Add streaming support for Responses API (#405)

* Add streaming support for Responses API.

* Update examples/responses-stream/src/main.rs

* Update examples/responses-stream/src/main.rs

* Update examples/responses-stream/src/main.rs

* Delete async-openai/tests/responses.rs

---------

Co-authored-by: Himanshu Neema <himanshun.iitkgp@gmail.com>

Author: kazzix14

async-openai/src/responses.rs

@@ -1,13 +1,13 @@
 use crate::{
     config::Config,
     error::OpenAIError,
-    types::responses::{CreateResponse, Response},
+    types::responses::{CreateResponse, Response, ResponseStream},
     Client,
 };
 
 /// Given text input or a list of context items, the model will generate a response.
 ///
-/// Related guide: [Responses API](https://platform.openai.com/docs/guides/responses)
+/// Related guide: [Responses](https://platform.openai.com/docs/api-reference/responses)
 pub struct Responses<'c, C: Config> {
     client: &'c Client<C>,
 }
@@ -26,4 +26,30 @@ impl<'c, C: Config> Responses<'c, C> {
     pub async fn create(&self, request: CreateResponse) -> Result<Response, OpenAIError> {
         self.client.post("/responses", request).await
     }
+
+    /// Creates a model response for the given input with streaming.
+    ///
+    /// Response events will be sent as server-sent events as they become available,
+    #[crate::byot(
+        T0 = serde::Serialize,
+        R = serde::de::DeserializeOwned,
+        stream = "true",
+        where_clause = "R: std::marker::Send + 'static"
+    )]
+    #[allow(unused_mut)]
+    pub async fn create_stream(
+        &self,
+        mut request: CreateResponse,
+    ) -> Result<ResponseStream, OpenAIError> {
+        #[cfg(not(feature = "byot"))]
+        {
+            if matches!(request.stream, Some(false)) {
+                return Err(OpenAIError::InvalidArgument(
+                    "When stream is false, use Responses::create".into(),
+                ));
+            }
+            request.stream = Some(true);
+        }
+        Ok(self.client.post_stream("/responses", request).await)
+    }
 }