prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  Concat reverse indexing not working correctly
  The following works correctly:
  
  ```
  a_data = np.zeros((10, 5))
  b_data = np.zeros((10, 3))
  a = Variable(a_data)
  b = Variable(b_data)
  F.concat((a, b), axis=1)
  ```
  
  However, if I change the last line into:
  
  ```
  F.concat((a, b), axis=-1)
  ```
  
  it gives the error:
  
  ```
  Invalid operation is performed in: Concat (Forward)
  
  Expect: in_types[0].shape[1] == in_types[1].shape[1]
  Actual: 5 != 3
  ```
  
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `chainer/functions/array/concat.py`
  Content:
  ```
  1 import numpy
  2 
  3 from chainer import cuda
  4 from chainer import function
  5 from chainer.utils import type_check
  6 
  7 
  8 class Concat(function.Function):
  9 
  10     """Concatenate multiple tensors towards specified axis."""
  11 
  12     # concat along the channel dimension by default
  13     def __init__(self, axis=1):
  14         self.axis = axis
  15 
  16     def check_type_forward(self, in_types):
  17         type_check.expect(in_types.size() > 0)
  18         type_check.expect(in_types[0].ndim >
  19                           type_check.Variable(self.axis, 'axis'))
  20 
  21         ndim = in_types[0].ndim.eval()
  22         for i in range(1, in_types.size().eval()):
  23             type_check.expect(
  24                 in_types[0].dtype == in_types[i].dtype,
  25                 in_types[0].ndim == in_types[i].ndim,
  26             )
  27             for d in range(0, ndim):
  28                 if d == self.axis:
  29                     continue
  30                 type_check.expect(in_types[0].shape[d] == in_types[i].shape[d])
  31 
  32     def forward(self, xs):
  33         xp = cuda.get_array_module(*xs)
  34         return xp.concatenate(xs, axis=self.axis),
  35 
  36     def backward(self, xs, gy):
  37         if not xs[:-1]:
  38             return gy
  39 
  40         xp = cuda.get_array_module(*xs)
  41         sizes = numpy.array([x.shape[self.axis] for x in xs[:-1]]).cumsum()
  42         return xp.split(gy[0], sizes, axis=self.axis)
  43 
  44 
  45 def concat(xs, axis=1):
  46     """Concatenates given variables along an axis.
  47 
  48     Args:
  49         xs (tuple of Variables): Variables to be concatenated.
  50         axis (int): Axis that the input arrays are concatenated along.
  51 
  52     Returns:
  53         ~chainer.Variable: Output variable.
  54 
  55     """
  56     return Concat(axis=axis)(*xs)
  57
  ```
  
  Path: `cupy/manipulation/split.py`
  Content:
  ```
  1 import numpy
  2 import six
  3 
  4 
  5 def array_split(ary, indices_or_sections, axis=0):
  6     """Splits an array into multiple sub arrays along a given axis.
  7 
  8     This function is almost equivalent to :func:`cupy.split`. The only
  9     difference is that this function allows an integer sections that does not
  10     evenly divide the axis.
  11 
  12     .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.array_split`
  13 
  14     """
  15     if ary.ndim <= axis:
  16         raise IndexError('Axis exceeds ndim')
  17     size = ary.shape[axis]
  18 
  19     if numpy.isscalar(indices_or_sections):
  20         each_size = (size - 1) // indices_or_sections + 1
  21         indices = [i * each_size
  22                    for i in six.moves.range(1, indices_or_sections)]
  23     else:
  24         indices = indices_or_sections
  25 
  26     if len(indices) == 0:
  27         return [ary]
  28 
  29     skip = (slice(None),) * axis
  30     ret = []
  31     i = 0
  32     for index in indices:
  33         ret.append(ary[skip + (slice(i, index),)])
  34         i = index
  35     ret.append(ary[skip + (slice(i, size),)])
  36 
  37     return ret
  38 
  39 
  40 def dsplit(ary, indices_or_sections):
  41     """Splits an array into multiple sub arrays along the third axis.
  42 
  43     This is equivalent to ``split`` with ``axis=2``.
  44 
  45     .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.dsplit`
  46 
  47     """
  48     if ary.ndim <= 2:
  49         raise ValueError('Cannot dsplit an array with less than 3 dimensions')
  50     return split(ary, indices_or_sections, 2)
  51 
  52 
  53 def hsplit(ary, indices_or_sections):
  54     """Splits an array into multiple sub arrays horizontally.
  55 
  56     This is equivalent to ``split`` with ``axis=0`` if ``ary`` has one
  57     dimension, and otherwise that with ``axis=1``.
  58 
  59     .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.hsplit`
  60 
  61     """
  62     if ary.ndim == 0:
  63         raise ValueError('Cannot hsplit a zero-dimensional array')
  64     if ary.ndim == 1:
  65         return split(ary, indices_or_sections, 0)
  66     else:
  67         return split(ary, indices_or_sections, 1)
  68 
  69 
  70 def split(ary, indices_or_sections, axis=0):
  71     """Splits an array into multiple sub arrays along a given axis.
  72 
  73     Args:
  74         ary (cupy.ndarray): Array to split.
  75         indices_or_sections (int or sequence of ints): A value indicating how
  76             to divide the axis. If it is an integer, then is treated as the
  77             number of sections, and the axis is evenly divided. Otherwise,
  78             the integers indicate indices to split at. Note that the sequence
  79             on the device memory is not allowed.
  80         axis (int): Axis along which the array is split.
  81 
  82     Returns:
  83         A list of sub arrays. Each array is a view of the corresponding input
  84         array.
  85 
  86     .. seealso:: :func:`numpy.split`
  87 
  88     """
  89     if ary.ndim <= axis:
  90         raise IndexError('Axis exceeds ndim')
  91     size = ary.shape[axis]
  92 
  93     if numpy.isscalar(indices_or_sections):
  94         if size % indices_or_sections != 0:
  95             raise ValueError(
  96                 'indices_or_sections must divide the size along the axes.\n'
  97                 'If you want to split the array into non-equally-sized '
  98                 'arrays, use array_split instead.')
  99     return array_split(ary, indices_or_sections, axis)
  100 
  101 
  102 def vsplit(ary, indices_or_sections):
  103     """Splits an array into multiple sub arrays along the first axis.
  104 
  105     This is equivalent to ``split`` with ``axis=0``.
  106 
  107     .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.dsplit`
  108 
  109     """
  110     if ary.ndim <= 1:
  111         raise ValueError('Cannot vsplit an array with less than 2 dimensions')
  112     return split(ary, indices_or_sections, 0)
  113
  ```
  --- END FILES ---
  
  Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.
  
  Here is an example:
  
  ```diff
  diff --git a/examples/server_async.py b/examples/server_async.py
  --- a/examples/server_async.py
  +++ b/examples/server_async.py
  @@ -313,4 +313,4 @@
   
   
   if __name__ == "__main__":
  -    asyncio.run(run_async_server("."), debug=True)
  +    asyncio.run(run_async_server(), debug=True)
  diff --git a/examples/server_sync.py b/examples/server_sync.py
  --- a/examples/server_sync.py
  +++ b/examples/server_sync.py
  @@ -313,5 +313,5 @@
   
   
   if __name__ == "__main__":
  -    server = run_sync_server(".")
  +    server = run_sync_server()
       server.shutdown()
  
  ```
  
golden_diff: diff --git a/chainer/functions/array/concat.py b/chainer/functions/array/concat.py
  --- a/chainer/functions/array/concat.py
  +++ b/chainer/functions/array/concat.py
  @@ -18,14 +18,19 @@
           type_check.expect(in_types[0].ndim >
                             type_check.Variable(self.axis, 'axis'))
   
  +        type_check.expect(
  +            -in_types[0].ndim <= self.axis,
  +            self.axis < in_types[0].ndim
  +        )
           ndim = in_types[0].ndim.eval()
  +        axis = self.axis % ndim
           for i in range(1, in_types.size().eval()):
               type_check.expect(
                   in_types[0].dtype == in_types[i].dtype,
                   in_types[0].ndim == in_types[i].ndim,
               )
               for d in range(0, ndim):
  -                if d == self.axis:
  +                if d == axis:
                       continue
                   type_check.expect(in_types[0].shape[d] == in_types[i].shape[d])
   
  diff --git a/cupy/manipulation/split.py b/cupy/manipulation/split.py
  --- a/cupy/manipulation/split.py
  +++ b/cupy/manipulation/split.py
  @@ -12,8 +12,10 @@
       .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.array_split`
   
       """
  -    if ary.ndim <= axis:
  +    ndim = ary.ndim
  +    if -ndim > axis or ndim <= axis:
           raise IndexError('Axis exceeds ndim')
  +    axis %= ndim
       size = ary.shape[axis]
   
       if numpy.isscalar(indices_or_sections):
  @@ -33,7 +35,6 @@
           ret.append(ary[skip + (slice(i, index),)])
           i = index
       ret.append(ary[skip + (slice(i, size),)])
  -
       return ret
  
verification_info: {"golden_diff": "diff --git a/chainer/functions/array/concat.py b/chainer/functions/array/concat.py\n--- a/chainer/functions/array/concat.py\n+++ b/chainer/functions/array/concat.py\n@@ -18,14 +18,19 @@\n         type_check.expect(in_types[0].ndim >\n                           type_check.Variable(self.axis, 'axis'))\n \n+        type_check.expect(\n+            -in_types[0].ndim <= self.axis,\n+            self.axis < in_types[0].ndim\n+        )\n         ndim = in_types[0].ndim.eval()\n+        axis = self.axis % ndim\n         for i in range(1, in_types.size().eval()):\n             type_check.expect(\n                 in_types[0].dtype == in_types[i].dtype,\n                 in_types[0].ndim == in_types[i].ndim,\n             )\n             for d in range(0, ndim):\n-                if d == self.axis:\n+                if d == axis:\n                     continue\n                 type_check.expect(in_types[0].shape[d] == in_types[i].shape[d])\n \ndiff --git a/cupy/manipulation/split.py b/cupy/manipulation/split.py\n--- a/cupy/manipulation/split.py\n+++ b/cupy/manipulation/split.py\n@@ -12,8 +12,10 @@\n     .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.array_split`\n \n     \"\"\"\n-    if ary.ndim <= axis:\n+    ndim = ary.ndim\n+    if -ndim > axis or ndim <= axis:\n         raise IndexError('Axis exceeds ndim')\n+    axis %= ndim\n     size = ary.shape[axis]\n \n     if numpy.isscalar(indices_or_sections):\n@@ -33,7 +35,6 @@\n         ret.append(ary[skip + (slice(i, index),)])\n         i = index\n     ret.append(ary[skip + (slice(i, size),)])\n-\n     return ret\n", "issue": "Concat reverse indexing not working correctly\nThe following works correctly:\n\n```\na_data = np.zeros((10, 5))\nb_data = np.zeros((10, 3))\na = Variable(a_data)\nb = Variable(b_data)\nF.concat((a, b), axis=1)\n```\n\nHowever, if I change the last line into:\n\n```\nF.concat((a, b), axis=-1)\n```\n\nit gives the error:\n\n```\nInvalid operation is performed in: Concat (Forward)\n\nExpect: in_types[0].shape[1] == in_types[1].shape[1]\nActual: 5 != 3\n```\n\n", "before_files": [{"content": "import numpy\n\nfrom chainer import cuda\nfrom chainer import function\nfrom chainer.utils import type_check\n\n\nclass Concat(function.Function):\n\n    \"\"\"Concatenate multiple tensors towards specified axis.\"\"\"\n\n    # concat along the channel dimension by default\n    def __init__(self, axis=1):\n        self.axis = axis\n\n    def check_type_forward(self, in_types):\n        type_check.expect(in_types.size() > 0)\n        type_check.expect(in_types[0].ndim >\n                          type_check.Variable(self.axis, 'axis'))\n\n        ndim = in_types[0].ndim.eval()\n        for i in range(1, in_types.size().eval()):\n            type_check.expect(\n                in_types[0].dtype == in_types[i].dtype,\n                in_types[0].ndim == in_types[i].ndim,\n            )\n            for d in range(0, ndim):\n                if d == self.axis:\n                    continue\n                type_check.expect(in_types[0].shape[d] == in_types[i].shape[d])\n\n    def forward(self, xs):\n        xp = cuda.get_array_module(*xs)\n        return xp.concatenate(xs, axis=self.axis),\n\n    def backward(self, xs, gy):\n        if not xs[:-1]:\n            return gy\n\n        xp = cuda.get_array_module(*xs)\n        sizes = numpy.array([x.shape[self.axis] for x in xs[:-1]]).cumsum()\n        return xp.split(gy[0], sizes, axis=self.axis)\n\n\ndef concat(xs, axis=1):\n    \"\"\"Concatenates given variables along an axis.\n\n    Args:\n        xs (tuple of Variables): Variables to be concatenated.\n        axis (int): Axis that the input arrays are concatenated along.\n\n    Returns:\n        ~chainer.Variable: Output variable.\n\n    \"\"\"\n    return Concat(axis=axis)(*xs)\n", "path": "chainer/functions/array/concat.py"}, {"content": "import numpy\nimport six\n\n\ndef array_split(ary, indices_or_sections, axis=0):\n    \"\"\"Splits an array into multiple sub arrays along a given axis.\n\n    This function is almost equivalent to :func:`cupy.split`. The only\n    difference is that this function allows an integer sections that does not\n    evenly divide the axis.\n\n    .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.array_split`\n\n    \"\"\"\n    if ary.ndim <= axis:\n        raise IndexError('Axis exceeds ndim')\n    size = ary.shape[axis]\n\n    if numpy.isscalar(indices_or_sections):\n        each_size = (size - 1) // indices_or_sections + 1\n        indices = [i * each_size\n                   for i in six.moves.range(1, indices_or_sections)]\n    else:\n        indices = indices_or_sections\n\n    if len(indices) == 0:\n        return [ary]\n\n    skip = (slice(None),) * axis\n    ret = []\n    i = 0\n    for index in indices:\n        ret.append(ary[skip + (slice(i, index),)])\n        i = index\n    ret.append(ary[skip + (slice(i, size),)])\n\n    return ret\n\n\ndef dsplit(ary, indices_or_sections):\n    \"\"\"Splits an array into multiple sub arrays along the third axis.\n\n    This is equivalent to ``split`` with ``axis=2``.\n\n    .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.dsplit`\n\n    \"\"\"\n    if ary.ndim <= 2:\n        raise ValueError('Cannot dsplit an array with less than 3 dimensions')\n    return split(ary, indices_or_sections, 2)\n\n\ndef hsplit(ary, indices_or_sections):\n    \"\"\"Splits an array into multiple sub arrays horizontally.\n\n    This is equivalent to ``split`` with ``axis=0`` if ``ary`` has one\n    dimension, and otherwise that with ``axis=1``.\n\n    .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.hsplit`\n\n    \"\"\"\n    if ary.ndim == 0:\n        raise ValueError('Cannot hsplit a zero-dimensional array')\n    if ary.ndim == 1:\n        return split(ary, indices_or_sections, 0)\n    else:\n        return split(ary, indices_or_sections, 1)\n\n\ndef split(ary, indices_or_sections, axis=0):\n    \"\"\"Splits an array into multiple sub arrays along a given axis.\n\n    Args:\n        ary (cupy.ndarray): Array to split.\n        indices_or_sections (int or sequence of ints): A value indicating how\n            to divide the axis. If it is an integer, then is treated as the\n            number of sections, and the axis is evenly divided. Otherwise,\n            the integers indicate indices to split at. Note that the sequence\n            on the device memory is not allowed.\n        axis (int): Axis along which the array is split.\n\n    Returns:\n        A list of sub arrays. Each array is a view of the corresponding input\n        array.\n\n    .. seealso:: :func:`numpy.split`\n\n    \"\"\"\n    if ary.ndim <= axis:\n        raise IndexError('Axis exceeds ndim')\n    size = ary.shape[axis]\n\n    if numpy.isscalar(indices_or_sections):\n        if size % indices_or_sections != 0:\n            raise ValueError(\n                'indices_or_sections must divide the size along the axes.\\n'\n                'If you want to split the array into non-equally-sized '\n                'arrays, use array_split instead.')\n    return array_split(ary, indices_or_sections, axis)\n\n\ndef vsplit(ary, indices_or_sections):\n    \"\"\"Splits an array into multiple sub arrays along the first axis.\n\n    This is equivalent to ``split`` with ``axis=0``.\n\n    .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.dsplit`\n\n    \"\"\"\n    if ary.ndim <= 1:\n        raise ValueError('Cannot vsplit an array with less than 2 dimensions')\n    return split(ary, indices_or_sections, 0)\n", "path": "cupy/manipulation/split.py"}], "after_files": [{"content": "import numpy\n\nfrom chainer import cuda\nfrom chainer import function\nfrom chainer.utils import type_check\n\n\nclass Concat(function.Function):\n\n    \"\"\"Concatenate multiple tensors towards specified axis.\"\"\"\n\n    # concat along the channel dimension by default\n    def __init__(self, axis=1):\n        self.axis = axis\n\n    def check_type_forward(self, in_types):\n        type_check.expect(in_types.size() > 0)\n        type_check.expect(in_types[0].ndim >\n                          type_check.Variable(self.axis, 'axis'))\n\n        type_check.expect(\n            -in_types[0].ndim <= self.axis,\n            self.axis < in_types[0].ndim\n        )\n        ndim = in_types[0].ndim.eval()\n        axis = self.axis % ndim\n        for i in range(1, in_types.size().eval()):\n            type_check.expect(\n                in_types[0].dtype == in_types[i].dtype,\n                in_types[0].ndim == in_types[i].ndim,\n            )\n            for d in range(0, ndim):\n                if d == axis:\n                    continue\n                type_check.expect(in_types[0].shape[d] == in_types[i].shape[d])\n\n    def forward(self, xs):\n        xp = cuda.get_array_module(*xs)\n        return xp.concatenate(xs, axis=self.axis),\n\n    def backward(self, xs, gy):\n        if not xs[:-1]:\n            return gy\n\n        xp = cuda.get_array_module(*xs)\n        sizes = numpy.array([x.shape[self.axis] for x in xs[:-1]]).cumsum()\n        return xp.split(gy[0], sizes, axis=self.axis)\n\n\ndef concat(xs, axis=1):\n    \"\"\"Concatenates given variables along an axis.\n\n    Args:\n        xs (tuple of Variables): Variables to be concatenated.\n        axis (int): Axis that the input arrays are concatenated along.\n\n    Returns:\n        ~chainer.Variable: Output variable.\n\n    \"\"\"\n    return Concat(axis=axis)(*xs)\n", "path": "chainer/functions/array/concat.py"}, {"content": "import numpy\nimport six\n\n\ndef array_split(ary, indices_or_sections, axis=0):\n    \"\"\"Splits an array into multiple sub arrays along a given axis.\n\n    This function is almost equivalent to :func:`cupy.split`. The only\n    difference is that this function allows an integer sections that does not\n    evenly divide the axis.\n\n    .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.array_split`\n\n    \"\"\"\n    ndim = ary.ndim\n    if -ndim > axis or ndim <= axis:\n        raise IndexError('Axis exceeds ndim')\n    axis %= ndim\n    size = ary.shape[axis]\n\n    if numpy.isscalar(indices_or_sections):\n        each_size = (size - 1) // indices_or_sections + 1\n        indices = [i * each_size\n                   for i in six.moves.range(1, indices_or_sections)]\n    else:\n        indices = indices_or_sections\n\n    if len(indices) == 0:\n        return [ary]\n\n    skip = (slice(None),) * axis\n    ret = []\n    i = 0\n    for index in indices:\n        ret.append(ary[skip + (slice(i, index),)])\n        i = index\n    ret.append(ary[skip + (slice(i, size),)])\n    return ret\n\n\ndef dsplit(ary, indices_or_sections):\n    \"\"\"Splits an array into multiple sub arrays along the third axis.\n\n    This is equivalent to ``split`` with ``axis=2``.\n\n    .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.dsplit`\n\n    \"\"\"\n    if ary.ndim <= 2:\n        raise ValueError('Cannot dsplit an array with less than 3 dimensions')\n    return split(ary, indices_or_sections, 2)\n\n\ndef hsplit(ary, indices_or_sections):\n    \"\"\"Splits an array into multiple sub arrays horizontally.\n\n    This is equivalent to ``split`` with ``axis=0`` if ``ary`` has one\n    dimension, and otherwise that with ``axis=1``.\n\n    .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.hsplit`\n\n    \"\"\"\n    if ary.ndim == 0:\n        raise ValueError('Cannot hsplit a zero-dimensional array')\n    if ary.ndim == 1:\n        return split(ary, indices_or_sections, 0)\n    else:\n        return split(ary, indices_or_sections, 1)\n\n\ndef split(ary, indices_or_sections, axis=0):\n    \"\"\"Splits an array into multiple sub arrays along a given axis.\n\n    Args:\n        ary (cupy.ndarray): Array to split.\n        indices_or_sections (int or sequence of ints): A value indicating how\n            to divide the axis. If it is an integer, then is treated as the\n            number of sections, and the axis is evenly divided. Otherwise,\n            the integers indicate indices to split at. Note that the sequence\n            on the device memory is not allowed.\n        axis (int): Axis along which the array is split.\n\n    Returns:\n        A list of sub arrays. Each array is a view of the corresponding input\n        array.\n\n    .. seealso:: :func:`numpy.split`\n\n    \"\"\"\n    if ary.ndim <= axis:\n        raise IndexError('Axis exceeds ndim')\n    size = ary.shape[axis]\n\n    if numpy.isscalar(indices_or_sections):\n        if size % indices_or_sections != 0:\n            raise ValueError(\n                'indices_or_sections must divide the size along the axes.\\n'\n                'If you want to split the array into non-equally-sized '\n                'arrays, use array_split instead.')\n    return array_split(ary, indices_or_sections, axis)\n\n\ndef vsplit(ary, indices_or_sections):\n    \"\"\"Splits an array into multiple sub arrays along the first axis.\n\n    This is equivalent to ``split`` with ``axis=0``.\n\n    .. seealso:: :func:`cupy.split` for more detail, :func:`numpy.dsplit`\n\n    \"\"\"\n    if ary.ndim <= 1:\n        raise ValueError('Cannot vsplit an array with less than 2 dimensions')\n    return split(ary, indices_or_sections, 0)\n", "path": "cupy/manipulation/split.py"}]}