prompt: You will be provided with a partial code base and an issue statement explaining a problem to resolve.
  <issue>
  `split_axis` doesn't support empty sections
  This code causes a TypeError.
  `functions.split_axis(x, [], 0)`
  
  
  </issue>
  <code>
  [start of chainer/functions/array/split_axis.py]
  1 import collections
  2 
  3 import six
  4 
  5 import chainer
  6 from chainer import cuda
  7 from chainer import function
  8 from chainer.utils import type_check
  9 
  10 
  11 class SplitAxis(function.Function):
  12 
  13     """Function that splits multiple arrays along the specified axis."""
  14 
  15     def __init__(self, indices_or_sections, axis):
  16         if not isinstance(indices_or_sections, (int, collections.Iterable)):
  17             raise TypeError('indices_or_sections must be integer or 1-D array')
  18         self.indices_or_sections = indices_or_sections
  19         self.axis = axis
  20 
  21     def check_type_forward(self, in_types):
  22         type_check.expect(in_types.size() == 1)
  23         type_check.expect(in_types[0].ndim > self.axis)
  24 
  25         if isinstance(self.indices_or_sections, collections.Iterable):
  26             max_index = type_check.Variable(
  27                 self.indices_or_sections[-1], 'max_index')
  28             type_check.expect(in_types[0].shape[self.axis] > max_index)
  29         else:
  30             sections = type_check.Variable(
  31                 self.indices_or_sections, 'sections')
  32             type_check.expect(in_types[0].shape[self.axis] % sections == 0)
  33 
  34     def forward(self, x):
  35         if isinstance(self.indices_or_sections, collections.Iterable):
  36             cdimx = x[0].shape[self.axis]
  37             ind = list(self.indices_or_sections)
  38             ind.append(cdimx)
  39             prev_i = 0
  40             for i in ind:
  41                 cdimy = max(0, min(i, cdimx) - prev_i)
  42                 if cdimy == 0:
  43                     raise ValueError('Not support if shape contains 0')
  44                 prev_i = i
  45         xp = cuda.get_array_module(*x)
  46         return tuple(xp.split(x[0], self.indices_or_sections, self.axis))
  47 
  48     def backward(self, x, gys):
  49         xp = cuda.get_array_module(*x)
  50         if any(gy is None for gy in gys):
  51             gx = xp.zeros_like(x[0])
  52             gxs = xp.split(gx, self.indices_or_sections, self.axis)
  53             for gxi, gy in six.moves.zip(gxs, gys):
  54                 if gy is None:
  55                     continue
  56                 gxi[:] = gy
  57             return gx,
  58         else:
  59             return xp.concatenate(gys, axis=self.axis),
  60 
  61 
  62 def split_axis(x, indices_or_sections, axis, force_tuple=False):
  63     """Splits given variables along an axis.
  64 
  65     Args:
  66         x (tuple of Variables): Variables to be split.
  67         indices_or_sections (int or 1-D array): If this argument is an integer,
  68             N, the array will be divided into N equal arrays along axis.
  69             If it is a 1-D array of sorted integers, it
  70             indicates the positions where the array is split.
  71         axis (int): Axis that the input array is split along.
  72         force_tuple (bool): If ``True``, this method returns a tuple even when
  73             the number of outputs is one.
  74 
  75     Returns:
  76         tuple or Variable: Tuple of :class:`~chainer.Variable` objects
  77              if the number of outputs is more than 1 or
  78              :class:`~chainer.Variable` otherwise.
  79              When ``force_tuple`` is ``True``, returned value is always a tuple
  80              regardless of the number of outputs.
  81 
  82     .. note::
  83         This function raises :class:`ValueError` if at least
  84         one of the outputs is split to zero-size
  85         (i.e. ``axis``-th value of its shape is zero).
  86 
  87     """
  88     res = SplitAxis(indices_or_sections, axis)(x)
  89     if force_tuple and isinstance(res, chainer.Variable):
  90         res = (res,)
  91     return res
  92
  [end of chainer/functions/array/split_axis.py]
  </code>
  
  I need you to solve this issue by generating a single patch file that I can apply directly to this repository using git apply. Please respond with a single patch file in the following format.
  <patch>
  diff --git a/file.py b/file.py
  --- a/file.py
  +++ b/file.py
  @@ -1,27 +1,35 @@
   def euclidean(a, b):
  -    while b:
  -        a, b = b, a % b
  -    return a
  +    if b == 0:
  +        return a
  +    return euclidean(b, a % b)
   
   
   def bresenham(x0, y0, x1, y1):
       points = []
       dx = abs(x1 - x0)
       dy = abs(y1 - y0)
  -    sx = 1 if x0 < x1 else -1
  -    sy = 1 if y0 < y1 else -1
  -    err = dx - dy
  +    x, y = x0, y0
  +    sx = -1 if x0 > x1 else 1
  +    sy = -1 if y0 > y1 else 1
   
  -    while True:
  -        points.append((x0, y0))
  -        if x0 == x1 and y0 == y1:
  -            break
  -        e2 = 2 * err
  -        if e2 > -dy:
  -            err -= dy
  -            x0 += sx
  -        if e2 < dx:
  -            err += dx
  -            y0 += sy
  +    if dx > dy:
  +        err = dx / 2.0
  +        while x != x1:
  +            points.append((x, y))
  +            err -= dy
  +            if err < 0:
  +                y += sy
  +                err += dx
  +            x += sx
  +    else:
  +        err = dy / 2.0
  +        while y != y1:
  +            points.append((x, y))
  +            err -= dx
  +            if err < 0:
  +                x += sx
  +                err += dy
  +            y += sy
  +
  +    points.append((x, y))
       return points
  
  </patch>
golden_diff: diff --git a/chainer/functions/array/split_axis.py b/chainer/functions/array/split_axis.py
  --- a/chainer/functions/array/split_axis.py
  +++ b/chainer/functions/array/split_axis.py
  @@ -23,9 +23,10 @@
           type_check.expect(in_types[0].ndim > self.axis)
   
           if isinstance(self.indices_or_sections, collections.Iterable):
  -            max_index = type_check.Variable(
  -                self.indices_or_sections[-1], 'max_index')
  -            type_check.expect(in_types[0].shape[self.axis] > max_index)
  +            if len(self.indices_or_sections) > 0:
  +                max_index = type_check.Variable(
  +                    self.indices_or_sections[-1], 'max_index')
  +                type_check.expect(in_types[0].shape[self.axis] > max_index)
           else:
               sections = type_check.Variable(
                   self.indices_or_sections, 'sections')
  
verification_info: {"golden_diff": "diff --git a/chainer/functions/array/split_axis.py b/chainer/functions/array/split_axis.py\n--- a/chainer/functions/array/split_axis.py\n+++ b/chainer/functions/array/split_axis.py\n@@ -23,9 +23,10 @@\n         type_check.expect(in_types[0].ndim > self.axis)\n \n         if isinstance(self.indices_or_sections, collections.Iterable):\n-            max_index = type_check.Variable(\n-                self.indices_or_sections[-1], 'max_index')\n-            type_check.expect(in_types[0].shape[self.axis] > max_index)\n+            if len(self.indices_or_sections) > 0:\n+                max_index = type_check.Variable(\n+                    self.indices_or_sections[-1], 'max_index')\n+                type_check.expect(in_types[0].shape[self.axis] > max_index)\n         else:\n             sections = type_check.Variable(\n                 self.indices_or_sections, 'sections')\n", "issue": "`split_axis` doesn't support empty sections\nThis code causes a TypeError.\n`functions.split_axis(x, [], 0)`\n\n", "before_files": [{"content": "import collections\n\nimport six\n\nimport chainer\nfrom chainer import cuda\nfrom chainer import function\nfrom chainer.utils import type_check\n\n\nclass SplitAxis(function.Function):\n\n    \"\"\"Function that splits multiple arrays along the specified axis.\"\"\"\n\n    def __init__(self, indices_or_sections, axis):\n        if not isinstance(indices_or_sections, (int, collections.Iterable)):\n            raise TypeError('indices_or_sections must be integer or 1-D array')\n        self.indices_or_sections = indices_or_sections\n        self.axis = axis\n\n    def check_type_forward(self, in_types):\n        type_check.expect(in_types.size() == 1)\n        type_check.expect(in_types[0].ndim > self.axis)\n\n        if isinstance(self.indices_or_sections, collections.Iterable):\n            max_index = type_check.Variable(\n                self.indices_or_sections[-1], 'max_index')\n            type_check.expect(in_types[0].shape[self.axis] > max_index)\n        else:\n            sections = type_check.Variable(\n                self.indices_or_sections, 'sections')\n            type_check.expect(in_types[0].shape[self.axis] % sections == 0)\n\n    def forward(self, x):\n        if isinstance(self.indices_or_sections, collections.Iterable):\n            cdimx = x[0].shape[self.axis]\n            ind = list(self.indices_or_sections)\n            ind.append(cdimx)\n            prev_i = 0\n            for i in ind:\n                cdimy = max(0, min(i, cdimx) - prev_i)\n                if cdimy == 0:\n                    raise ValueError('Not support if shape contains 0')\n                prev_i = i\n        xp = cuda.get_array_module(*x)\n        return tuple(xp.split(x[0], self.indices_or_sections, self.axis))\n\n    def backward(self, x, gys):\n        xp = cuda.get_array_module(*x)\n        if any(gy is None for gy in gys):\n            gx = xp.zeros_like(x[0])\n            gxs = xp.split(gx, self.indices_or_sections, self.axis)\n            for gxi, gy in six.moves.zip(gxs, gys):\n                if gy is None:\n                    continue\n                gxi[:] = gy\n            return gx,\n        else:\n            return xp.concatenate(gys, axis=self.axis),\n\n\ndef split_axis(x, indices_or_sections, axis, force_tuple=False):\n    \"\"\"Splits given variables along an axis.\n\n    Args:\n        x (tuple of Variables): Variables to be split.\n        indices_or_sections (int or 1-D array): If this argument is an integer,\n            N, the array will be divided into N equal arrays along axis.\n            If it is a 1-D array of sorted integers, it\n            indicates the positions where the array is split.\n        axis (int): Axis that the input array is split along.\n        force_tuple (bool): If ``True``, this method returns a tuple even when\n            the number of outputs is one.\n\n    Returns:\n        tuple or Variable: Tuple of :class:`~chainer.Variable` objects\n             if the number of outputs is more than 1 or\n             :class:`~chainer.Variable` otherwise.\n             When ``force_tuple`` is ``True``, returned value is always a tuple\n             regardless of the number of outputs.\n\n    .. note::\n        This function raises :class:`ValueError` if at least\n        one of the outputs is split to zero-size\n        (i.e. ``axis``-th value of its shape is zero).\n\n    \"\"\"\n    res = SplitAxis(indices_or_sections, axis)(x)\n    if force_tuple and isinstance(res, chainer.Variable):\n        res = (res,)\n    return res\n", "path": "chainer/functions/array/split_axis.py"}]}