core[patch]: fix loss of partially initialized variables during prompt composition (#30096)

**Description:**
This PR addresses the loss of partially initialised variables when
composing different prompts. I.e. it allows the following snippet to
run:

```python
from langchain_core.prompts import ChatPromptTemplate

prompt = ChatPromptTemplate.from_messages([('system', 'Prompt {x} {y}')]).partial(x='1')
appendix = ChatPromptTemplate.from_messages([('system', 'Appendix {z}')])

(prompt + appendix).invoke({'y': '2', 'z': '3'})
```

Previously, this would have raised a `KeyError`, stating that variable
`x` remains undefined.

**Issue**
References issue #30049

**Todo**
- [x] **Add tests and docs**: If you're adding a new integration, please
include
1. a test for the integration, preferably unit tests that do not rely on
network access,
2. an example notebook showing its use. It lives in
`docs/docs/integrations` directory.


- [x] **Lint and test**: Run `make format`, `make lint` and `make test`
from the root of the package(s) you've modified. See contribution
guidelines for more: https://python.langchain.com/docs/contributing/

---------

Co-authored-by: Eugene Yurtsev <[email protected]>

Author: Thommy257

Diff for: libs/core/langchain_core/prompts/chat.py

@@ -1040,19 +1040,34 @@ def __add__(self, other: Any) -> ChatPromptTemplate:
         Returns:
             Combined prompt template.
         """
+        partials = {**self.partial_variables}
+
+        # Need to check that other has partial variables since it may not be
+        # a ChatPromptTemplate.
+        if hasattr(other, "partial_variables") and other.partial_variables:
+            partials.update(other.partial_variables)
+
         # Allow for easy combining
         if isinstance(other, ChatPromptTemplate):
-            return ChatPromptTemplate(messages=self.messages + other.messages)  # type: ignore[call-arg]
+            return ChatPromptTemplate(messages=self.messages + other.messages).partial(
+                **partials
+            )  # type: ignore[call-arg]
         elif isinstance(
             other, (BaseMessagePromptTemplate, BaseMessage, BaseChatPromptTemplate)
         ):
-            return ChatPromptTemplate(messages=self.messages + [other])  # type: ignore[call-arg]
+            return ChatPromptTemplate(messages=self.messages + [other]).partial(
+                **partials
+            )  # type: ignore[call-arg]
         elif isinstance(other, (list, tuple)):
             _other = ChatPromptTemplate.from_messages(other)
-            return ChatPromptTemplate(messages=self.messages + _other.messages)  # type: ignore[call-arg]
+            return ChatPromptTemplate(messages=self.messages + _other.messages).partial(
+                **partials
+            )  # type: ignore[call-arg]
         elif isinstance(other, str):
             prompt = HumanMessagePromptTemplate.from_template(other)
-            return ChatPromptTemplate(messages=self.messages + [prompt])  # type: ignore[call-arg]
+            return ChatPromptTemplate(messages=self.messages + [prompt]).partial(
+                **partials
+            )  # type: ignore[call-arg]
         else:
             msg = f"Unsupported operand type for +: {type(other)}"
             raise NotImplementedError(msg)

Diff for: libs/core/tests/unit_tests/prompts/test_chat.py

@@ -582,6 +582,23 @@ def test_chat_message_partial() -> None:
     assert template2.format(input="hello") == get_buffer_string(expected)
 
 
+def test_chat_message_partial_composition() -> None:
+    """Test composition of partially initialized messages."""
+    prompt = ChatPromptTemplate.from_messages([("system", "Prompt {x} {y}")]).partial(
+        x="1"
+    )
+
+    appendix = ChatPromptTemplate.from_messages([("system", "Appendix {z}")])
+
+    res = (prompt + appendix).format_messages(y="2", z="3")
+    expected = [
+        SystemMessage(content="Prompt 1 2"),
+        SystemMessage(content="Appendix 3"),
+    ]
+
+    assert res == expected
+
+
 async def test_chat_tmpl_from_messages_multipart_text() -> None:
     template = ChatPromptTemplate.from_messages(
         [