 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)

    @property
    def has_finite_limits(self):
        """
        Returns True if the limits are known to be finite, either by the
        explicit bounds, assumptions on the bounds, or assumptions on the
        variables.  False if known to be infinite, based on the bounds.
        None if not enough information is available to determine.

        Examples
        ========

        >>> from sympy import Sum, Integral, Product, oo, Symbol
        >>> x = Symbol('x')
        >>> Sum(x, (x, 1, 8)).has_finite_limits
        True

        >>> Integral(x, (x, 1, oo)).has_finite_limits
        False

        >>> M = Symbol('M')
        >>> Sum(x, (x, 1, M)).has_finite_limits

        >>> N = Symbol('N', integer=True)
        >>> Product(x, (x, 1, N)).has_finite_limits
        True

        See Also
        ========

        has_reversed_limits

        """

        ret_None = False
        for lim in self.limits:
            if len(lim) == 3:
                if any(l.is_infinite for l in lim[1:]):
                    # Any of the bounds are +/-oo
                    return False
                elif any(l.is_infinite is None for l in lim[1:]):
                    # Maybe there are assumptions on the variable?
                    if lim[0].is_infinite is None:
                        ret_None = True
            else:
                if lim[0].is_infinite is None:
                    ret_None = True

        if ret_None:
            return None
        return True

    @property
    def has_reversed_limits(self):
        """
        Returns True if the limits are known to be in reversed order, either
        by the explicit bounds, assumptions on the bounds, or assumptions on the
        variables.  False if known to be in normal order, based on the bounds.
        None if not enough information is available to determine.

        Examples
        ========

        >>> from sympy import Sum, Integral, Product, oo, Symbol
        >>> x = Symbol('x')
        >>> Sum(x, (x, 8, 1)).has_reversed_limits
        True

        >>> Sum(x, (x, 1, oo)).has_reversed_limits
        False

        >>> M = Symbol('M')
        >>> Integral(x, (x, 1, M)).has_reversed_limits

        >>> N = Symbol('N', integer=True, positive=True)
        >>> Sum(x, (x, 1, N)).has_reversed_limits
        False

        >>> Product(x, (x, 2, N)).has_reversed_limits

        >>> Product(x, (x, 2, N)).subs(N, N + 2).has_reversed_limits
        False

        See Also
        ========

        sympy.concrete.expr_with_intlimits.ExprWithIntLimits.has_empty_sequence

        """
        ret_None = False
        for lim in self.limits:
            if len(lim) == 3:
                var, a, b = lim
                dif = b - a
                if dif.is_extended_negative:
                    return True
                elif dif.is_extended_nonnegative:
                    continue
                else:
                    ret_None = True
            else:
                return None
        if ret_None:
            return None
        return False


class AddWithLimits(ExprWithLimits):
    r"""Represents unevaluated oriented additions.
        Parent class for Integral and Sum.
    """

    __slots__ = ()

    def __new__(cls, function, *symbols, **assumptions):
        from sympy.concrete.summations import Sum
        pre = _common_new(cls, function, *symbols,
            discrete=issubclass(cls, Sum), **assumptions)
        if isinstance(pre, tuple):
            function, limits, orientation = pre
        else:
            return pre

        obj = Expr.__new__(cls, **assumptions)
        arglist = [orientation*function]  # orientation not used in ExprWithLimits
        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 set(self.variables) & w.free_symbols)
                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)
        force = hints.get('force', False)
        if (summand.is_Add and (force or summand.is_commutative and
                 self.has_finite_limits is not False)):
            return Add(*[self.func(i, *self.limits) for i in summand.args])
        elif isinstance(summand, MatrixBase):
            return summand.applyfunc(lambda x: self.func(x, *self.limits))
        elif summand != self.function:
            return self.func(summand, *self.limits)
        return self