





'''simple docstring'''




def        _lowerCAmelCase			(  __snake_case  : str			)     ->     List[str]:
     __A						: Any      =       0
     __A						: Union[str, Any]      =       len(__snake_case			)

     for i in range(n - 1			):
          for j in range(i + 1 ,							__snake_case			):
               if arr[i] > arr[j]:
                    num_inversions += 1

     return num_inversions






def        _lowerCAmelCase			(  __snake_case  : Tuple			)     ->     Any:
     if len(__snake_case			) <= 1:
          return arr, 0
     __A						: int      =       len(__snake_case			) // 2
     __A						: str      =       arr[0:mid]
     __A						: Union[str, Any]      =       arr[mid:]

     __A ,__A						: str      =       count_inversions_recursive(__snake_case			)
     __A ,__A						: Any      =       count_inversions_recursive(__snake_case			)
     __A ,__A						: int      =       _count_cross_inversions(__snake_case ,							__snake_case			)

     __A						: str      =       inversion_p + inversions_q + cross_inversions
     return c, num_inversions






def        _lowerCAmelCase			(  __snake_case  : Any ,							__snake_case  : Optional[Any]			)     ->     Dict:
     __A						: Dict      =       []
     __A						: List[str]      =       0
     while i < len(__snake_case			) and j < len(__snake_case			):
          if p[i] > q[j]:
               # if P[1] > Q[j], then P[k] > Q[k] for all  i < k <= len(P)
               # These are all inversions. The claim emerges from the
               # property that P is sorted.
               num_inversion += len(__snake_case			) - i
               r.append(q[j]			)
               j += 1
          else:
               r.append(p[i]			)
               i += 1

     if i < len(__snake_case			):
          r.extend(p[i:]			)
     else:
          r.extend(q[j:]			)

     return r, num_inversion






def        _lowerCAmelCase			(  )     ->     str:
     __A						: str      =       [10, 2, 1, 5, 5, 2, 11]

     # this arr has 8 inversions:
     # (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)

     __A						: Optional[Any]      =       count_inversions_bf(__snake_case			)
     __A ,__A						: Optional[Any]      =       count_inversions_recursive(__snake_case			)

     assert num_inversions_bf == num_inversions_recursive == 8

     print('number of inversions = ' ,							__snake_case			)

     # testing an array with zero inversion (a sorted arr_1)

     arr_a.sort()
     __A						: Optional[Any]      =       count_inversions_bf(__snake_case			)
     __A ,__A						: List[str]      =       count_inversions_recursive(__snake_case			)

     assert num_inversions_bf == num_inversions_recursive == 0
     print('number of inversions = ' ,							__snake_case			)

     # an empty list should also have zero inversions
     __A						: Optional[Any]      =       []
     __A						: Any      =       count_inversions_bf(__snake_case			)
     __A ,__A						: List[Any]      =       count_inversions_recursive(__snake_case			)

     assert num_inversions_bf == num_inversions_recursive == 0
     print('number of inversions = ' ,							__snake_case			)


if __name__ == "__main__":
  main()






'''simple docstring'''




lowercase__    :      Dict						=    {
    '''Pillow''': '''Pillow''',
    '''accelerate''': '''accelerate>=0.11.0''',
    '''compel''': '''compel==0.1.8''',
    '''black''': '''black~=23.1''',
    '''datasets''': '''datasets''',
    '''filelock''': '''filelock''',
    '''flax''': '''flax>=0.4.1''',
    '''hf-doc-builder''': '''hf-doc-builder>=0.3.0''',
    '''huggingface-hub''': '''huggingface-hub>=0.13.2''',
    '''requests-mock''': '''requests-mock==1.10.0''',
    '''importlib_metadata''': '''importlib_metadata''',
    '''invisible-watermark''': '''invisible-watermark''',
    '''isort''': '''isort>=5.5.4''',
    '''jax''': '''jax>=0.2.8,!=0.3.2''',
    '''jaxlib''': '''jaxlib>=0.1.65''',
    '''Jinja2''': '''Jinja2''',
    '''k-diffusion''': '''k-diffusion>=0.0.12''',
    '''torchsde''': '''torchsde''',
    '''note_seq''': '''note_seq''',
    '''librosa''': '''librosa''',
    '''numpy''': '''numpy''',
    '''omegaconf''': '''omegaconf''',
    '''parameterized''': '''parameterized''',
    '''protobuf''': '''protobuf>=3.20.3,<4''',
    '''pytest''': '''pytest''',
    '''pytest-timeout''': '''pytest-timeout''',
    '''pytest-xdist''': '''pytest-xdist''',
    '''ruff''': '''ruff>=0.0.241''',
    '''safetensors''': '''safetensors''',
    '''sentencepiece''': '''sentencepiece>=0.1.91,!=0.1.92''',
    '''scipy''': '''scipy''',
    '''onnx''': '''onnx''',
    '''regex''': '''regex!=2019.12.17''',
    '''requests''': '''requests''',
    '''tensorboard''': '''tensorboard''',
    '''torch''': '''torch>=1.4''',
    '''torchvision''': '''torchvision''',
    '''transformers''': '''transformers>=4.25.1''',
    '''urllib3''': '''urllib3<=2.0.0''',
}