<gh_stars>1-10
from __future__ import print_function, division
import functools, itertools
from sympy.core.sympify import sympify
from sympy.core.expr import Expr
from sympy.core import Basic
from sympy.tensor.array import ImmutableDenseNDimArray
from sympy import Symbol
from sympy.core.numbers import Integer


class ArrayComprehension(Basic):
    """
    Generate a list comprehension
    If there is a symbolic dimension, for example, say [i for i in range(1, N)] where
    N is a Symbol, then the expression will not be expanded to an array. Otherwise,
    calling the doit() function will launch the expansion.

    Examples
    ========

    >>> from sympy.tensor.array import ArrayComprehension
    >>> from sympy import symbols
    >>> i, j, k = symbols('i j k')
    >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
    >>> a
    ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
    >>> a.doit()
    [[11, 12, 13], [21, 22, 23], [31, 32, 33], [41, 42, 43]]
    >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k))
    >>> b.doit()
    ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k))
    """
    def __new__(cls, function, *symbols, **assumptions):
        if any(len(l) != 3 or None for l in symbols):
            raise ValueError('ArrayComprehension requires values lower and upper bound'
                              ' for the expression')
        arglist = [sympify(function)]
        arglist.extend(cls._check_limits_validity(function, symbols))
        obj = Basic.__new__(cls, *arglist, **assumptions)
        obj._limits = obj._args[1:]
        obj._shape = cls._calculate_shape_from_limits(obj._limits)
        obj._rank = len(obj._shape)
        obj._loop_size = cls._calculate_loop_size(obj._shape)
        return obj

    @property
    def function(self):
        """The function applied across limits

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.function
        10*i + j
        """
        return self._args[0]

    @property
    def limits(self):
        """
        The list of limits that will be applied while expanding the array

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.limits
        ((i, 1, 4), (j, 1, 3))
        """
        return self._limits

    @property
    def free_symbols(self):
        """
        The set of the free_symbols in the array
        Variables appeared in the bounds are supposed to be excluded
        from the free symbol set.

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.free_symbols
        set()
        >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3))
        >>> b.free_symbols
        {k}
        """
        expr_free_sym = self.function.free_symbols
        for var, inf, sup in self._limits:
            expr_free_sym.discard(var)
            curr_free_syms = inf.free_symbols.union(sup.free_symbols)
            expr_free_sym = expr_free_sym.union(curr_free_syms)
        return expr_free_sym

    @property
    def variables(self):
        """The tuples of the variables in the limits

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.variables
        [i, j]
        """
        return [l[0] for l in self._limits]

    @property
    def bound_symbols(self):
        """The list of dummy variables

        Note
        ====

        Note that all variables are dummy variables since a limit without
        lower bound or upper bound is not accepted.
        """
        return [l[0] for l in self._limits if len(l) != 1]

    @property
    def shape(self):
        """
        The shape of the expanded array, which may have symbols

        Note
        ====

        Both the lower and the upper bounds are included while
        calculating the shape.

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.shape
        (4, 3)
        >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3))
        >>> b.shape
        (4, k + 3)
        """
        return self._shape

    @property
    def is_shape_numeric(self):
        """
        Test if the array is shape-numeric which means there is no symbolic
        dimension

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.is_shape_numeric
        True
        >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3))
        >>> b.is_shape_numeric
        False
        """
        for _, inf, sup in self._limits:
            if Basic(inf, sup).atoms(Symbol):
                return False
        return True

    def rank(self):
        """The rank of the expanded array

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.rank()
        2
        """
        return self._rank

    def __len__(self):
        """
        The length of the expanded array which means the number
        of elements in the array.

        Raises
        ======

        ValueError : When the length of the array is symbolic

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> len(a)
        12
        """
        if self._loop_size.free_symbols:
            raise ValueError('Symbolic length is not supported')
        return self._loop_size

    @classmethod
    def _check_limits_validity(cls, function, limits):
        limits = sympify(limits)
        for var, inf, sup in limits:
            if any((not isinstance(i, Expr)) or i.atoms(Symbol, Integer) != i.atoms()
                                                                for i in [inf, sup]):
                raise TypeError('Bounds should be an Expression(combination of Integer and Symbol)')
            if (inf > sup) == True:
                raise ValueError('Lower bound should be inferior to upper bound')
            if var in inf.free_symbols or var in sup.free_symbols:
                raise ValueError('Variable should not be part of its bounds')
        return limits

    @classmethod
    def _calculate_shape_from_limits(cls, limits):
        return tuple([sup - inf + 1 for _, inf, sup in limits])

    @classmethod
    def _calculate_loop_size(cls, shape):
        if not shape:
            return 0
        loop_size = 1
        for l in shape:
            loop_size = loop_size * l

        return loop_size

    def doit(self):
        if not self.is_shape_numeric:
            return self

        return self._expand_array()

    def _expand_array(self):
        res = []
        for values in itertools.product(*[range(inf, sup+1)
                                        for var, inf, sup
                                        in self._limits]):
            res.append(self._get_element(values))

        return ImmutableDenseNDimArray(res, self.shape)

    def _get_element(self, values):
        temp = self.function
        for var, val in zip(self.variables, values):
            temp = temp.subs(var, val)
        return temp

    def tolist(self):
        """Transform the expanded array to a list

        Raises
        ======

        ValueError : When there is a symbolic dimension

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.tolist()
        [[11, 12, 13], [21, 22, 23], [31, 32, 33], [41, 42, 43]]
        """
        if self.is_shape_numeric:
            return self._expand_array().tolist()

        raise ValueError("A symbolic array cannot be expanded to a list")

    def tomatrix(self):
        """Transform the expanded array to a matrix

        Raises
        ======

        ValueError : When there is a symbolic dimension
        ValueError : When the rank of the expanded array is not equal to 2

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.tomatrix()
        Matrix([
        [11, 12, 13],
        [21, 22, 23],
        [31, 32, 33],
        [41, 42, 43]])
        """
        from sympy.matrices import Matrix

        if not self.is_shape_numeric:
            raise ValueError("A symbolic array cannot be expanded to a matrix")
        if self._rank != 2:
            raise ValueError('Dimensions must be of size of 2')

        return Matrix(self._expand_array().tomatrix())


def isLambda(v):
    LAMBDA = lambda: 0
    return isinstance(v, type(LAMBDA)) and v.__name__ == LAMBDA.__name__

class ArrayComprehensionMap(ArrayComprehension):
    '''
    A subclass of ArrayComprehension dedicated to map external function lambda.

    Notes
    =====

    Only the lambda function is considered.
    At most one argument in lambda function is accepted in order to avoid ambiguity
    in value assignment.

    Examples
    ========

    >>> from sympy.tensor.array import ArrayComprehensionMap
    >>> from sympy import symbols
    >>> i, j, k = symbols('i j k')
    >>> a = ArrayComprehensionMap(lambda: 1, (i, 1, 4))
    >>> a.doit()
    [1, 1, 1, 1]
    >>> b = ArrayComprehensionMap(lambda a: a+1, (j, 1, 4))
    >>> b.doit()
    [2, 3, 4, 5]

    '''
    def __new__(cls, function, *symbols, **assumptions):
        if any(len(l) != 3 or None for l in symbols):
            raise ValueError('ArrayComprehension requires values lower and upper bound'
                              ' for the expression')

        if not isLambda(function):
            raise ValueError('Data type not supported')

        arglist = cls._check_limits_validity(function, symbols)
        obj = Basic.__new__(cls, *arglist, **assumptions)
        obj._limits = obj._args
        obj._shape = cls._calculate_shape_from_limits(obj._limits)
        obj._rank = len(obj._shape)
        obj._loop_size = cls._calculate_loop_size(obj._shape)
        obj._lambda = function
        return obj

    @property
    def func(self):
        class _(ArrayComprehensionMap):
            def __new__(cls, *args, **kwargs):
                return ArrayComprehensionMap(self._lambda, *args, **kwargs)
        return _

    def _get_element(self, values):
        temp = self._lambda
        if self._lambda.__code__.co_argcount == 0:
            temp = temp()
        elif self._lambda.__code__.co_argcount == 1:
            temp = temp(functools.reduce(lambda a, b: a*b, values))
        return temp

from __future__ import print_function, division
import functools, itertools
from sympy.core.sympify import sympify
from sympy.core.expr import Expr
from sympy.core import Basic
from sympy.tensor.array import ImmutableDenseNDimArray
from sympy import Symbol
from sympy.core.numbers import Integer


class ArrayComprehension(Basic):
    """
    Generate a list comprehension
    If there is a symbolic dimension, for example, say [i for i in range(1, N)] where
    N is a Symbol, then the expression will not be expanded to an array. Otherwise,
    calling the doit() function will launch the expansion.

    Examples
    ========

    >>> from sympy.tensor.array import ArrayComprehension
    >>> from sympy import symbols
    >>> i, j, k = symbols('i j k')
    >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
    >>> a
    ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
    >>> a.doit()
    [[11, 12, 13], [21, 22, 23], [31, 32, 33], [41, 42, 43]]
    >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k))
    >>> b.doit()
    ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k))
    """
    def __new__(cls, function, *symbols, **assumptions):
        if any(len(l) != 3 or None for l in symbols):
            raise ValueError('ArrayComprehension requires values lower and upper bound'
                              ' for the expression')
        arglist = [sympify(function)]
        arglist.extend(cls._check_limits_validity(function, symbols))
        obj = Basic.__new__(cls, *arglist, **assumptions)
        obj._limits = obj._args[1:]
        obj._shape = cls._calculate_shape_from_limits(obj._limits)
        obj._rank = len(obj._shape)
        obj._loop_size = cls._calculate_loop_size(obj._shape)
        return obj

    @property
    def function(self):
        """The function applied across limits

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.function
        10*i + j
        """
        return self._args[0]

    @property
    def limits(self):
        """
        The list of limits that will be applied while expanding the array

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.limits
        ((i, 1, 4), (j, 1, 3))
        """
        return self._limits

    @property
    def free_symbols(self):
        """
        The set of the free_symbols in the array
        Variables appeared in the bounds are supposed to be excluded
        from the free symbol set.

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.free_symbols
        set()
        >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3))
        >>> b.free_symbols
        {k}
        """
        expr_free_sym = self.function.free_symbols
        for var, inf, sup in self._limits:
            expr_free_sym.discard(var)
            curr_free_syms = inf.free_symbols.union(sup.free_symbols)
            expr_free_sym = expr_free_sym.union(curr_free_syms)
        return expr_free_sym

    @property
    def variables(self):
        """The tuples of the variables in the limits

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.variables
        [i, j]
        """
        return [l[0] for l in self._limits]

    @property
    def bound_symbols(self):
        """The list of dummy variables

        Note
        ====

        Note that all variables are dummy variables since a limit without
        lower bound or upper bound is not accepted.
        """
        return [l[0] for l in self._limits if len(l) != 1]

    @property
    def shape(self):
        """
        The shape of the expanded array, which may have symbols

        Note
        ====

        Both the lower and the upper bounds are included while
        calculating the shape.

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.shape
        (4, 3)
        >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3))
        >>> b.shape
        (4, k + 3)
        """
        return self._shape

    @property
    def is_shape_numeric(self):
        """
        Test if the array is shape-numeric which means there is no symbolic
        dimension

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.is_shape_numeric
        True
        >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3))
        >>> b.is_shape_numeric
        False
        """
        for _, inf, sup in self._limits:
            if Basic(inf, sup).atoms(Symbol):
                return False
        return True

    def rank(self):
        """The rank of the expanded array

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j, k = symbols('i j k')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.rank()
        2
        """
        return self._rank

    def __len__(self):
        """
        The length of the expanded array which means the number
        of elements in the array.

        Raises
        ======

        ValueError : When the length of the array is symbolic

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> len(a)
        12
        """
        if self._loop_size.free_symbols:
            raise ValueError('Symbolic length is not supported')
        return self._loop_size

    @classmethod
    def _check_limits_validity(cls, function, limits):
        limits = sympify(limits)
        for var, inf, sup in limits:
            if any((not isinstance(i, Expr)) or i.atoms(Symbol, Integer) != i.atoms()
                                                                for i in [inf, sup]):
                raise TypeError('Bounds should be an Expression(combination of Integer and Symbol)')
            if (inf > sup) == True:
                raise ValueError('Lower bound should be inferior to upper bound')
            if var in inf.free_symbols or var in sup.free_symbols:
                raise ValueError('Variable should not be part of its bounds')
        return limits

    @classmethod
    def _calculate_shape_from_limits(cls, limits):
        return tuple([sup - inf + 1 for _, inf, sup in limits])

    @classmethod
    def _calculate_loop_size(cls, shape):
        if not shape:
            return 0
        loop_size = 1
        for l in shape:
            loop_size = loop_size * l

        return loop_size

    def doit(self):
        if not self.is_shape_numeric:
            return self

        return self._expand_array()

    def _expand_array(self):
        res = []
        for values in itertools.product(*[range(inf, sup+1)
                                        for var, inf, sup
                                        in self._limits]):
            res.append(self._get_element(values))

        return ImmutableDenseNDimArray(res, self.shape)

    def _get_element(self, values):
        temp = self.function
        for var, val in zip(self.variables, values):
            temp = temp.subs(var, val)
        return temp

    def tolist(self):
        """Transform the expanded array to a list

        Raises
        ======

        ValueError : When there is a symbolic dimension

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.tolist()
        [[11, 12, 13], [21, 22, 23], [31, 32, 33], [41, 42, 43]]
        """
        if self.is_shape_numeric:
            return self._expand_array().tolist()

        raise ValueError("A symbolic array cannot be expanded to a list")

    def tomatrix(self):
        """Transform the expanded array to a matrix

        Raises
        ======

        ValueError : When there is a symbolic dimension
        ValueError : When the rank of the expanded array is not equal to 2

        Examples
        ========

        >>> from sympy.tensor.array import ArrayComprehension
        >>> from sympy import symbols
        >>> i, j = symbols('i j')
        >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))
        >>> a.tomatrix()
        Matrix([
        [11, 12, 13],
        [21, 22, 23],
        [31, 32, 33],
        [41, 42, 43]])
        """
        from sympy.matrices import Matrix

        if not self.is_shape_numeric:
            raise ValueError("A symbolic array cannot be expanded to a matrix")
        if self._rank != 2:
            raise ValueError('Dimensions must be of size of 2')

        return Matrix(self._expand_array().tomatrix())


def isLambda(v):
    LAMBDA = lambda: 0
    return isinstance(v, type(LAMBDA)) and v.__name__ == LAMBDA.__name__

class ArrayComprehensionMap(ArrayComprehension):
    '''
    A subclass of ArrayComprehension dedicated to map external function lambda.

    Notes
    =====

    Only the lambda function is considered.
    At most one argument in lambda function is accepted in order to avoid ambiguity
    in value assignment.

    Examples
    ========

    >>> from sympy.tensor.array import ArrayComprehensionMap
    >>> from sympy import symbols
    >>> i, j, k = symbols('i j k')
    >>> a = ArrayComprehensionMap(lambda: 1, (i, 1, 4))
    >>> a.doit()
    [1, 1, 1, 1]
    >>> b = ArrayComprehensionMap(lambda a: a+1, (j, 1, 4))
    >>> b.doit()
    [2, 3, 4, 5]

    '''
    def __new__(cls, function, *symbols, **assumptions):
        if any(len(l) != 3 or None for l in symbols):
            raise ValueError('ArrayComprehension requires values lower and upper bound'
                              ' for the expression')

        if not isLambda(function):
            raise ValueError('Data type not supported')

        arglist = cls._check_limits_validity(function, symbols)
        obj = Basic.__new__(cls, *arglist, **assumptions)
        obj._limits = obj._args
        obj._shape = cls._calculate_shape_from_limits(obj._limits)
        obj._rank = len(obj._shape)
        obj._loop_size = cls._calculate_loop_size(obj._shape)
        obj._lambda = function
        return obj

    @property
    def func(self):
        class _(ArrayComprehensionMap):
            def __new__(cls, *args, **kwargs):
                return ArrayComprehensionMap(self._lambda, *args, **kwargs)
        return _

    def _get_element(self, values):
        temp = self._lambda
        if self._lambda.__code__.co_argcount == 0:
            temp = temp()
        elif self._lambda.__code__.co_argcount == 1:
            temp = temp(functools.reduce(lambda a, b: a*b, values))
        return temp
Generate a list comprehension     If there is a symbolic dimension, for example, say [i for i in range(1, N)] where     N is a Symbol, then the expression will not be expanded to an array. Otherwise,     calling the doit() function will launch the expansion.      Examples     ========      >>> from sympy.tensor.array import ArrayComprehension     >>> from sympy import symbols     >>> i, j, k = symbols('i j k')     >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))     >>> a     ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))     >>> a.doit()     [[11, 12, 13], [21, 22, 23], [31, 32, 33], [41, 42, 43]]     >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k))     >>> b.doit()     ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k)) The function applied across limits          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j = symbols('i j')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> a.function         10*i + j The list of limits that will be applied while expanding the array          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j = symbols('i j')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> a.limits         ((i, 1, 4), (j, 1, 3)) The set of the free_symbols in the array         Variables appeared in the bounds are supposed to be excluded         from the free symbol set.          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j, k = symbols('i j k')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> a.free_symbols         set()         >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3))         >>> b.free_symbols         {k} The tuples of the variables in the limits          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j, k = symbols('i j k')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> a.variables         [i, j] The list of dummy variables          Note         ====          Note that all variables are dummy variables since a limit without         lower bound or upper bound is not accepted. The shape of the expanded array, which may have symbols          Note         ====          Both the lower and the upper bounds are included while         calculating the shape.          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j, k = symbols('i j k')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> a.shape         (4, 3)         >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3))         >>> b.shape         (4, k + 3) Test if the array is shape-numeric which means there is no symbolic         dimension          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j, k = symbols('i j k')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> a.is_shape_numeric         True         >>> b = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, k+3))         >>> b.is_shape_numeric         False The rank of the expanded array          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j, k = symbols('i j k')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> a.rank()         2 The length of the expanded array which means the number         of elements in the array.          Raises         ======          ValueError : When the length of the array is symbolic          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j = symbols('i j')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> len(a)         12 Transform the expanded array to a list          Raises         ======          ValueError : When there is a symbolic dimension          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j = symbols('i j')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> a.tolist()         [[11, 12, 13], [21, 22, 23], [31, 32, 33], [41, 42, 43]] Transform the expanded array to a matrix          Raises         ======          ValueError : When there is a symbolic dimension         ValueError : When the rank of the expanded array is not equal to 2          Examples         ========          >>> from sympy.tensor.array import ArrayComprehension         >>> from sympy import symbols         >>> i, j = symbols('i j')         >>> a = ArrayComprehension(10*i + j, (i, 1, 4), (j, 1, 3))         >>> a.tomatrix()         Matrix([         [11, 12, 13],         [21, 22, 23],         [31, 32, 33],         [41, 42, 43]]) A subclass of ArrayComprehension dedicated to map external function lambda.      Notes     =====      Only the lambda function is considered.     At most one argument in lambda function is accepted in order to avoid ambiguity     in value assignment.      Examples     ========      >>> from sympy.tensor.array import ArrayComprehensionMap     >>> from sympy import symbols     >>> i, j, k = symbols('i j k')     >>> a = ArrayComprehensionMap(lambda: 1, (i, 1, 4))     >>> a.doit()     [1, 1, 1, 1]     >>> b = ArrayComprehensionMap(lambda a: a+1, (j, 1, 4))     >>> b.doit()     [2, 3, 4, 5]