fix: add proper type annotations for embedding() function (#12262)

Fixes #9526

The embedding() function can return either an EmbeddingResponse or a
Coroutine[Any, Any, EmbeddingResponse] depending on whether aembedding=True
is passed. This adds @overload decorators to properly indicate the return
type based on the aembedding parameter.

This resolves type checking errors when using embedding() with aembedding=True
where the type checker couldn't determine that a coroutine was being returned.

litellm/main.py

@@ -2870,9 +2870,9 @@ def completion(  # type: ignore # noqa: PLR0915
                     "aws_region_name" not in optional_params
                     or optional_params["aws_region_name"] is None
                 ):
-                    optional_params["aws_region_name"] = (
-                        aws_bedrock_client.meta.region_name
-                    )
+                    optional_params[
+                        "aws_region_name"
+                    ] = aws_bedrock_client.meta.region_name
 
             bedrock_route = BedrockModelInfo.get_bedrock_route(model)
             if bedrock_route == "converse":
@@ -3498,6 +3498,82 @@ async def aembedding(*args, **kwargs) -> EmbeddingResponse:
         )
 
 
+# Overload for when aembedding=True (returns coroutine)
+@overload
+def embedding(
+    model,
+    input=[],
+    # Optional params
+    dimensions: Optional[int] = None,
+    encoding_format: Optional[str] = None,
+    timeout=600,  # default to 10 minutes
+    # set api_base, api_version, api_key
+    api_base: Optional[str] = None,
+    api_version: Optional[str] = None,
+    api_key: Optional[str] = None,
+    api_type: Optional[str] = None,
+    caching: bool = False,
+    user: Optional[str] = None,
+    custom_llm_provider=None,
+    litellm_call_id=None,
+    logger_fn=None,
+    *,
+    aembedding: Literal[True],
+    **kwargs,
+) -> Coroutine[Any, Any, EmbeddingResponse]:
+    ...
+
+
+# Overload for when aembedding=False or not specified (returns EmbeddingResponse)
+@overload
+def embedding(
+    model,
+    input=[],
+    # Optional params
+    dimensions: Optional[int] = None,
+    encoding_format: Optional[str] = None,
+    timeout=600,  # default to 10 minutes
+    # set api_base, api_version, api_key
+    api_base: Optional[str] = None,
+    api_version: Optional[str] = None,
+    api_key: Optional[str] = None,
+    api_type: Optional[str] = None,
+    caching: bool = False,
+    user: Optional[str] = None,
+    custom_llm_provider=None,
+    litellm_call_id=None,
+    logger_fn=None,
+    *,
+    aembedding: Literal[False] = False,
+    **kwargs,
+) -> EmbeddingResponse:
+    ...
+
+
+# Overload for when aembedding is not specified at all (returns EmbeddingResponse)
+@overload
+def embedding(
+    model,
+    input=[],
+    # Optional params
+    dimensions: Optional[int] = None,
+    encoding_format: Optional[str] = None,
+    timeout=600,  # default to 10 minutes
+    # set api_base, api_version, api_key
+    api_base: Optional[str] = None,
+    api_version: Optional[str] = None,
+    api_key: Optional[str] = None,
+    api_type: Optional[str] = None,
+    caching: bool = False,
+    user: Optional[str] = None,
+    custom_llm_provider=None,
+    litellm_call_id=None,
+    logger_fn=None,
+    **kwargs,
+) -> EmbeddingResponse:
+    ...
+
+
 @client
 def embedding(  # noqa: PLR0915
     model,
@@ -4651,9 +4727,9 @@ def adapter_completion(
     new_kwargs = translation_obj.translate_completion_input_params(kwargs=kwargs)
 
     response: Union[ModelResponse, CustomStreamWrapper] = completion(**new_kwargs)  # type: ignore
-    translated_response: Optional[Union[BaseModel, AdapterCompletionStreamWrapper]] = (
-        None
-    )
+    translated_response: Optional[
+        Union[BaseModel, AdapterCompletionStreamWrapper]
+    ] = None
     if isinstance(response, ModelResponse):
         translated_response = translation_obj.translate_completion_output_params(
             response=response
@@ -5644,9 +5720,9 @@ def stream_chunk_builder(  # noqa: PLR0915
         ]
 
         if len(content_chunks) > 0:
-            response["choices"][0]["message"]["content"] = (
-                processor.get_combined_content(content_chunks)
-            )
+            response["choices"][0]["message"][
+                "content"
+            ] = processor.get_combined_content(content_chunks)
 
         reasoning_chunks = [
             chunk
@@ -5657,9 +5733,9 @@ def stream_chunk_builder(  # noqa: PLR0915
         ]
 
         if len(reasoning_chunks) > 0:
-            response["choices"][0]["message"]["reasoning_content"] = (
-                processor.get_combined_reasoning_content(reasoning_chunks)
-            )
+            response["choices"][0]["message"][
+                "reasoning_content"
+            ] = processor.get_combined_reasoning_content(reasoning_chunks)
 
         audio_chunks = [
             chunk