MTtranslateService/Lib/site-packages/sympy/tensor/array/expressions/array_expressions.py

import collections.abc
import operator
from collections import defaultdict, Counter
from functools import reduce
import itertools
from itertools import accumulate
from typing import Optional, List, Dict as tDict, Tuple as tTuple

import typing

from sympy.core.numbers import Integer
from sympy.core.relational import Equality
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import (Function, Lambda)
from sympy.core.mul import Mul
from sympy.core.singleton import S
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import (Dummy, Symbol)
from sympy.matrices.common import MatrixCommon
from sympy.matrices.expressions.diagonal import diagonalize_vector
from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy.matrices.expressions.special import ZeroMatrix
from sympy.tensor.array.arrayop import (permutedims, tensorcontraction, tensordiagonal, tensorproduct)
from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray
from sympy.tensor.array.ndim_array import NDimArray
from sympy.tensor.indexed import (Indexed, IndexedBase)
from sympy.matrices.expressions.matexpr import MatrixElement
from sympy.tensor.array.expressions.utils import _apply_recursively_over_nested_lists, _sort_contraction_indices, \
    _get_mapping_from_subranks, _build_push_indices_up_func_transformation, _get_contraction_links, \
    _build_push_indices_down_func_transformation
from sympy.combinatorics import Permutation
from sympy.combinatorics.permutations import _af_invert
from sympy.core.sympify import _sympify


class _ArrayExpr(Expr):
    shape : tTuple[Expr, ...]


class ArraySymbol(_ArrayExpr):
    """
    Symbol representing an array expression
    """

    def __new__(cls, symbol, shape: typing.Iterable) -> "ArraySymbol":
        if isinstance(symbol, str):
            symbol = Symbol(symbol)
        # symbol = _sympify(symbol)
        shape = Tuple(*map(_sympify, shape))
        obj = Expr.__new__(cls, symbol, shape)
        return obj

    @property
    def name(self):
        return self._args[0]

    @property
    def shape(self):
        return self._args[1]

    def __getitem__(self, item):
        return ArrayElement(self, item)

    def as_explicit(self):
        if not all(i.is_Integer for i in self.shape):
            raise ValueError("cannot express explicit array with symbolic shape")
        data = [self[i] for i in itertools.product(*[range(j) for j in self.shape])]
        return ImmutableDenseNDimArray(data).reshape(*self.shape)


class ArrayElement(_ArrayExpr):
    """
    An element of an array.
    """

    _diff_wrt = True
    is_symbol = True
    is_commutative = True

    def __new__(cls, name, indices):
        if isinstance(name, str):
            name = Symbol(name)
        name = _sympify(name)
        if not isinstance(indices, collections.abc.Iterable):
            indices = (indices,)
        indices = _sympify(tuple(indices))
        if hasattr(name, "shape"):
            if any((i >= s) == True for i, s in zip(indices, name.shape)):
                raise ValueError("shape is out of bounds")
        if any((i < 0) == True for i in indices):
            raise ValueError("shape contains negative values")
        obj = Expr.__new__(cls, name, indices)
        return obj

    @property
    def name(self):
        return self._args[0]

    @property
    def indices(self):
        return self._args[1]

    def _eval_derivative(self, s):
        if not isinstance(s, ArrayElement):
            return S.Zero

        if s == self:
            return S.One

        if s.name != self.name:
            return S.Zero

        return Mul.fromiter(KroneckerDelta(i, j) for i, j in zip(self.indices, s.indices))


class ZeroArray(_ArrayExpr):
    """
    Symbolic array of zeros. Equivalent to ``ZeroMatrix`` for matrices.
    """

    def __new__(cls, *shape):
        if len(shape) == 0:
            return S.Zero
        shape = map(_sympify, shape)
        obj = Expr.__new__(cls, *shape)
        return obj

    @property
    def shape(self):
        return self._args

    def as_explicit(self):
        if not all(i.is_Integer for i in self.shape):
            raise ValueError("Cannot return explicit form for symbolic shape.")
        return ImmutableDenseNDimArray.zeros(*self.shape)


class OneArray(_ArrayExpr):
    """
    Symbolic array of ones.
    """

    def __new__(cls, *shape):
        if len(shape) == 0:
            return S.One
        shape = map(_sympify, shape)
        obj = Expr.__new__(cls, *shape)
        return obj

    @property
    def shape(self):
        return self._args

    def as_explicit(self):
        if not all(i.is_Integer for i in self.shape):
            raise ValueError("Cannot return explicit form for symbolic shape.")
        return ImmutableDenseNDimArray([S.One for i in range(reduce(operator.mul, self.shape))]).reshape(*self.shape)


class _CodegenArrayAbstract(Basic):

    @property
    def subranks(self):
        """
        Returns the ranks of the objects in the uppermost tensor product inside
        the current object.  In case no tensor products are contained, return
        the atomic ranks.

        Examples
        ========

        >>> from sympy.tensor.array import tensorproduct, tensorcontraction
        >>> from sympy import MatrixSymbol
        >>> M = MatrixSymbol("M", 3, 3)
        >>> N = MatrixSymbol("N", 3, 3)
        >>> P = MatrixSymbol("P", 3, 3)

        Important: do not confuse the rank of the matrix with the rank of an array.

        >>> tp = tensorproduct(M, N, P)
        >>> tp.subranks
        [2, 2, 2]

        >>> co = tensorcontraction(tp, (1, 2), (3, 4))
        >>> co.subranks
        [2, 2, 2]
        """
        return self._subranks[:]

    def subrank(self):
        """
        The sum of ``subranks``.
        """
        return sum(self.subranks)

    @property
    def shape(self):
        return self._shape


class ArrayTensorProduct(_CodegenArrayAbstract):
    r"""
    Class to represent the tensor product of array-like objects.
    """

    def __new__(cls, *args, **kwargs):
        args = [_sympify(arg) for arg in args]

        canonicalize = kwargs.pop("canonicalize", False)

        ranks = [get_rank(arg) for arg in args]

        obj = Basic.__new__(cls, *args)
        obj._subranks = ranks
        shapes = [get_shape(i) for i in args]

        if any(i is None for i in shapes):
            obj._shape = None
        else:
            obj._shape = tuple(j for i in shapes for j in i)
        if canonicalize:
            return obj._canonicalize()
        return obj

    def _canonicalize(self):
        args = self.args
        args = self._flatten(args)

        ranks = [get_rank(arg) for arg in args]

        # Check if there are nested permutation and lift them up:
        permutation_cycles = []
        for i, arg in enumerate(args):
            if not isinstance(arg, PermuteDims):
                continue
            permutation_cycles.extend([[k + sum(ranks[:i]) for k in j] for j in arg.permutation.cyclic_form])
            args[i] = arg.expr
        if permutation_cycles:
            return _permute_dims(_array_tensor_product(*args), Permutation(sum(ranks)-1)*Permutation(permutation_cycles))

        if len(args) == 1:
            return args[0]

        # If any object is a ZeroArray, return a ZeroArray:
        if any(isinstance(arg, (ZeroArray, ZeroMatrix)) for arg in args):
            shapes = reduce(operator.add, [get_shape(i) for i in args], ())
            return ZeroArray(*shapes)

        # If there are contraction objects inside, transform the whole
        # expression into `ArrayContraction`:
        contractions = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayContraction)}
        if contractions:
            ranks = [_get_subrank(arg) if isinstance(arg, ArrayContraction) else get_rank(arg) for arg in args]
            cumulative_ranks = list(accumulate([0] + ranks))[:-1]
            tp = _array_tensor_product(*[arg.expr if isinstance(arg, ArrayContraction) else arg for arg in args])
            contraction_indices = [tuple(cumulative_ranks[i] + k for k in j) for i, arg in contractions.items() for j in arg.contraction_indices]
            return _array_contraction(tp, *contraction_indices)

        diagonals = {i: arg for i, arg in enumerate(args) if isinstance(arg, ArrayDiagonal)}
        if diagonals:
            inverse_permutation = []
            last_perm = []
            ranks = [get_rank(arg) for arg in args]
            cumulative_ranks = list(accumulate([0] + ranks))[:-1]
            for i, arg in enumerate(args):
                if isinstance(arg, ArrayDiagonal):
                    i1 = get_rank(arg) - len(arg.diagonal_indices)
                    i2 = len(arg.diagonal_indices)
                    inverse_permutation.extend([cumulative_ranks[i] + j for j in range(i1)])
                    last_perm.extend([cumulative_ranks[i] + j for j in range(i1, i1 + i2)])
                else:
                    inverse_permutation.extend([cumulative_ranks[i] + j for j in range(get_rank(arg))])
            inverse_permutation.extend(last_perm)
            tp = _array_tensor_product(*[arg.expr if isinstance(arg, ArrayDiagonal) else arg for arg in args])
            ranks2 = [_get_subrank(arg) if isinstance(arg, ArrayDiagonal) else get_rank(arg) for arg in args]
            cumulative_ranks2 = list(accumulate([0] + ranks2))[:-1]
            diagonal_indices = [tuple(cumulative_ranks2[i] + k for k in j) for i, arg in diagonals.items() for j in arg.diagonal_indices]
            return _permute_dims(_array_diagonal(tp, *diagonal_indices), _af_invert(inverse_permutation))

        return self.func(*args, canonicalize=False)

    def doit(self, **kwargs):
        deep = kwargs.get("deep", True)
        if deep:
            return self.func(*[arg.doit(**kwargs) for arg in self.args])._canonicalize()
        else:
            return self._canonicalize()

    @classmethod
    def _flatten(cls, args):
        args = [i for arg in args for i in (arg.args if isinstance(arg, cls) else [arg])]
        return args

    def as_explicit(self):
        return tensorproduct(*[arg.as_explicit() if hasattr(arg, "as_explicit") else arg for arg in self.args])


class ArrayAdd(_CodegenArrayAbstract):
    r"""
    Class for elementwise array additions.
    """

    def __new__(cls, *args, **kwargs):
        args = [_sympify(arg) for arg in args]
        ranks = [get_rank(arg) for arg in args]
        ranks = list(set(ranks))
        if len(ranks) != 1:
            raise ValueError("summing arrays of different ranks")
        shapes = [arg.shape for arg in args]
        if len({i for i in shapes if i is not None}) > 1:
            raise ValueError("mismatching shapes in addition")

        canonicalize = kwargs.pop("canonicalize", False)

        obj = Basic.__new__(cls, *args)
        obj._subranks = ranks
        if any(i is None for i in shapes):
            obj._shape = None
        else:
            obj._shape = shapes[0]
        if canonicalize:
            return obj._canonicalize()
        return obj

    def _canonicalize(self):
        args = self.args

        # Flatten:
        args = self._flatten_args(args)

        shapes = [get_shape(arg) for arg in args]
        args = [arg for arg in args if not isinstance(arg, (ZeroArray, ZeroMatrix))]
        if len(args) == 0:
            if any(i for i in shapes if i is None):
                raise NotImplementedError("cannot handle addition of ZeroMatrix/ZeroArray and undefined shape object")
            return ZeroArray(*shapes[0])
        elif len(args) == 1:
            return args[0]
        return self.func(*args, canonicalize=False)

    def doit(self, **kwargs):
        deep = kwargs.get("deep", True)
        if deep:
            return self.func(*[arg.doit(**kwargs) for arg in self.args])._canonicalize()
        else:
            return self._canonicalize()

    @classmethod
    def _flatten_args(cls, args):
        new_args = []
        for arg in args:
            if isinstance(arg, ArrayAdd):
                new_args.extend(arg.args)
            else:
                new_args.append(arg)
        return new_args

    def as_explicit(self):
        return reduce(operator.add, [arg.as_explicit() for arg in self.args])


class PermuteDims(_CodegenArrayAbstract):
    r"""
    Class to represent permutation of axes of arrays.

    Examples
    ========

    >>> from sympy.tensor.array import permutedims
    >>> from sympy import MatrixSymbol
    >>> M = MatrixSymbol("M", 3, 3)
    >>> cg = permutedims(M, [1, 0])

    The object ``cg`` represents the transposition of ``M``, as the permutation
    ``[1, 0]`` will act on its indices by switching them:

    `M_{ij} \Rightarrow M_{ji}`

    This is evident when transforming back to matrix form:

    >>> from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
    >>> convert_array_to_matrix(cg)
    M.T

    >>> N = MatrixSymbol("N", 3, 2)
    >>> cg = permutedims(N, [1, 0])
    >>> cg.shape
    (2, 3)

    Permutations of tensor products are simplified in order to achieve a
    standard form:

    >>> from sympy.tensor.array import tensorproduct
    >>> M = MatrixSymbol("M", 4, 5)
    >>> tp = tensorproduct(M, N)
    >>> tp.shape
    (4, 5, 3, 2)
    >>> perm1 = permutedims(tp, [2, 3, 1, 0])

    The args ``(M, N)`` have been sorted and the permutation has been
    simplified, the expression is equivalent:

    >>> perm1.expr.args
    (N, M)
    >>> perm1.shape
    (3, 2, 5, 4)
    >>> perm1.permutation
    (2 3)

    The permutation in its array form has been simplified from
    ``[2, 3, 1, 0]`` to ``[0, 1, 3, 2]``, as the arguments of the tensor
    product `M` and `N` have been switched:

    >>> perm1.permutation.array_form
    [0, 1, 3, 2]

    We can nest a second permutation:

    >>> perm2 = permutedims(perm1, [1, 0, 2, 3])
    >>> perm2.shape
    (2, 3, 5, 4)
    >>> perm2.permutation.array_form
    [1, 0, 3, 2]
    """

    def __new__(cls, expr, permutation, **kwargs):
        from sympy.combinatorics import Permutation
        expr = _sympify(expr)
        permutation = Permutation(permutation)
        permutation_size = permutation.size
        expr_rank = get_rank(expr)
        if permutation_size != expr_rank:
            raise ValueError("Permutation size must be the length of the shape of expr")

        canonicalize = kwargs.pop("canonicalize", False)

        obj = Basic.__new__(cls, expr, permutation)
        obj._subranks = [get_rank(expr)]
        shape = get_shape(expr)
        if shape is None:
            obj._shape = None
        else:
            obj._shape = tuple(shape[permutation(i)] for i in range(len(shape)))
        if canonicalize:
            return obj._canonicalize()
        return obj

    def _canonicalize(self):
        expr = self.expr
        permutation = self.permutation
        if isinstance(expr, PermuteDims):
            subexpr = expr.expr
            subperm = expr.permutation
            permutation = permutation * subperm
            expr = subexpr
        if isinstance(expr, ArrayContraction):
            expr, permutation = self._PermuteDims_denestarg_ArrayContraction(expr, permutation)
        if isinstance(expr, ArrayTensorProduct):
            expr, permutation = self._PermuteDims_denestarg_ArrayTensorProduct(expr, permutation)
        if isinstance(expr, (ZeroArray, ZeroMatrix)):
            return ZeroArray(*[expr.shape[i] for i in permutation.array_form])
        plist = permutation.array_form
        if plist == sorted(plist):
            return expr
        return self.func(expr, permutation, canonicalize=False)

    def doit(self, **kwargs):
        deep = kwargs.get("deep", True)
        if deep:
            return self.func(*[arg.doit(**kwargs) for arg in self.args])._canonicalize()
        else:
            return self._canonicalize()

    @property
    def expr(self):
        return self.args[0]

    @property
    def permutation(self):
        return self.args[1]

    @classmethod
    def _PermuteDims_denestarg_ArrayTensorProduct(cls, expr, permutation):
        # Get the permutation in its image-form:
        perm_image_form = _af_invert(permutation.array_form)
        args = list(expr.args)
        # Starting index global position for every arg:
        cumul = list(accumulate([0] + expr.subranks))
        # Split `perm_image_form` into a list of list corresponding to the indices
        # of every argument:
        perm_image_form_in_components = [perm_image_form[cumul[i]:cumul[i+1]] for i in range(len(args))]
        # Create an index, target-position-key array:
        ps = [(i, sorted(comp)) for i, comp in enumerate(perm_image_form_in_components)]
        # Sort the array according to the target-position-key:
        # In this way, we define a canonical way to sort the arguments according
        # to the permutation.
        ps.sort(key=lambda x: x[1])
        # Read the inverse-permutation (i.e. image-form) of the args:
        perm_args_image_form = [i[0] for i in ps]
        # Apply the args-permutation to the `args`:
        args_sorted = [args[i] for i in perm_args_image_form]
        # Apply the args-permutation to the array-form of the permutation of the axes (of `expr`):
        perm_image_form_sorted_args = [perm_image_form_in_components[i] for i in perm_args_image_form]
        new_permutation = Permutation(_af_invert([j for i in perm_image_form_sorted_args for j in i]))
        return _array_tensor_product(*args_sorted), new_permutation

    @classmethod
    def _PermuteDims_denestarg_ArrayContraction(cls, expr, permutation):
        if not isinstance(expr, ArrayContraction):
            return expr, permutation
        if not isinstance(expr.expr, ArrayTensorProduct):
            return expr, permutation
        args = expr.expr.args
        subranks = [get_rank(arg) for arg in expr.expr.args]

        contraction_indices = expr.contraction_indices
        contraction_indices_flat = [j for i in contraction_indices for j in i]
        cumul = list(accumulate([0] + subranks))

        # Spread the permutation in its array form across the args in the corresponding
        # tensor-product arguments with free indices:
        permutation_array_blocks_up = []
        image_form = _af_invert(permutation.array_form)
        counter = 0
        for i, e in enumerate(subranks):
            current = []
            for j in range(cumul[i], cumul[i+1]):
                if j in contraction_indices_flat:
                    continue
                current.append(image_form[counter])
                counter += 1
            permutation_array_blocks_up.append(current)

        # Get the map of axis repositioning for every argument of tensor-product:
        index_blocks = [[j for j in range(cumul[i], cumul[i+1])] for i, e in enumerate(expr.subranks)]
        index_blocks_up = expr._push_indices_up(expr.contraction_indices, index_blocks)
        inverse_permutation = permutation**(-1)
        index_blocks_up_permuted = [[inverse_permutation(j) for j in i if j is not None] for i in index_blocks_up]

        # Sorting key is a list of tuple, first element is the index of `args`, second element of
        # the tuple is the sorting key to sort `args` of the tensor product:
        sorting_keys = list(enumerate(index_blocks_up_permuted))
        sorting_keys.sort(key=lambda x: x[1])

        # Now we can get the permutation acting on the args in its image-form:
        new_perm_image_form = [i[0] for i in sorting_keys]
        # Apply the args-level permutation to various elements:
        new_index_blocks = [index_blocks[i] for i in new_perm_image_form]
        new_index_perm_array_form = _af_invert([j for i in new_index_blocks for j in i])
        new_args = [args[i] for i in new_perm_image_form]
        new_contraction_indices = [tuple(new_index_perm_array_form[j] for j in i) for i in contraction_indices]
        new_expr = _array_contraction(_array_tensor_product(*new_args), *new_contraction_indices)
        new_permutation = Permutation(_af_invert([j for i in [permutation_array_blocks_up[k] for k in new_perm_image_form] for j in i]))
        return new_expr, new_permutation

    @classmethod
    def _check_permutation_mapping(cls, expr, permutation):
        subranks = expr.subranks
        index2arg = [i for i, arg in enumerate(expr.args) for j in range(expr.subranks[i])]
        permuted_indices = [permutation(i) for i in range(expr.subrank())]
        new_args = list(expr.args)
        arg_candidate_index = index2arg[permuted_indices[0]]
        current_indices = []
        new_permutation = []
        inserted_arg_cand_indices = set([])
        for i, idx in enumerate(permuted_indices):
            if index2arg[idx] != arg_candidate_index:
                new_permutation.extend(current_indices)
                current_indices = []
                arg_candidate_index = index2arg[idx]
            current_indices.append(idx)
            arg_candidate_rank = subranks[arg_candidate_index]
            if len(current_indices) == arg_candidate_rank:
                new_permutation.extend(sorted(current_indices))
                local_current_indices = [j - min(current_indices) for j in current_indices]
                i1 = index2arg[i]
                new_args[i1] = _permute_dims(new_args[i1], Permutation(local_current_indices))
                inserted_arg_cand_indices.add(arg_candidate_index)
                current_indices = []
        new_permutation.extend(current_indices)

        # TODO: swap args positions in order to simplify the expression:
        # TODO: this should be in a function
        args_positions = list(range(len(new_args)))
        # Get possible shifts:
        maps = {}
        cumulative_subranks = [0] + list(accumulate(subranks))
        for i in range(0, len(subranks)):
            s = set([index2arg[new_permutation[j]] for j in range(cumulative_subranks[i], cumulative_subranks[i+1])])
            if len(s) != 1:
                continue
            elem = next(iter(s))
            if i != elem:
                maps[i] = elem

        # Find cycles in the map:
        lines = []
        current_line = []
        while maps:
            if len(current_line) == 0:
                k, v = maps.popitem()
                current_line.append(k)
            else:
                k = current_line[-1]
                if k not in maps:
                    current_line = []
                    continue
                v = maps.pop(k)
            if v in current_line:
                lines.append(current_line)
                current_line = []
                continue
            current_line.append(v)
        for line in lines:
            for i, e in enumerate(line):
                args_positions[line[(i + 1) % len(line)]] = e

        # TODO: function in order to permute the args:
        permutation_blocks = [[new_permutation[cumulative_subranks[i] + j] for j in range(e)] for i, e in enumerate(subranks)]
        new_args = [new_args[i] for i in args_positions]
        new_permutation_blocks = [permutation_blocks[i] for i in args_positions]
        new_permutation2 = [j for i in new_permutation_blocks for j in i]
        return _array_tensor_product(*new_args), Permutation(new_permutation2)  # **(-1)

    @classmethod
    def _check_if_there_are_closed_cycles(cls, expr, permutation):
        args = list(expr.args)
        subranks = expr.subranks
        cyclic_form = permutation.cyclic_form
        cumulative_subranks = [0] + list(accumulate(subranks))
        cyclic_min = [min(i) for i in cyclic_form]
        cyclic_max = [max(i) for i in cyclic_form]
        cyclic_keep = []
        for i, cycle in enumerate(cyclic_form):
            flag = True
            for j in range(0, len(cumulative_subranks) - 1):
                if cyclic_min[i] >= cumulative_subranks[j] and cyclic_max[i] < cumulative_subranks[j+1]:
                    # Found a sinkable cycle.
                    args[j] = _permute_dims(args[j], Permutation([[k - cumulative_subranks[j] for k in cyclic_form[i]]]))
                    flag = False
                    break
            if flag:
                cyclic_keep.append(cyclic_form[i])
        return _array_tensor_product(*args), Permutation(cyclic_keep, size=permutation.size)

    def nest_permutation(self):
        r"""
        DEPRECATED.
        """
        ret = self._nest_permutation(self.expr, self.permutation)
        if ret is None:
            return self
        return ret

    @classmethod
    def _nest_permutation(cls, expr, permutation):
        if isinstance(expr, ArrayTensorProduct):
            return _permute_dims(*cls._check_if_there_are_closed_cycles(expr, permutation))
        elif isinstance(expr, ArrayContraction):
            # Invert tree hierarchy: put the contraction above.
            cycles = permutation.cyclic_form
            newcycles = ArrayContraction._convert_outer_indices_to_inner_indices(expr, *cycles)
            newpermutation = Permutation(newcycles)
            new_contr_indices = [tuple(newpermutation(j) for j in i) for i in expr.contraction_indices]
            return _array_contraction(PermuteDims(expr.expr, newpermutation), *new_contr_indices)
        elif isinstance(expr, ArrayAdd):
            return _array_add(*[PermuteDims(arg, permutation) for arg in expr.args])
        return None

    def as_explicit(self):
        return permutedims(self.expr.as_explicit(), self.permutation)


class ArrayDiagonal(_CodegenArrayAbstract):
    r"""
    Class to represent the diagonal operator.

    Explanation
    ===========

    In a 2-dimensional array it returns the diagonal, this looks like the
    operation:

    `A_{ij} \rightarrow A_{ii}`

    The diagonal over axes 1 and 2 (the second and third) of the tensor product
    of two 2-dimensional arrays `A \otimes B` is

    `\Big[ A_{ab} B_{cd} \Big]_{abcd} \rightarrow \Big[ A_{ai} B_{id} \Big]_{adi}`

    In this last example the array expression has been reduced from
    4-dimensional to 3-dimensional. Notice that no contraction has occurred,
    rather there is a new index `i` for the diagonal, contraction would have
    reduced the array to 2 dimensions.

    Notice that the diagonalized out dimensions are added as new dimensions at
    the end of the indices.
    """

    def __new__(cls, expr, *diagonal_indices, **kwargs):
        expr = _sympify(expr)
        diagonal_indices = [Tuple(*sorted(i)) for i in diagonal_indices]
        canonicalize = kwargs.get("canonicalize", False)

        shape = get_shape(expr)
        if shape is not None:
            cls._validate(expr, *diagonal_indices, **kwargs)
            # Get new shape:
            positions, shape = cls._get_positions_shape(shape, diagonal_indices)
        else:
            positions = None
        if len(diagonal_indices) == 0:
            return expr
        obj = Basic.__new__(cls, expr, *diagonal_indices)
        obj._positions = positions
        obj._subranks = _get_subranks(expr)
        obj._shape = shape
        if canonicalize:
            return obj._canonicalize()
        return obj

    def _canonicalize(self):
        expr = self.expr
        diagonal_indices = self.diagonal_indices
        trivial_diags = [i for i in diagonal_indices if len(i) == 1]
        if len(trivial_diags) > 0:
            trivial_pos = {e[0]: i for i, e in enumerate(diagonal_indices) if len(e) == 1}
            diag_pos = {e: i for i, e in enumerate(diagonal_indices) if len(e) > 1}
            diagonal_indices_short = [i for i in diagonal_indices if len(i) > 1]
            rank1 = get_rank(self)
            rank2 = len(diagonal_indices)
            rank3 = rank1 - rank2
            inv_permutation = []
            counter1: int = 0
            indices_down = ArrayDiagonal._push_indices_down(diagonal_indices_short, list(range(rank1)), get_rank(expr))
            for i in indices_down:
                if i in trivial_pos:
                    inv_permutation.append(rank3 + trivial_pos[i])
                elif isinstance(i, (Integer, int)):
                    inv_permutation.append(counter1)
                    counter1 += 1
                else:
                    inv_permutation.append(rank3 + diag_pos[i])
            permutation = _af_invert(inv_permutation)
            if len(diagonal_indices_short) > 0:
                return _permute_dims(_array_diagonal(expr, *diagonal_indices_short), permutation)
            else:
                return _permute_dims(expr, permutation)
        if isinstance(expr, ArrayAdd):
            return self._ArrayDiagonal_denest_ArrayAdd(expr, *diagonal_indices)
        if isinstance(expr, ArrayDiagonal):
            return self._ArrayDiagonal_denest_ArrayDiagonal(expr, *diagonal_indices)
        if isinstance(expr, PermuteDims):
            return self._ArrayDiagonal_denest_PermuteDims(expr, *diagonal_indices)
        if isinstance(expr, (ZeroArray, ZeroMatrix)):
            positions, shape = self._get_positions_shape(expr.shape, diagonal_indices)
            return ZeroArray(*shape)
        return self.func(expr, *diagonal_indices, canonicalize=False)

    def doit(self, **kwargs):
        deep = kwargs.get("deep", True)
        if deep:
            return self.func(*[arg.doit(**kwargs) for arg in self.args])._canonicalize()
        else:
            return self._canonicalize()

    @staticmethod
    def _validate(expr, *diagonal_indices, **kwargs):
        # Check that no diagonalization happens on indices with mismatched
        # dimensions:
        shape = get_shape(expr)
        for i in diagonal_indices:
            if any(j >= len(shape) for j in i):
                raise ValueError("index is larger than expression shape")
            if len({shape[j] for j in i}) != 1:
                raise ValueError("diagonalizing indices of different dimensions")
            if not kwargs.get("allow_trivial_diags", False) and len(i) <= 1:
                raise ValueError("need at least two axes to diagonalize")
            if len(set(i)) != len(i):
                raise ValueError("axis index cannot be repeated")

    @staticmethod
    def _remove_trivial_dimensions(shape, *diagonal_indices):
        return [tuple(j for j in i) for i in diagonal_indices if shape[i[0]] != 1]

    @property
    def expr(self):
        return self.args[0]

    @property
    def diagonal_indices(self):
        return self.args[1:]

    @staticmethod
    def _flatten(expr, *outer_diagonal_indices):
        inner_diagonal_indices = expr.diagonal_indices
        all_inner = [j for i in inner_diagonal_indices for j in i]
        all_inner.sort()
        # TODO: add API for total rank and cumulative rank:
        total_rank = _get_subrank(expr)
        inner_rank = len(all_inner)
        outer_rank = total_rank - inner_rank
        shifts = [0 for i in range(outer_rank)]
        counter = 0
        pointer = 0
        for i in range(outer_rank):
            while pointer < inner_rank and counter >= all_inner[pointer]:
                counter += 1
                pointer += 1
            shifts[i] += pointer
            counter += 1
        outer_diagonal_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_diagonal_indices)
        diagonal_indices = inner_diagonal_indices + outer_diagonal_indices
        return _array_diagonal(expr.expr, *diagonal_indices)

    @classmethod
    def _ArrayDiagonal_denest_ArrayAdd(cls, expr, *diagonal_indices):
        return _array_add(*[_array_diagonal(arg, *diagonal_indices) for arg in expr.args])

    @classmethod
    def _ArrayDiagonal_denest_ArrayDiagonal(cls, expr, *diagonal_indices):
        return cls._flatten(expr, *diagonal_indices)

    @classmethod
    def _ArrayDiagonal_denest_PermuteDims(cls, expr: PermuteDims, *diagonal_indices):
        back_diagonal_indices = [[expr.permutation(j) for j in i] for i in diagonal_indices]
        nondiag = [i for i in range(get_rank(expr)) if not any(i in j for j in diagonal_indices)]
        back_nondiag = [expr.permutation(i) for i in nondiag]
        remap = {e: i for i, e in enumerate(sorted(back_nondiag))}
        new_permutation1 = [remap[i] for i in back_nondiag]
        shift = len(new_permutation1)
        diag_block_perm = [i + shift for i in range(len(back_diagonal_indices))]
        new_permutation = new_permutation1 + diag_block_perm
        return _permute_dims(
            _array_diagonal(
                expr.expr,
                *back_diagonal_indices
            ),
            new_permutation
        )

    def _push_indices_down_nonstatic(self, indices):
        transform = lambda x: self._positions[x] if x < len(self._positions) else None
        return _apply_recursively_over_nested_lists(transform, indices)

    def _push_indices_up_nonstatic(self, indices):

        def transform(x):
            for i, e in enumerate(self._positions):
                if (isinstance(e, int) and x == e) or (isinstance(e, tuple) and x in e):
                    return i

        return _apply_recursively_over_nested_lists(transform, indices)

    @classmethod
    def _push_indices_down(cls, diagonal_indices, indices, rank):
        positions, shape = cls._get_positions_shape(range(rank), diagonal_indices)
        transform = lambda x: positions[x] if x < len(positions) else None
        return _apply_recursively_over_nested_lists(transform, indices)

    @classmethod
    def _push_indices_up(cls, diagonal_indices, indices, rank):
        positions, shape = cls._get_positions_shape(range(rank), diagonal_indices)

        def transform(x):
            for i, e in enumerate(positions):
                if (isinstance(e, int) and x == e) or (isinstance(e, (tuple, Tuple)) and (x in e)):
                    return i

        return _apply_recursively_over_nested_lists(transform, indices)

    @classmethod
    def _get_positions_shape(cls, shape, diagonal_indices):
        data1 = tuple((i, shp) for i, shp in enumerate(shape) if not any(i in j for j in diagonal_indices))
        pos1, shp1 = zip(*data1) if data1 else ((), ())
        data2 = tuple((i, shape[i[0]]) for i in diagonal_indices)
        pos2, shp2 = zip(*data2) if data2 else ((), ())
        positions = pos1 + pos2
        shape = shp1 + shp2
        return positions, shape

    def as_explicit(self):
        return tensordiagonal(self.expr.as_explicit(), *self.diagonal_indices)


class ArrayElementwiseApplyFunc(_CodegenArrayAbstract):

    def __new__(cls, function, element):

        if not isinstance(function, Lambda):
            d = Dummy('d')
            function = Lambda(d, function(d))

        obj = _CodegenArrayAbstract.__new__(cls, function, element)
        obj._subranks = _get_subranks(element)
        return obj

    @property
    def function(self):
        return self.args[0]

    @property
    def expr(self):
        return self.args[1]

    @property
    def shape(self):
        return self.expr.shape

    def _get_function_fdiff(self):
        d = Dummy("d")
        function = self.function(d)
        fdiff = function.diff(d)
        if isinstance(fdiff, Function):
            fdiff = type(fdiff)
        else:
            fdiff = Lambda(d, fdiff)
        return fdiff


class ArrayContraction(_CodegenArrayAbstract):
    r"""
    This class is meant to represent contractions of arrays in a form easily
    processable by the code printers.
    """

    def __new__(cls, expr, *contraction_indices, **kwargs):
        contraction_indices = _sort_contraction_indices(contraction_indices)
        expr = _sympify(expr)

        canonicalize = kwargs.get("canonicalize", False)

        obj = Basic.__new__(cls, expr, *contraction_indices)
        obj._subranks = _get_subranks(expr)
        obj._mapping = _get_mapping_from_subranks(obj._subranks)

        free_indices_to_position = {i: i for i in range(sum(obj._subranks)) if all(i not in cind for cind in contraction_indices)}
        obj._free_indices_to_position = free_indices_to_position

        shape = get_shape(expr)
        cls._validate(expr, *contraction_indices)
        if shape:
            shape = tuple(shp for i, shp in enumerate(shape) if not any(i in j for j in contraction_indices))
        obj._shape = shape
        if canonicalize:
            return obj._canonicalize()
        return obj

    def _canonicalize(self):
        expr = self.expr
        contraction_indices = self.contraction_indices

        if len(contraction_indices) == 0:
            return expr

        if isinstance(expr, ArrayContraction):
            return self._ArrayContraction_denest_ArrayContraction(expr, *contraction_indices)

        if isinstance(expr, (ZeroArray, ZeroMatrix)):
            return self._ArrayContraction_denest_ZeroArray(expr, *contraction_indices)

        if isinstance(expr, PermuteDims):
            return self._ArrayContraction_denest_PermuteDims(expr, *contraction_indices)

        if isinstance(expr, ArrayTensorProduct):
            expr, contraction_indices = self._sort_fully_contracted_args(expr, contraction_indices)
            expr, contraction_indices = self._lower_contraction_to_addends(expr, contraction_indices)
            if len(contraction_indices) == 0:
                return expr

        if isinstance(expr, ArrayDiagonal):
            return self._ArrayContraction_denest_ArrayDiagonal(expr, *contraction_indices)

        if isinstance(expr, ArrayAdd):
            return self._ArrayContraction_denest_ArrayAdd(expr, *contraction_indices)

        # Check single index contractions on 1-dimensional axes:
        contraction_indices = [i for i in contraction_indices if len(i) > 1 or get_shape(expr)[i[0]] != 1]
        if len(contraction_indices) == 0:
            return expr

        return self.func(expr, *contraction_indices, canonicalize=False)

    def doit(self, **kwargs):
        deep = kwargs.get("deep", True)
        if deep:
            return self.func(*[arg.doit(**kwargs) for arg in self.args])._canonicalize()
        else:
            return self._canonicalize()

    def __mul__(self, other):
        if other == 1:
            return self
        else:
            raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.")

    def __rmul__(self, other):
        if other == 1:
            return self
        else:
            raise NotImplementedError("Product of N-dim arrays is not uniquely defined. Use another method.")

    @staticmethod
    def _validate(expr, *contraction_indices):
        shape = get_shape(expr)
        if shape is None:
            return

        # Check that no contraction happens when the shape is mismatched:
        for i in contraction_indices:
            if len({shape[j] for j in i if shape[j] != -1}) != 1:
                raise ValueError("contracting indices of different dimensions")

    @classmethod
    def _push_indices_down(cls, contraction_indices, indices):
        flattened_contraction_indices = [j for i in contraction_indices for j in i]
        flattened_contraction_indices.sort()
        transform = _build_push_indices_down_func_transformation(flattened_contraction_indices)
        return _apply_recursively_over_nested_lists(transform, indices)

    @classmethod
    def _push_indices_up(cls, contraction_indices, indices):
        flattened_contraction_indices = [j for i in contraction_indices for j in i]
        flattened_contraction_indices.sort()
        transform = _build_push_indices_up_func_transformation(flattened_contraction_indices)
        return _apply_recursively_over_nested_lists(transform, indices)

    @classmethod
    def _lower_contraction_to_addends(cls, expr, contraction_indices):
        if isinstance(expr, ArrayAdd):
            raise NotImplementedError()
        if not isinstance(expr, ArrayTensorProduct):
            return expr, contraction_indices
        subranks = expr.subranks
        cumranks = list(accumulate([0] + subranks))
        contraction_indices_remaining = []
        contraction_indices_args = [[] for i in expr.args]
        backshift = set([])
        for i, contraction_group in enumerate(contraction_indices):
            for j in range(len(expr.args)):
                if not isinstance(expr.args[j], ArrayAdd):
                    continue
                if all(cumranks[j] <= k < cumranks[j+1] for k in contraction_group):
                    contraction_indices_args[j].append([k - cumranks[j] for k in contraction_group])
                    backshift.update(contraction_group)
                    break
            else:
                contraction_indices_remaining.append(contraction_group)
        if len(contraction_indices_remaining) == len(contraction_indices):
            return expr, contraction_indices
        total_rank = get_rank(expr)
        shifts = list(accumulate([1 if i in backshift else 0 for i in range(total_rank)]))
        contraction_indices_remaining = [Tuple.fromiter(j - shifts[j] for j in i) for i in contraction_indices_remaining]
        ret = _array_tensor_product(*[
            _array_contraction(arg, *contr) for arg, contr in zip(expr.args, contraction_indices_args)
        ])
        return ret, contraction_indices_remaining

    def split_multiple_contractions(self):
        """
        Recognize multiple contractions and attempt at rewriting them as paired-contractions.

        This allows some contractions involving more than two indices to be
        rewritten as multiple contractions involving two indices, thus allowing
        the expression to be rewritten as a matrix multiplication line.

        Examples:

        * `A_ij b_j0 C_jk` ===> `A*DiagMatrix(b)*C`

        Care for:
        - matrix being diagonalized (i.e. `A_ii`)
        - vectors being diagonalized (i.e. `a_i0`)

        Multiple contractions can be split into matrix multiplications if
        not more than two arguments are non-diagonals or non-vectors.
        Vectors get diagonalized while diagonal matrices remain diagonal.
        The non-diagonal matrices can be at the beginning or at the end
        of the final matrix multiplication line.
        """

        editor = _EditArrayContraction(self)

        contraction_indices = self.contraction_indices

        onearray_insert = []

        for indl, links in enumerate(contraction_indices):
            if len(links) <= 2:
                continue

            # Check multiple contractions:
            #
            # Examples:
            #
            # * `A_ij b_j0 C_jk` ===> `A*DiagMatrix(b)*C \otimes OneArray(1)` with permutation (1 2)
            #
            # Care for:
            # - matrix being diagonalized (i.e. `A_ii`)
            # - vectors being diagonalized (i.e. `a_i0`)

            # Multiple contractions can be split into matrix multiplications if
            # not more than three arguments are non-diagonals or non-vectors.
            #
            # Vectors get diagonalized while diagonal matrices remain diagonal.
            # The non-diagonal matrices can be at the beginning or at the end
            # of the final matrix multiplication line.

            positions = editor.get_mapping_for_index(indl)

            # Also consider the case of diagonal matrices being contracted:
            current_dimension = self.expr.shape[links[0]]

            not_vectors: tTuple[_ArgE, int] = []
            vectors: tTuple[_ArgE, int] = []
            for arg_ind, rel_ind in positions:
                arg = editor.args_with_ind[arg_ind]
                mat = arg.element
                abs_arg_start, abs_arg_end = editor.get_absolute_range(arg)
                other_arg_pos = 1-rel_ind
                other_arg_abs = abs_arg_start + other_arg_pos
                if ((1 not in mat.shape) or
                    ((current_dimension == 1) is True and mat.shape != (1, 1)) or
                    any(other_arg_abs in l for li, l in enumerate(contraction_indices) if li != indl)
                ):
                    not_vectors.append((arg, rel_ind))
                else:
                    vectors.append((arg, rel_ind))
            if len(not_vectors) > 2:
                # If more than two arguments in the multiple contraction are
                # non-vectors and non-diagonal matrices, we cannot find a way
                # to split this contraction into a matrix multiplication line:
                continue
            # Three cases to handle:
            # - zero non-vectors
            # - one non-vector
            # - two non-vectors
            for v, rel_ind in vectors:
                v.element = diagonalize_vector(v.element)
            vectors_to_loop = not_vectors[:1] + vectors + not_vectors[1:]
            first_not_vector, rel_ind = vectors_to_loop[0]
            new_index = first_not_vector.indices[rel_ind]

            for v, rel_ind in vectors_to_loop[1:-1]:
                v.indices[rel_ind] = new_index
                new_index = editor.get_new_contraction_index()
                assert v.indices.index(None) == 1 - rel_ind
                v.indices[v.indices.index(None)] = new_index
                onearray_insert.append(v)

            last_vec, rel_ind = vectors_to_loop[-1]
            last_vec.indices[rel_ind] = new_index

        for v in onearray_insert:
            editor.insert_after(v, _ArgE(OneArray(1), [None]))

        return editor.to_array_contraction()

    def flatten_contraction_of_diagonal(self):
        if not isinstance(self.expr, ArrayDiagonal):
            return self
        contraction_down = self.expr._push_indices_down(self.expr.diagonal_indices, self.contraction_indices)
        new_contraction_indices = []
        diagonal_indices = self.expr.diagonal_indices[:]
        for i in contraction_down:
            contraction_group = list(i)
            for j in i:
                diagonal_with = [k for k in diagonal_indices if j in k]
                contraction_group.extend([l for k in diagonal_with for l in k])
                diagonal_indices = [k for k in diagonal_indices if k not in diagonal_with]
            new_contraction_indices.append(sorted(set(contraction_group)))

        new_contraction_indices = ArrayDiagonal._push_indices_up(diagonal_indices, new_contraction_indices)
        return _array_contraction(
            _array_diagonal(
                self.expr.expr,
                *diagonal_indices
            ),
            *new_contraction_indices
        )

    @staticmethod
    def _get_free_indices_to_position_map(free_indices, contraction_indices):
        free_indices_to_position = {}
        flattened_contraction_indices = [j for i in contraction_indices for j in i]
        counter = 0
        for ind in free_indices:
            while counter in flattened_contraction_indices:
                counter += 1
            free_indices_to_position[ind] = counter
            counter += 1
        return free_indices_to_position

    @staticmethod
    def _get_index_shifts(expr):
        """
        Get the mapping of indices at the positions before the contraction
        occurs.

        Examples
        ========

        >>> from sympy.tensor.array import tensorproduct, tensorcontraction
        >>> from sympy import MatrixSymbol
        >>> M = MatrixSymbol("M", 3, 3)
        >>> N = MatrixSymbol("N", 3, 3)
        >>> cg = tensorcontraction(tensorproduct(M, N), [1, 2])
        >>> cg._get_index_shifts(cg)
        [0, 2]

        Indeed, ``cg`` after the contraction has two dimensions, 0 and 1. They
        need to be shifted by 0 and 2 to get the corresponding positions before
        the contraction (that is, 0 and 3).
        """
        inner_contraction_indices = expr.contraction_indices
        all_inner = [j for i in inner_contraction_indices for j in i]
        all_inner.sort()
        # TODO: add API for total rank and cumulative rank:
        total_rank = _get_subrank(expr)
        inner_rank = len(all_inner)
        outer_rank = total_rank - inner_rank
        shifts = [0 for i in range(outer_rank)]
        counter = 0
        pointer = 0
        for i in range(outer_rank):
            while pointer < inner_rank and counter >= all_inner[pointer]:
                counter += 1
                pointer += 1
            shifts[i] += pointer
            counter += 1
        return shifts

    @staticmethod
    def _convert_outer_indices_to_inner_indices(expr, *outer_contraction_indices):
        shifts = ArrayContraction._get_index_shifts(expr)
        outer_contraction_indices = tuple(tuple(shifts[j] + j for j in i) for i in outer_contraction_indices)
        return outer_contraction_indices

    @staticmethod
    def _flatten(expr, *outer_contraction_indices):
        inner_contraction_indices = expr.contraction_indices
        outer_contraction_indices = ArrayContraction._convert_outer_indices_to_inner_indices(expr, *outer_contraction_indices)
        contraction_indices = inner_contraction_indices + outer_contraction_indices
        return _array_contraction(expr.expr, *contraction_indices)

    @classmethod
    def _ArrayContraction_denest_ArrayContraction(cls, expr, *contraction_indices):
        return cls._flatten(expr, *contraction_indices)

    @classmethod
    def _ArrayContraction_denest_ZeroArray(cls, expr, *contraction_indices):
        contraction_indices_flat = [j for i in contraction_indices for j in i]
        shape = [e for i, e in enumerate(expr.shape) if i not in contraction_indices_flat]
        return ZeroArray(*shape)

    @classmethod
    def _ArrayContraction_denest_ArrayAdd(cls, expr, *contraction_indices):
        return _array_add(*[_array_contraction(i, *contraction_indices) for i in expr.args])

    @classmethod
    def _ArrayContraction_denest_PermuteDims(cls, expr, *contraction_indices):
        permutation = expr.permutation
        plist = permutation.array_form
        new_contraction_indices = [tuple(permutation(j) for j in i) for i in contraction_indices]
        new_plist = [i for i in plist if not any(i in j for j in new_contraction_indices)]
        new_plist = cls._push_indices_up(new_contraction_indices, new_plist)
        return _permute_dims(
            _array_contraction(expr.expr, *new_contraction_indices),
            Permutation(new_plist)
        )

    @classmethod
    def _ArrayContraction_denest_ArrayDiagonal(cls, expr: 'ArrayDiagonal', *contraction_indices):
        diagonal_indices = list(expr.diagonal_indices)
        down_contraction_indices = expr._push_indices_down(expr.diagonal_indices, contraction_indices, get_rank(expr.expr))
        # Flatten diagonally contracted indices:
        down_contraction_indices = [[k for j in i for k in (j if isinstance(j, (tuple, Tuple)) else [j])] for i in down_contraction_indices]
        new_contraction_indices = []
        for contr_indgrp in down_contraction_indices:
            ind = contr_indgrp[:]
            for j, diag_indgrp in enumerate(diagonal_indices):
                if diag_indgrp is None:
                    continue
                if any(i in diag_indgrp for i in contr_indgrp):
                    ind.extend(diag_indgrp)
                    diagonal_indices[j] = None
            new_contraction_indices.append(sorted(set(ind)))

        new_diagonal_indices_down = [i for i in diagonal_indices if i is not None]
        new_diagonal_indices = ArrayContraction._push_indices_up(new_contraction_indices, new_diagonal_indices_down)
        return _array_diagonal(
            _array_contraction(expr.expr, *new_contraction_indices),
            *new_diagonal_indices
        )

    @classmethod
    def _sort_fully_contracted_args(cls, expr, contraction_indices):
        if expr.shape is None:
            return expr, contraction_indices
        cumul = list(accumulate([0] + expr.subranks))
        index_blocks = [list(range(cumul[i], cumul[i+1])) for i in range(len(expr.args))]
        contraction_indices_flat = {j for i in contraction_indices for j in i}
        fully_contracted = [all(j in contraction_indices_flat for j in range(cumul[i], cumul[i+1])) for i, arg in enumerate(expr.args)]
        new_pos = sorted(range(len(expr.args)), key=lambda x: (0, default_sort_key(expr.args[x])) if fully_contracted[x] else (1,))
        new_args = [expr.args[i] for i in new_pos]
        new_index_blocks_flat = [j for i in new_pos for j in index_blocks[i]]
        index_permutation_array_form = _af_invert(new_index_blocks_flat)
        new_contraction_indices = [tuple(index_permutation_array_form[j] for j in i) for i in contraction_indices]
        new_contraction_indices = _sort_contraction_indices(new_contraction_indices)
        return _array_tensor_product(*new_args), new_contraction_indices

    def _get_contraction_tuples(self):
        r"""
        Return tuples containing the argument index and position within the
        argument of the index position.

        Examples
        ========

        >>> from sympy import MatrixSymbol
        >>> from sympy.abc import N
        >>> from sympy.tensor.array import tensorproduct, tensorcontraction
        >>> A = MatrixSymbol("A", N, N)
        >>> B = MatrixSymbol("B", N, N)

        >>> cg = tensorcontraction(tensorproduct(A, B), (1, 2))
        >>> cg._get_contraction_tuples()
        [[(0, 1), (1, 0)]]

        Notes
        =====

        Here the contraction pair `(1, 2)` meaning that the 2nd and 3rd indices
        of the tensor product `A\otimes B` are contracted, has been transformed
        into `(0, 1)` and `(1, 0)`, identifying the same indices in a different
        notation. `(0, 1)` is the second index (1) of the first argument (i.e.
                0 or `A`). `(1, 0)` is the first index (i.e. 0) of the second
        argument (i.e. 1 or `B`).
        """
        mapping = self._mapping
        return [[mapping[j] for j in i] for i in self.contraction_indices]

    @staticmethod
    def _contraction_tuples_to_contraction_indices(expr, contraction_tuples):
        # TODO: check that `expr` has `.subranks`:
        ranks = expr.subranks
        cumulative_ranks = [0] + list(accumulate(ranks))
        return [tuple(cumulative_ranks[j]+k for j, k in i) for i in contraction_tuples]

    @property
    def free_indices(self):
        return self._free_indices[:]

    @property
    def free_indices_to_position(self):
        return dict(self._free_indices_to_position)

    @property
    def expr(self):
        return self.args[0]

    @property
    def contraction_indices(self):
        return self.args[1:]

    def _contraction_indices_to_components(self):
        expr = self.expr
        if not isinstance(expr, ArrayTensorProduct):
            raise NotImplementedError("only for contractions of tensor products")
        ranks = expr.subranks
        mapping = {}
        counter = 0
        for i, rank in enumerate(ranks):
            for j in range(rank):
                mapping[counter] = (i, j)
                counter += 1
        return mapping

    def sort_args_by_name(self):
        """
        Sort arguments in the tensor product so that their order is lexicographical.

        Examples
        ========

        >>> from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
        >>> from sympy import MatrixSymbol
        >>> from sympy.abc import N
        >>> A = MatrixSymbol("A", N, N)
        >>> B = MatrixSymbol("B", N, N)
        >>> C = MatrixSymbol("C", N, N)
        >>> D = MatrixSymbol("D", N, N)

        >>> cg = convert_matrix_to_array(C*D*A*B)
        >>> cg
        ArrayContraction(ArrayTensorProduct(A, D, C, B), (0, 3), (1, 6), (2, 5))
        >>> cg.sort_args_by_name()
        ArrayContraction(ArrayTensorProduct(A, D, B, C), (0, 3), (1, 4), (2, 7))
        """
        expr = self.expr
        if not isinstance(expr, ArrayTensorProduct):
            return self
        args = expr.args
        sorted_data = sorted(enumerate(args), key=lambda x: default_sort_key(x[1]))
        pos_sorted, args_sorted = zip(*sorted_data)
        reordering_map = {i: pos_sorted.index(i) for i, arg in enumerate(args)}
        contraction_tuples = self._get_contraction_tuples()
        contraction_tuples = [[(reordering_map[j], k) for j, k in i] for i in contraction_tuples]
        c_tp = _array_tensor_product(*args_sorted)
        new_contr_indices = self._contraction_tuples_to_contraction_indices(
                c_tp,
                contraction_tuples
        )
        return _array_contraction(c_tp, *new_contr_indices)

    def _get_contraction_links(self):
        r"""
        Returns a dictionary of links between arguments in the tensor product
        being contracted.

        See the example for an explanation of the values.

        Examples
        ========

        >>> from sympy import MatrixSymbol
        >>> from sympy.abc import N
        >>> from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
        >>> A = MatrixSymbol("A", N, N)
        >>> B = MatrixSymbol("B", N, N)
        >>> C = MatrixSymbol("C", N, N)
        >>> D = MatrixSymbol("D", N, N)

        Matrix multiplications are pairwise contractions between neighboring
        matrices:

        `A_{ij} B_{jk} C_{kl} D_{lm}`

        >>> cg = convert_matrix_to_array(A*B*C*D)
        >>> cg
        ArrayContraction(ArrayTensorProduct(B, C, A, D), (0, 5), (1, 2), (3, 6))

        >>> cg._get_contraction_links()
        {0: {0: (2, 1), 1: (1, 0)}, 1: {0: (0, 1), 1: (3, 0)}, 2: {1: (0, 0)}, 3: {0: (1, 1)}}

        This dictionary is interpreted as follows: argument in position 0 (i.e.
        matrix `A`) has its second index (i.e. 1) contracted to `(1, 0)`, that
        is argument in position 1 (matrix `B`) on the first index slot of `B`,
        this is the contraction provided by the index `j` from `A`.

        The argument in position 1 (that is, matrix `B`) has two contractions,
        the ones provided by the indices `j` and `k`, respectively the first
        and second indices (0 and 1 in the sub-dict).  The link `(0, 1)` and
        `(2, 0)` respectively. `(0, 1)` is the index slot 1 (the 2nd) of
        argument in position 0 (that is, `A_{\ldot j}`), and so on.
        """
        args, dlinks = _get_contraction_links([self], self.subranks, *self.contraction_indices)
        return dlinks

    def as_explicit(self):
        return tensorcontraction(self.expr.as_explicit(), *self.contraction_indices)


class Reshape(_CodegenArrayAbstract):
    """
    Reshape the dimensions of an array expression.

    Examples
    ========

    >>> from sympy.tensor.array.expressions import ArraySymbol, Reshape
    >>> A = ArraySymbol("A", (6,))
    >>> A.shape
    (6,)
    >>> Reshape(A, (3, 2)).shape
    (3, 2)

    Check the component-explicit forms:

    >>> A.as_explicit()
    [A[0], A[1], A[2], A[3], A[4], A[5]]
    >>> Reshape(A, (3, 2)).as_explicit()
    [[A[0], A[1]], [A[2], A[3]], [A[4], A[5]]]

    """

    def __new__(cls, expr, shape):
        expr = _sympify(expr)
        if not isinstance(shape, Tuple):
            shape = Tuple(*shape)
        if Equality(Mul.fromiter(expr.shape), Mul.fromiter(shape)) == False:
            raise ValueError("shape mismatch")
        obj = Expr.__new__(cls, expr, shape)
        obj._shape = tuple(shape)
        obj._expr = expr
        return obj

    @property
    def shape(self):
        return self._shape

    @property
    def expr(self):
        return self._expr

    def doit(self, *args, **kwargs):
        if kwargs.get("deep", True):
            expr = self.expr.doit(*args, **kwargs)
        else:
            expr = self.expr
        if isinstance(expr, (MatrixCommon, NDimArray)):
            return expr.reshape(*self.shape)
        return Reshape(expr, self.shape)

    def as_explicit(self):
        ee = self.expr.as_explicit()
        if isinstance(ee, MatrixCommon):
            from sympy import Array
            ee = Array(ee)
        elif isinstance(ee, MatrixExpr):
            return self
        return ee.reshape(*self.shape)


class _ArgE:
    """
    The ``_ArgE`` object contains references to the array expression
    (``.element``) and a list containing the information about index
    contractions (``.indices``).

    Index contractions are numbered and contracted indices show the number of
    the contraction. Uncontracted indices have ``None`` value.

    For example:
    ``_ArgE(M, [None, 3])``
    This object means that expression ``M`` is part of an array contraction
    and has two indices, the first is not contracted (value ``None``),
    the second index is contracted to the 4th (i.e. number ``3``) group of the
    array contraction object.
    """
    indices: List[Optional[int]]

    def __init__(self, element, indices: Optional[List[Optional[int]]] = None):
        self.element = element
        if indices is None:
            self.indices = [None for i in range(get_rank(element))]
        else:
            self.indices = indices

    def __str__(self):
        return "_ArgE(%s, %s)" % (self.element, self.indices)

    __repr__ = __str__


class _IndPos:
    """
    Index position, requiring two integers in the constructor:

    - arg: the position of the argument in the tensor product,
    - rel: the relative position of the index inside the argument.
    """
    def __init__(self, arg: int, rel: int):
        self.arg = arg
        self.rel = rel

    def __str__(self):
        return "_IndPos(%i, %i)" % (self.arg, self.rel)

    __repr__ = __str__

    def __iter__(self):
        yield from [self.arg, self.rel]


class _EditArrayContraction:
    """
    Utility class to help manipulate array contraction objects.

    This class takes as input an ``ArrayContraction`` object and turns it into
    an editable object.

    The field ``args_with_ind`` of this class is a list of ``_ArgE`` objects
    which can be used to easily edit the contraction structure of the
    expression.

    Once editing is finished, the ``ArrayContraction`` object may be recreated
    by calling the ``.to_array_contraction()`` method.
    """

    def __init__(self, base_array: typing.Union[ArrayContraction, ArrayDiagonal, ArrayTensorProduct]):

        expr: Basic
        diagonalized: tTuple[tTuple[int, ...], ...]
        contraction_indices: List[tTuple[int]]
        if isinstance(base_array, ArrayContraction):
            mapping = _get_mapping_from_subranks(base_array.subranks)
            expr = base_array.expr
            contraction_indices = base_array.contraction_indices
            diagonalized = ()
        elif isinstance(base_array, ArrayDiagonal):

            if isinstance(base_array.expr, ArrayContraction):
                mapping = _get_mapping_from_subranks(base_array.expr.subranks)
                expr = base_array.expr.expr
                diagonalized = ArrayContraction._push_indices_down(base_array.expr.contraction_indices, base_array.diagonal_indices)
                contraction_indices = base_array.expr.contraction_indices
            elif isinstance(base_array.expr, ArrayTensorProduct):
                mapping = {}
                expr = base_array.expr
                diagonalized = base_array.diagonal_indices
                contraction_indices = []
            else:
                mapping = {}
                expr = base_array.expr
                diagonalized = base_array.diagonal_indices
                contraction_indices = []

        elif isinstance(base_array, ArrayTensorProduct):
            expr = base_array
            contraction_indices = []
            diagonalized = ()
        else:
            raise NotImplementedError()

        if isinstance(expr, ArrayTensorProduct):
            args = list(expr.args)
        else:
            args = [expr]

        args_with_ind: List[_ArgE] = [_ArgE(arg) for arg in args]
        for i, contraction_tuple in enumerate(contraction_indices):
            for j in contraction_tuple:
                arg_pos, rel_pos = mapping[j]
                args_with_ind[arg_pos].indices[rel_pos] = i
        self.args_with_ind: List[_ArgE] = args_with_ind
        self.number_of_contraction_indices: int = len(contraction_indices)
        self._track_permutation: Optional[List[List[int]]] = None

        mapping = _get_mapping_from_subranks(base_array.subranks)

        # Trick: add diagonalized indices as negative indices into the editor object:
        for i, e in enumerate(diagonalized):
            for j in e:
                arg_pos, rel_pos = mapping[j]
                self.args_with_ind[arg_pos].indices[rel_pos] = -1 - i

    def insert_after(self, arg: _ArgE, new_arg: _ArgE):
        pos = self.args_with_ind.index(arg)
        self.args_with_ind.insert(pos + 1, new_arg)

    def get_new_contraction_index(self):
        self.number_of_contraction_indices += 1
        return self.number_of_contraction_indices - 1

    def refresh_indices(self):
        updates: tDict[int, int] = {}
        for arg_with_ind in self.args_with_ind:
            updates.update({i: -1 for i in arg_with_ind.indices if i is not None})
        for i, e in enumerate(sorted(updates)):
            updates[e] = i
        self.number_of_contraction_indices: int = len(updates)
        for arg_with_ind in self.args_with_ind:
            arg_with_ind.indices = [updates.get(i, None) for i in arg_with_ind.indices]

    def merge_scalars(self):
        scalars = []
        for arg_with_ind in self.args_with_ind:
            if len(arg_with_ind.indices) == 0:
                scalars.append(arg_with_ind)
        for i in scalars:
            self.args_with_ind.remove(i)
        scalar = Mul.fromiter([i.element for i in scalars])
        if len(self.args_with_ind) == 0:
            self.args_with_ind.append(_ArgE(scalar))
        else:
            from sympy.tensor.array.expressions.conv_array_to_matrix import _a2m_tensor_product
            self.args_with_ind[0].element = _a2m_tensor_product(scalar, self.args_with_ind[0].element)

    def to_array_contraction(self):

        # Count the ranks of the arguments:
        counter = 0
        # Create a collector for the new diagonal indices:
        diag_indices = defaultdict(list)

        count_index_freq = Counter()
        for arg_with_ind in self.args_with_ind:
            count_index_freq.update(Counter(arg_with_ind.indices))

        free_index_count = count_index_freq[None]

        # Construct the inverse permutation:
        inv_perm1 = []
        inv_perm2 = []
        # Keep track of which diagonal indices have already been processed:
        done = set([])

        # Counter for the diagonal indices:
        counter4 = 0

        for arg_with_ind in self.args_with_ind:
            # If some diagonalization axes have been removed, they should be
            # permuted in order to keep the permutation.
            # Add permutation here
            counter2 = 0  # counter for the indices
            for i in arg_with_ind.indices:
                if i is None:
                    inv_perm1.append(counter4)
                    counter2 += 1
                    counter4 += 1
                    continue
                if i >= 0:
                    continue
                # Reconstruct the diagonal indices:
                diag_indices[-1 - i].append(counter + counter2)
                if count_index_freq[i] == 1 and i not in done:
                    inv_perm1.append(free_index_count - 1 - i)
                    done.add(i)
                elif i not in done:
                    inv_perm2.append(free_index_count - 1 - i)
                    done.add(i)
                counter2 += 1
            # Remove negative indices to restore a proper editor object:
            arg_with_ind.indices = [i if i is not None and i >= 0 else None for i in arg_with_ind.indices]
            counter += len([i for i in arg_with_ind.indices if i is None or i < 0])

        inverse_permutation = inv_perm1 + inv_perm2
        permutation = _af_invert(inverse_permutation)

        # Get the diagonal indices after the detection of HadamardProduct in the expression:
        diag_indices_filtered = [tuple(v) for v in diag_indices.values() if len(v) > 1]

        self.merge_scalars()
        self.refresh_indices()
        args = [arg.element for arg in self.args_with_ind]
        contraction_indices = self.get_contraction_indices()
        expr = _array_contraction(_array_tensor_product(*args), *contraction_indices)
        expr2 = _array_diagonal(expr, *diag_indices_filtered)
        if self._track_permutation is not None:
            permutation2 = _af_invert([j for i in self._track_permutation for j in i])
            expr2 = _permute_dims(expr2, permutation2)

        expr3 = _permute_dims(expr2, permutation)
        return expr3

    def get_contraction_indices(self) -> List[List[int]]:
        contraction_indices: List[List[int]] = [[] for i in range(self.number_of_contraction_indices)]
        current_position: int = 0
        for i, arg_with_ind in enumerate(self.args_with_ind):
            for j in arg_with_ind.indices:
                if j is not None:
                    contraction_indices[j].append(current_position)
                current_position += 1
        return contraction_indices

    def get_mapping_for_index(self, ind) -> List[_IndPos]:
        if ind >= self.number_of_contraction_indices:
            raise ValueError("index value exceeding the index range")
        positions: List[_IndPos] = []
        for i, arg_with_ind in enumerate(self.args_with_ind):
            for j, arg_ind in enumerate(arg_with_ind.indices):
                if ind == arg_ind:
                    positions.append(_IndPos(i, j))
        return positions

    def get_contraction_indices_to_ind_rel_pos(self) -> List[List[_IndPos]]:
        contraction_indices: List[List[_IndPos]] = [[] for i in range(self.number_of_contraction_indices)]
        for i, arg_with_ind in enumerate(self.args_with_ind):
            for j, ind in enumerate(arg_with_ind.indices):
                if ind is not None:
                    contraction_indices[ind].append(_IndPos(i, j))
        return contraction_indices

    def count_args_with_index(self, index: int) -> int:
        """
        Count the number of arguments that have the given index.
        """
        counter: int = 0
        for arg_with_ind in self.args_with_ind:
            if index in arg_with_ind.indices:
                counter += 1
        return counter

    def get_args_with_index(self, index: int) -> List[_ArgE]:
        """
        Get a list of arguments having the given index.
        """
        ret: List[_ArgE] = [i for i in self.args_with_ind if index in i.indices]
        return ret

    @property
    def number_of_diagonal_indices(self):
        data = set([])
        for arg in self.args_with_ind:
            data.update({i for i in arg.indices if i is not None and i < 0})
        return len(data)

    def track_permutation_start(self):
        permutation = []
        perm_diag = []
        counter: int = 0
        counter2: int = -1
        for arg_with_ind in self.args_with_ind:
            perm = []
            for i in arg_with_ind.indices:
                if i is not None:
                    if i < 0:
                        perm_diag.append(counter2)
                        counter2 -= 1
                    continue
                perm.append(counter)
                counter += 1
            permutation.append(perm)
        max_ind = max([max(i) if i else -1 for i in permutation]) if permutation else -1
        perm_diag = [max_ind - i for i in perm_diag]
        self._track_permutation = permutation + [perm_diag]

    def track_permutation_merge(self, destination: _ArgE, from_element: _ArgE):
        index_destination = self.args_with_ind.index(destination)
        index_element = self.args_with_ind.index(from_element)
        self._track_permutation[index_destination].extend(self._track_permutation[index_element]) # type: ignore
        self._track_permutation.pop(index_element) # type: ignore

    def get_absolute_free_range(self, arg: _ArgE) -> typing.Tuple[int, int]:
        """
        Return the range of the free indices of the arg as absolute positions
        among all free indices.
        """
        counter = 0
        for arg_with_ind in self.args_with_ind:
            number_free_indices = len([i for i in arg_with_ind.indices if i is None])
            if arg_with_ind == arg:
                return counter, counter + number_free_indices
            counter += number_free_indices
        raise IndexError("argument not found")

    def get_absolute_range(self, arg: _ArgE) -> typing.Tuple[int, int]:
        """
        Return the absolute range of indices for arg, disregarding dummy
        indices.
        """
        counter = 0
        for arg_with_ind in self.args_with_ind:
            number_indices = len(arg_with_ind.indices)
            if arg_with_ind == arg:
                return counter, counter + number_indices
            counter += number_indices
        raise IndexError("argument not found")


def get_rank(expr):
    if isinstance(expr, (MatrixExpr, MatrixElement)):
        return 2
    if isinstance(expr, _CodegenArrayAbstract):
        return len(expr.shape)
    if isinstance(expr, NDimArray):
        return expr.rank()
    if isinstance(expr, Indexed):
        return expr.rank
    if isinstance(expr, IndexedBase):
        shape = expr.shape
        if shape is None:
            return -1
        else:
            return len(shape)
    if hasattr(expr, "shape"):
        return len(expr.shape)
    return 0


def _get_subrank(expr):
    if isinstance(expr, _CodegenArrayAbstract):
        return expr.subrank()
    return get_rank(expr)


def _get_subranks(expr):
    if isinstance(expr, _CodegenArrayAbstract):
        return expr.subranks
    else:
        return [get_rank(expr)]


def get_shape(expr):
    if hasattr(expr, "shape"):
        return expr.shape
    return ()


def nest_permutation(expr):
    if isinstance(expr, PermuteDims):
        return expr.nest_permutation()
    else:
        return expr


def _array_tensor_product(*args, **kwargs):
    return ArrayTensorProduct(*args, canonicalize=True, **kwargs)


def _array_contraction(expr, *contraction_indices, **kwargs):
    return ArrayContraction(expr, *contraction_indices, canonicalize=True, **kwargs)


def _array_diagonal(expr, *diagonal_indices, **kwargs):
    return ArrayDiagonal(expr, *diagonal_indices, canonicalize=True, **kwargs)


def _permute_dims(expr, permutation, **kwargs):
    return PermuteDims(expr, permutation, canonicalize=True, **kwargs)


def _array_add(*args, **kwargs):
    return ArrayAdd(*args, canonicalize=True, **kwargs)